Nordström's theory of gravitation

25: 2945:

we lose the conservation law. So which interpretation is correct? In other words, which metric is the one which according to Nordström can be measured locally by physical experiments? The answer is: the curved spacetime is the physically observable one in this theory (as in all metric theories of gravitation); the flat background is a mere mathematical fiction which is however of inestimable value for such purposes as writing down the general vacuum solution, or studying the weak field limit.

6672: 7997: 7773: 1794:), and on the right hand stands a purely physical quantity, the trace of the stress–energy tensor. Einstein gleefully pointed out that this equation now takes the form which he had earlier proposed with von Laue, and gives a concrete example of a class of theories which he had studied with Grossmann. 2944:

gravitational field energy, but suffers from a not very memorable force law. In the curved spacetime formulation the motion of test particles is described (the world line of a free test particle is a timelike geodesic, and by an obvious limit, the world line of a laser pulse is a null geodesic), but

2239:

is the ordinary wave operator on flat spacetime (expressed in cylindrical, polar spherical, or double null coordinates, respectively). But the general solution of the ordinary three-dimensional wave equation is well known, and can be given rather explicit form. Specifically, for certain charts such

6204:

In other words, nearly circular orbits will exhibit a radial oscillation. However, unlike what happens in Newtonian gravitation, the period of this oscillation will not quite match the orbital period. This will result in slow precession of the periastria (points of closest approach) of our nearly

4335:

which is the same result as for the Schwarzschild vacuum. This expresses conservation of orbital angular momentum of test particles moving in the equatorial plane, and shows that the period of a nearly circular orbit (as observed by a distant observer) will be same as for the Schwarzschild vacuum.

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In any Lorentzian manifold (with appropriate tensor fields describing any matter and physical fields) which stands as a solution to Nordström's field equations, the conformal part of the Riemann tensor (i.e. the Weyl tensor) always vanishes. The Ricci scalar also vanishes identically in any vacuum

1313:. Lorentz, delighted by his former student's success, arranged for Fokker to pursue post-doctoral study with Einstein in Prague. The result was a historic paper which appeared in 1914, in which Einstein and Fokker observed that the Lagrangian for Nordström's equation of motion for test particles, 2960:

We will use the first to obtain the predictions of Nordström's theory for the four classic solar system tests of relativistic gravitation theories (in the ambient field of an isolated spherically symmetric object), and we will use the second to compare gravitational radiation in Nordström's theory

930:

and the second theory of Nordström were worthy of consideration. (Mie, who was in the audience, rose to protest, but Einstein explained his criteria and Mie was forced to admit that his own theory did not meet them.) Einstein considered the special case when the only matter present is a cloud of

5118:

Thus, the particle must maintain radially outward to maintain its position, with a magnitude given by the familiar Newtonian expression (but again we must bear in mind that the radial coordinate here cannot quite be identified with a flat space radial coordinate). Put in other words, this is the

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where we justify our choice of integration constants by the fact that this is the unique choice giving the correct Newtonian limit. This gives the solution in terms of coordinates which directly exhibit the fact that this spacetime is conformally equivalent to Minkowski spacetime, but the radial

2948:

At this point, we could show that in the limit of slowly moving test particles and slowly evolving weak gravitational fields, Nordström's theory of gravitation reduces to the Newtonian theory of gravitation. Rather than showing this in detail, we will proceed to a detailed study of the two most

1844:, i.e. with metric tensor having the form of the product of some scalar function with the metric tensor of flat spacetime. This is exactly the special form of the metric proposed in Nordström's second theory, so the entire content of this theory can be summarized in the following two equations: 10864: 6497: 112:

Neither of Nordström's theories are in agreement with observation and experiment. Nonetheless, the first remains of interest insofar as it led to the second. The second remains of interest both as an important milestone on the road to the current theory of gravitation,

350:

had sketched a vector theory of gravitation as early as 1908, but in 1912, Abraham pointed out that no such theory would admit stable planetary orbits. This was one reason why Nordström turned to scalar theories of gravitation (while Einstein explored tensor theories).

6340: 5123:

order in m, in the Schwarzschild vacuum the magnitude of the radially outward acceleration of a static observer is m r + m^2 r; here too, the second term expresses the fact that Einstein gravity is slightly stronger "at corresponding points" than Nordström gravity.

3447: 10619: 1189: 9467: 2765:

shows how variations in the trace of the stress–energy tensor in our spacetime model can generate a nonzero trace-free Ricci tensor, and thus nonzero semi-traceless curvature, which can propagate into a vacuum region. This is critically important, because

9022:

of relativistic theories of gravitation. Of the three theories appearing in the table, only general relativity is in agreement with the results of experiments and observations in the solar system. Nordström's theory gives the correct result only for the

6192: 2453: 117:, and as a simple example of a self-consistent relativistic theory of gravitation. As an example, this theory is particularly useful in the context of pedagogical discussions of how to derive and test the predictions of a metric theory of gravitation. 9900: 5305: 3777: 10196: 10088: 3681: 2785:

which is generated (via another first order covariant differential equation) by variations in the stress–energy tensor and which then propagates into vacuum regions, rendering gravitation a long-range force capable of propagating through a vacuum.

7549: 4705:

It makes sense to ask how much force is required to hold a test particle with a given mass over the massive object which we assume is the source of this static spherically symmetric gravitational field. To find out, we need only adopt the simple

406:, which means that any "news" concerning redistribution of matter in one location is transmitted at the speed of light to other locations. Correspondingly, the simplest guess for a suitable equation of motion for test particles might seem to be 10921:

of gravitational radiation in this theory. At this point, it would be natural to discuss the analog for Nordström's theory of gravitation of the standard linearized gravitational wave theory in general relativity, but we shall not pursue this.

5739: 2240:

as cylindrical or polar spherical charts on flat spacetime (which induce corresponding charts on our curved Lorentzian manifold), we can write the general solution in terms of a power series, and we can write the general solution of certain

6753: 4019: 7433: 5895: 4108: 8218: 7784: 7560: 9166: 5113: 9725: 6901: 6827: 2502:

Indeed, since every solution to the field equations of this theory is a spacetime which is among other things conformally equivalent to flat spacetime, null geodesics must agree with the null geodesics of the flat background, so

1289: 8887: 8413: 4270: 3080: 2102: 8816: 5005: 9011: 1034: 10869:

further exhibits the spin-0 character of the gravitational plane wave in Nordström's theory. (The tidal tensor and expansion tensor are three-dimensional tensors which "live" in the hyperplane elements orthogonal to

7070: 4571: 6486: 4471: 1398: 917: 829: 8087: 492:

of the test particle. This force law had earlier been proposed by Abraham, but it does not preserve the norm of the four-velocity as is required by the definition of proper time, so Nordström instead proposed

4926: 4865: 4787: 572: 3200: 9905:

Just as in special relativity (and general relativity), it is always possible to change coordinates, without disturbing the form of the solution, so that the wave propagates in any direction transverse to

2489:

In any metric theory of gravitation, all gravitational effects arise from the curvature of the metric. In a spacetime model in Nordström's theory (but not in general relativity), this depends only on the

1653: 6667:{\displaystyle \Delta \omega =\omega _{\rm {orb}}-\omega _{\rm {shm}}={\sqrt {\frac {m}{r^{3}}}}-{\sqrt {{\frac {m^{2}}{r^{4}}}+{\frac {m}{r^{3}}}}}\approx -{\frac {1}{2}}{\sqrt {\frac {m^{3}}{r^{5}}}}} 749: 10378: 2626: 2217: 1544: 325: 9247: 6211: 4162:

we find that test particle motion confined to the equatorial plane is possible, and in this case first integrals (first order ordinary differential equations) are readily obtained. First, we have

7291: 5500: 1712: 10665: 9639: 3281: 2727: 1786:. This is a historic result, because here for the first time we have a field equation in which on the left hand side stands a purely geometrical quantity (the Ricci scalar is the trace of the 9776: 9586: 8738: 8692: 5598: 1045: 10296: 10246: 1909: 1466: 1294:

Since Nordström's equation of motion for test particles in an ambient gravitational field also follows from a Lagrangian, this shows that Nordström's second theory can be derived from an

334:, and because, should the matter in some isolated object suddenly be redistributed by an explosion, the field equation requires that the potential everywhere in "space" must be "updated" 9304: 5361:. The first term agrees with the corresponding solution in the Newtonian theory of gravitation and the one in general relativity. The second term shows that the tidal forces are a bit 684:

In response to these criticisms, Nordström proposed his second theory in 1913. From the proportionality of inertial and gravitational mass, he deduced that the field equation should be

6074: 3261: 922:

Einstein took the first opportunity to proclaim his approval of the new theory. In a keynote address to the annual meeting of the Society of German Scientists and Physicians, given in

459: 10444: 6020: 5359: 10455: 8487:

signal emitted from Earth which grazes the limb of the Sun, bounces off Venus, and returns to Earth (once again grazing the limb of the Sun), the relativistic time delay is about 20

4695: 647: 396: 204: 6082: 4330: 2301: 461:

where the dot signifies differentiation with respect to proper time, subscripts following the comma denote partial differentiation with respect to the indexed coordinate, and where

9784: 3501: 1981: 5133: 3687: 11027: 7168: 2735:

Thus, according to Nordström's theory, in a vacuum region only the semi-traceless part of the Riemann tensor can be nonvanishing. Then our covariant differential constraint on

10094: 9986: 8473: 8278: 7121: 3926: 2143: 1194:

which he proposed should hold in general, and showed that the sum of the contributions to the stress–energy tensor from the gravitational field energy and from matter would be

4375: 3852: 3509: 7444: 7224: 10904: 6376: 5405: 4160: 1756:

where on the right hand side, we have taken the trace of the stress–energy tensor (with contributions from matter plus any non-gravitational fields) using the metric tensor

9931: 9542: 9507: 5946: 4630: 3897: 5606: 8310: 1751: 354:

Nordström's first attempt to propose a suitable relativistic scalar field equation of gravitation was the simplest and most natural choice imaginable: simply replace the

9285: 5771: 2920: 2534:

region (or even, any region free of matter but containing an electromagnetic field). Are there any further restrictions on the Riemann tensor in Nordström's theory?

1838: 11260: 9964: 6922:. In either theory, with more work, we can derive more general expressions, but we shall be satisfied here with treating the special case of nearly circular orbits. 3813: 6683: 3933: 3271:

Adopting polar spherical coordinates, and using the known spherically symmetric asymptotically vanishing solutions of the Laplace equation, we can write the desired

2872: 2763: 1784: 10383:

Notice that if f vanishes, this family becomes a family of mutually stationary test particles in flat (Minkowski) spacetime. With respect to the timelike geodesic

7992:{\displaystyle (\Delta t)_{2}=\int _{R}^{R_{2}}dt\approx {\frac {m+R_{2}}{R_{2}}}\,{\sqrt {R_{2}^{2}-R^{2}}}={\sqrt {R_{2}^{2}-R^{2}}}+m\,{\sqrt {1-(R/R_{2})^{2}}}} 7768:{\displaystyle (\Delta t)_{1}=\int _{R}^{R_{1}}dt\approx {\frac {m+R_{1}}{R_{1}}}\,{\sqrt {R_{1}^{2}-R^{2}}}={\sqrt {R_{1}^{2}-R^{2}}}+m\,{\sqrt {1-(R/R_{1})^{2}}}} 7302: 7188: 6403: 5779: 4025: 3453:

coordinate in this chart does not readily admit a direct geometric interpretation. Therefore, we adopt instead Schwarzschild coordinates, using the transformation

3103: 2476: 2293: 2269: 2237: 1564: 667: 486: 248: 224: 8920: 8098: 9041: 2830: 5016: 10636:; the behavior of analogous families of test particles which encounter a gravitational plane wave in general relativity is quite different, because these are 3857:

Just as happens in the corresponding static spherically symmetric asymptotically flat solution of general relativity, this solution admits a four-dimensional

9646: 6838: 6764: 1204: 4275:

where to first order in m we have the same result as for the Schwarzschild vacuum. This also shows that Nordström's theory agrees with the result of the

2514:(a solution in which there is no matter present, nor any non-gravitational fields except for an electromagnetic field) vanishes shows that in the general 8822: 8321: 4168: 4125:

The geodesic equations are readily obtained from the geodesic Lagrangian. As always, these are second order nonlinear ordinary differential equations.

2975: 2219:

is the line element for flat spacetime in any convenient coordinate chart (such as cylindrical, polar spherical, or double null coordinates), and where

1996: 8752: 11095: 4932: 8926: 6832:

This can be compared with the corresponding expression for the Schwarzschild vacuum solution in general relativity, which is (to first order in m)

4640:

distance interpretation of r agrees with flat space notions). In contrast, in the Schwarzschild vacuum we have to first order in m the expression

945: 581:, unlike the Newtonian field theory (or most metric theories of gravitation). Second, the proposed field equation is linear. But by analogy with 11397: 6955: 4482: 6411: 4383: 1316: 837: 2639:(other than the symmetric property, which this second rank tensor always enjoys). But taking account of the twice-contracted and detracted 577:

However, this theory is unacceptable for a variety of reasons. Two objections are theoretical. First, this theory is not derivable from a

10913:

The exact solution we are discussing here, which we interpret as a propagating gravitational plane wave, gives some basic insight into the

939:

in which the pressure is assumed to be negligible). He argued that the contribution of this matter to the stress–energy tensor should be:

757: 8499:

We can summarize the results we found above in the following table, in which the given expressions represent appropriate approximations:

8008: 4871: 11203: 2519: 2494:

of the stress–energy tensor. But the field energy of an electromagnetic field contributes a term to the stress–energy tensor which is

4793: 4715: 596:

Einstein and von Laue proposed that the problem might lie with the field equation, which, they suggested, should have the linear form

496: 3111: 589:, we should expect this energy to be equivalent to mass and therefore, to gravitate. This implies that the field equation should be 11125: 6205:

circular orbit, or more vividly, in a slow rotation of the long axis of a quasi-Keplerian nearly elliptical orbit. Specifically,

1573: 11701: 11521: 5119:"gravitational acceleration" measured by a static observer who uses a rocket engine to maintain his position. In contrast, to 687: 11680: 11629: 11188: 6335:{\displaystyle \omega _{\rm {shm}}\approx {\frac {m^{2}}{L^{4}}}\,{\sqrt {m^{2}+L^{2}}}={\frac {1}{r^{2}}}\,{\sqrt {m^{2}+mr}}} 10315: 9977:. (In both theories, special plane waves may have extra symmetries.) We'll say a bit more about why this is so in a moment. 1198:, as should be the case. Furthermore, he showed, the field equation of Nordström's second theory follows from the Lagrangian 11758: 11706: 11173: 11088: 10992: 7075:

Consider two events on a null geodesic, before and after its point of closest approach to the origin. Let these distances be

2554: 2518:

in Nordström's theory, the metric tensor has the same form as in a vacuum solution, so we need only write down and solve the

2148: 1475: 272: 9177: 11451: 11382: 11358: 11331: 10859:{\displaystyle E_{{\hat {p}}{\hat {q}}}={\frac {1}{2}}\,\exp(-4\,f(u))\;\left(f'(u)^{2}-f''(u)\right)\,{\rm {diag}}(0,1,1)} 7232: 5413: 3442:{\displaystyle ds^{2}=(1-m/\rho )\,\left(-dt^{2}+d\rho ^{2}+\rho ^{2}\,(d\theta ^{2}+\sin(\theta )^{2}\,d\phi ^{2})\right)} 1665: 6906:

Thus, in Nordström's theory, if the nearly elliptical orbit is transversed counterclockwise, the long axis slowly rotates

11505: 9593: 2768:

otherwise gravitation would not, according to this theory, be a long-range force capable of propagating through a vacuum

2660: 11686: 11234: 11217: 11158: 10946: 9733: 4707: 1184:{\displaystyle 4\pi \,\left(T_{\rm {grav}}\right)_{ab}=\phi _{,a}\,\phi _{,b}-1/2\,\eta _{ab}\,\phi _{,m}\,\phi ^{,m}} 101:, in 1912 and 1913 respectively. The first was quickly dismissed, but the second became the first known example of a 11339: 11168: 11059: 11036: 10931: 9549: 102: 68: 46: 9462:{\displaystyle ds^{2}=\exp(2f(u))\;\left(-2\,du\,dv+dx^{2}+dy^{2}\right),\;\;\;-\infty <u,\,v,\,x,\,y<\infty } 8701: 8655: 1039:

He then derived an expression for the stress–energy tensor of the gravitational field in Nordström's second theory,

39: 11081: 5508: 2732:

which constrains the semi-traceless part of the Riemann tensor (the one built out of the trace-free Ricci tensor).

2541:, splits the Riemann tensor into three pieces, which are each fourth-rank tensors, built out of, respectively, the 10252: 10202: 1850: 1414: 11140: 10936: 10614:{\displaystyle \theta _{{\hat {p}}{\hat {q}}}={\frac {1}{\sqrt {2}}}\,f'(u)\,\exp(-2\,f(u))\,{\rm {diag}}(0,1,1)} 10384: 4117:

of general relativity, and they simply express the fact that this spacetime is static and spherically symmetric.

6025: 3208: 409: 11446: 11368: 11135: 10398: 6937:

at a rate of about 7 seconds of arc per century, whereas according to general relativity, the perihelia should

6187:{\displaystyle {\ddot {\varepsilon }}=-{\frac {m^{4}}{L^{8}}}\,(m^{2}+L^{2})\,\varepsilon +O(\varepsilon ^{2})} 5951: 5313: 2448:{\displaystyle ds^{2}=\exp(2\,\psi )\,\eta _{ab}\,dx^{a}\,dx^{b}\approx (1+2\psi )\,\eta _{ab}\,dx^{a}\,dx^{b}} 11639: 11163: 9895:{\displaystyle (u,v,x,y)\longrightarrow (u,\;v+x\,\lambda +{\frac {u}{2}}\,\lambda ^{2},\;x+u\,\lambda ,\;y)} 4643: 599: 365: 174: 5300:{\displaystyle E_{ab}={\frac {m}{r^{3}}}\,{\rm {diag}}(-2,1,1)+{\frac {m^{2}}{r^{4}}}\,{\rm {diag}}(-1,1,1)} 4285: 3772:{\displaystyle -\infty <t<\infty ,\;0<r<\infty ,\;0<\theta <\pi ,\;-\pi <\phi <\pi } 11753: 10951: 2936:

Another feature of Nordström's theory is that it can be written as the theory of a certain scalar field in

10191:{\displaystyle {\vec {e}}_{1}={\frac {1}{\sqrt {2}}}\,\left(\partial _{v}-\exp(-2f)\,\partial _{u}\right)} 10083:{\displaystyle {\vec {e}}_{0}={\frac {1}{\sqrt {2}}}\,\left(\partial _{v}+\exp(-2f)\,\partial _{u}\right)} 3456: 2789:

We can tabulate the most basic differences between Nordström's theory and general relativity, as follows:

1929: 11424: 11318: 11308: 11183: 3676:{\displaystyle ds^{2}=(1+m/r)^{-2}\,(-dt^{2}+dr^{2})+r^{2}\,(d\theta ^{2}+\sin(\theta )^{2}\,d\phi ^{2})} 7544:{\displaystyle dt\approx {\frac {1}{\sqrt {r^{2}-R^{2}}}}\;\left(r+m\,{\frac {R^{2}}{r^{2}}}\right)\;dr} 7126: 4697:. In a sense, the extra term here results from the nonlinearity of the vacuum Einstein field equation. 2969:

The static vacuum solutions in Nordström's theory are the Lorentzian manifolds with metrics of the form

11624: 11266: 11178: 11130: 9024: 8421: 8226: 7078: 4276: 3904: 2245: 2110: 1310: 338:, which violates the principle that any "news" which has a physical effect (in this case, an effect on 4339: 3818: 11418: 11413: 11045: 7193: 5734:{\displaystyle 2{\dot {r}}{\ddot {r}}={\frac {d}{dr}}\left((E^{2}-V)\,(1+m/r)^{4}\right)\;{\dot {r}}} 926:

on September 23, 1913, Einstein surveyed the state of the art, declaring that only his own work with

578: 330:

This theory is not relativistic because the equation of motion refers to coordinate time rather than

10873: 6348: 5376: 4131: 11634: 11515: 11344: 9970: 9909: 9520: 9485: 5903: 4587: 4113:

These are exactly the same vector fields which arise in the Schwarzschild coordinate chart for the

3875: 3782:

Here, r now has the simple geometric interpretation that the surface area of the coordinate sphere

1791: 678: 585:, we should expect the gravitational field to carry energy, and on the basis of Einstein's work on 33: 1298:

and also shows that it obeys other properties we must demand from a self-consistent field theory.

11549: 11482: 8283: 2546: 1723: 227: 9255: 5747: 105:, in which the effects of gravitation are treated entirely in terms of the geometry of a curved 11466: 11350: 6748:{\displaystyle \Delta \phi =2\pi \,\Delta \omega \approx -\pi \,{\sqrt {\frac {m^{3}}{r^{5}}}}} 6198: 5373:

In our discussion of the geodesic equations, we showed that in the equatorial coordinate plane

4014:{\displaystyle -\cos(\phi )\,\partial _{\theta }+\cot(\theta )\,\sin(\phi )\,\partial _{\phi }} 2511: 674: 161: 50: 7428:{\displaystyle r^{2}\,d\phi ^{2}=\cot(\phi )^{2}\,dr^{2}={\frac {R^{2}}{r^{2}-R^{2}}}\,dr^{2}} 5890:{\displaystyle {\ddot {r}}={\frac {1}{2}}\,{\frac {d}{dr}}\left((E^{2}-V)\,(1+m/r)^{4}\right)} 4103:{\displaystyle \sin(\phi )\,\partial _{\theta }+\cot(\theta )\,\cos(\phi )\,\partial _{\phi }} 2889: 1807: 11598: 11577: 11570: 11510: 10917:

of gravitational radiation in Nordström's theory, but it does not yield any insight into the

9936: 3785: 1802: 1469: 125:

Nordström's theories arose at a time when several leading physicists, including Nordström in

8491:

according to Nordström's theory and about 240 microseconds according to general relativity.

8213:{\displaystyle \Delta t=m\,\left({\sqrt {1-(R/R_{1})^{2}}}+{\sqrt {1-(R/R_{2})^{2}}}\right)} 6758:

and to first order in m, the long axis of the nearly elliptical orbit rotates with the rate

2847: 2738: 1759: 593:. Another objection is more practical: this theory disagrees drastically with observation. 11665: 11584: 11563: 11556: 11431: 11286: 11244: 7173: 6381: 4114: 3088: 2651:, we see that in Nordström's theory, as a consequence of the field equations, we have the 2461: 2278: 2254: 2222: 1549: 652: 489: 464: 233: 209: 154: 9161:{\displaystyle ds^{2}=-2\,du\,dv+dx^{2}+dy^{2},\;\;\;-\infty <u,\,v,\,x,\,y<\infty } 8901: 8475:. For example, in the classic experiment in which, at a time when, as viewed from Earth, 8: 11046:

Lightman, Alan P.; Press, William H.; Price, Richard H. & Teukolsky, Saul A. (1975).

5108:{\displaystyle \nabla _{{\vec {e}}_{0}}{\vec {e}}_{0}={\frac {m}{r^{2}}}\,{\vec {e}}_{1}} 2940:, and in this form enjoys the expected conservation law for nongravitational mass-energy 2937: 2648: 2538: 1567: 1405: 263: 255: 82: 10941: 98: 11727: 11193: 11150: 11011: 9720:{\displaystyle x\,\partial _{v}+u\,\partial _{x},\;\;y\,\partial _{v}+u\,\partial _{y}} 3085:

where we can take the flat spacetime Laplace operator on the right. To first order in

2964: 2815: 2781:

in a region which is matter-free but contains an electromagnetic field), and it is the

114: 90: 6896:{\displaystyle {\frac {\Delta \phi }{\omega _{\rm {orb}}}}\approx {\frac {6\pi m}{r}}} 6822:{\displaystyle {\frac {\Delta \phi }{\omega _{\rm {orb}}}}\approx -{\frac {\pi m}{r}}} 11711: 11619: 11363: 11291: 11055: 11048: 11032: 10988: 10306: 2251:

In any solution to Nordström's field equations (vacuum or otherwise), if we consider

1401: 1284:{\displaystyle L={\frac {1}{8\pi }}\,\eta ^{ab}\,\phi _{,a}\,\phi _{,b}-\rho \,\phi } 586: 347: 160:

All of these researchers began by trying to suitably modify the existing theory, the

7226:(the equation of a straight line in polar coordinates) and differentiate to obtain 11693: 11387: 11281: 10980: 8882:{\displaystyle {\frac {\Delta \phi }{\omega _{\rm {orb}}}}={\frac {6\,\pi \,m}{R}}} 6930: 5600:

for a timelike geodesic. Differentiating with respect to proper time s, we obtain

4632:, which agrees perfectly with Newtonian theory (if we ignore the fact that now the 2640: 1295: 927: 582: 399: 359: 169: 9472:

which we can interpret in terms of the propagation of a gravitational plane wave.

8408:{\displaystyle \Delta t=2m+2m\,\log \left({\frac {4\,R_{1}\,R_{2}}{R^{2}}}\right)} 7554:

Thus the coordinate time from the first event to the event of closest approach is

4265:{\displaystyle {\dot {t}}=E\,\left(1+m/r\right)^{2}\approx E\,\left(1+2m/r\right)} 3075:{\displaystyle ds^{2}=\exp(2\psi )\,\eta _{ab}\,dx^{a}\,dx^{b},\;\;\Delta \psi =0} 2478:

with the Newtonian gravitational potential, and we can regard it as controlling a

2097:{\displaystyle ds^{2}=\exp(2\psi )\,\eta _{ab}\,dx^{a}\,dx^{b},\;\;\;\Box \psi =0} 11392: 11313: 11273: 11239: 10984: 8811:{\displaystyle {\frac {\Delta \phi }{\omega _{\rm {orb}}}}=-{\frac {\pi \,m}{R}}} 2773:

In general relativity, something somewhat analogous happens, but there it is the

1306: 146: 2637:

leaves the trace-free Ricci tensor entirely unconstrained by algebraic relations

11592: 5000:{\displaystyle {\vec {e}}_{3}={\frac {1}{r\,\sin(\theta )}}\,\partial _{\phi }} 3866: 2644: 2537:

To find out, note that an important identity from the theory of manifolds, the

2241: 1302: 343: 342:

motion far from the source of the field) cannot be transmitted faster than the

165: 11747: 11529: 11198: 9006:{\displaystyle 2\,m+2\,m\;\log \left({\frac {4\,R_{1}\,R_{2}}{R^{2}}}\right)} 8002:

Here the elapsed coordinate time expected from Newtonian theory is of course

1409: 1029:{\displaystyle \left(T_{\rm {matter}}\right)_{ab}=\phi \,\rho \,u_{a}\,u_{b}} 936: 403: 339: 251: 11487: 11296: 11007: 10907: 9479:, or equivalently, a six-dimensional Lie algebra of Killing vector fields: 2542: 1919:

Einstein was attracted to Nordström's second theory by its simplicity. The

1798: 1787: 1656: 259: 11073: 11456: 10975:. In Janssen, M.; Norton, J.D.; Renn, J.; Sauer, T.; Stachel, J. (eds.). 9514: 8488: 7065:{\displaystyle 0={\frac {-dt^{2}+dr^{2}}{(1+m/r)^{2}}}+r^{2}\,d\phi ^{2}} 4566:{\displaystyle V={\frac {L^{2}/r^{2}-\epsilon }{\left(1+m/r\right)^{2}}}} 3862: 2632: 2510:

Incidentally, the fact that the trace of the stress–energy tensor for an

331: 130: 10910:, so we can regard these tensors as defined on orthogonal hyperslices.) 9295:

traveling in the z direction. Therefore, Nordström's theory admits the

9027:; not surprisingly, Newton's theory flunks all four relativistic tests. 6481:{\displaystyle \omega _{\rm {orb}}={\frac {L}{r^{2}}}={\sqrt {m/r^{3}}}} 5010:

Then, the acceleration of the world line of our test particle is simply

4466:{\displaystyle {\frac {{\dot {r}}^{2}}{\left(1+m/r\right)^{4}}}=E^{2}-V} 751:, which is nonlinear. Nordström now took the equation of motion to be 11670: 10972: 10640:. This is due to the fact that Nordström's theory of gravitation is a 10388: 9292: 6926: 2500:

in Nordström's theory, electromagnetic field energy does not gravitate!

1393:{\displaystyle L=\phi ^{2}\,\eta _{ab}\,{\dot {u}}^{a}\,{\dot {u}}^{b}} 912:{\displaystyle \phi \,{\dot {u}}_{a}=-\phi _{,a}-{\dot {\phi }}\,u_{a}} 142: 138: 11016: 6022:. Evaluating the derivative, using our earlier results, and setting 398:. This has the result of changing the vacuum field equation from the 11675: 11657: 11461: 11438: 10644:, whereas Einstein's theory of gravitation (general relativity) is a 8092:

so the relativistic time delay, according to Nordström's theory, is

3858: 2953:

the spherically symmetric static asymptotically flat vacuum solutions

355: 106: 2965:

The static spherically symmetric asymptotically flat vacuum solution

2956:

the general vacuum gravitational plane wave solution in this theory.

824:{\displaystyle {\frac {d\left(\phi \,u_{a}\right)}{ds}}=-\phi _{,a}} 681:

describing the density, momentum, and stress of any matter present.

11732: 11542: 11535: 10392: 8082:{\displaystyle {\sqrt {R_{1}^{2}-R^{2}}}+{\sqrt {R_{2}^{2}-R^{2}}}} 6914:

six times faster. In the first case we may speak of a periastrion

1840:, which measures the deviation of a Lorentzian manifold from being 250:

is the density of matter, augmented by an equation of motion for a

126: 9933:. Note that our isometry group is transitive on the hypersurfaces 7438:

Plugging this into the line element and solving for dt, we obtain

4921:{\displaystyle {\vec {e}}_{2}={\frac {1}{r}}\,\partial _{\theta }} 11104: 4860:{\displaystyle {\vec {e}}_{1}=\left(1+m/r\right)\,\partial _{r}} 4782:{\displaystyle {\vec {e}}_{0}=\left(1+m/r\right)\,\partial _{t}} 567:{\displaystyle {\dot {u}}_{a}=-\phi _{,a}-{\dot {\phi }}\,u_{a}} 164:

version of Newton's theory of gravitation. In this theory, the

97:

distinct theories proposed by the Finnish theoretical physicist

11028:

Subtle is the Lord: The Science and the Life of Albert Einstein

10648:. On the other hand, gravitational waves in both theories are 6949:

Null geodesics in the equatorial plane of our solution satisfy

3195:{\displaystyle ds^{2}=(1+2\,\psi )\,\eta _{ab}\,dx^{a}\,dx^{b}} 923: 150: 11324: 8484: 8476: 134: 10301:

we find that the corresponding family of test particles are

3263:

is the metric of Minkowski spacetime (the flat background).

1648:{\displaystyle ds^{2}=\phi ^{2}\,\eta _{ab}\,dx^{a}\,dx^{b}} 254:

in an ambient gravitational field, which we can derive from

10624:

shows that our test particles are expanding or contracting

5368: 4580:. In the timelike case, we see from this that there exist 9778:

integrates to give the one parameter family of isometries

9641:(rotation about axis parallel to direction of propagation) 8504:

Comparison of Predictions in Three Theories of Gravitation

3861:

of isometries, or equivalently, a four-dimensional (real)

744:{\displaystyle \phi \,\Box \phi =-4\pi \,T_{\rm {matter}}} 10632:. This is exactly what we would expect for a transverse 10373:{\displaystyle \nabla _{{\vec {e}}_{0}}{\vec {e}}_{0}=0} 2794:

Comparison of Nordström's theory with General Relativity

10652:

waves. Electromagnetic plane waves are of course also

2621:{\displaystyle S_{ab}=R_{ab}-{\frac {1}{4}}\,R\,g_{ab}} 2458:

Thus, in the weak field approximation, we can identify

2212:{\displaystyle d\sigma ^{2}=\eta _{ab}\,dx^{a}\,dx^{b}} 1539:{\displaystyle d\sigma ^{2}=\eta _{ab}\,dx^{a}\,dx^{b}} 320:{\displaystyle {\frac {d{\vec {u}}}{dt}}=-\nabla \phi } 9242:{\displaystyle -2\,\psi _{uv}+\psi _{xx}+\psi _{yy}=0} 1570:, so that the line element of the curved spacetime is 10876: 10668: 10458: 10401: 10318: 10255: 10205: 10097: 9989: 9939: 9912: 9787: 9736: 9649: 9596: 9552: 9523: 9488: 9307: 9258: 9180: 9044: 9018:

The last four lines in this table list the so-called

8929: 8904: 8825: 8755: 8704: 8658: 8424: 8324: 8286: 8229: 8101: 8011: 7787: 7563: 7447: 7305: 7286:{\displaystyle 0=-r\sin \phi \,d\phi +\cos \phi \,dr} 7235: 7196: 7176: 7129: 7081: 6958: 6841: 6767: 6686: 6500: 6414: 6384: 6351: 6214: 6085: 6028: 5954: 5906: 5782: 5750: 5609: 5511: 5495:{\displaystyle {\dot {r}}^{2}=(E^{2}-V)\;(1+m/r)^{4}} 5416: 5379: 5316: 5136: 5019: 4935: 4874: 4796: 4718: 4646: 4590: 4485: 4386: 4342: 4288: 4171: 4134: 4028: 3936: 3907: 3878: 3821: 3788: 3690: 3512: 3459: 3284: 3211: 3114: 3091: 2978: 2892: 2850: 2818: 2741: 2663: 2557: 2464: 2304: 2281: 2257: 2225: 2151: 2113: 1999: 1932: 1853: 1810: 1790:, which is itself a kind of trace of the fourth rank 1762: 1726: 1717:

Therefore, Nordström's field equation becomes simply

1707:{\displaystyle R=-{\frac {6\,\Box \phi }{\phi ^{3}}}} 1668: 1576: 1552: 1478: 1417: 1319: 1207: 1048: 948: 840: 760: 690: 655: 602: 499: 467: 412: 368: 275: 236: 212: 177: 11010:(2004). "Scalar Gravitation and Extra Dimensions". 11047: 10898: 10858: 10613: 10438: 10372: 10290: 10240: 10190: 10082: 9958: 9925: 9894: 9770: 9719: 9634:{\displaystyle -y\,\partial _{x}+x\,\partial _{y}} 9633: 9588:(spatial translation orthogonal to the wavefronts) 9580: 9536: 9501: 9461: 9279: 9241: 9160: 9035:In the double null chart for Minkowski spacetime, 9030: 9005: 8914: 8881: 8810: 8732: 8686: 8467: 8418:which depends logarithmically on the small ratios 8407: 8315:The corresponding result in general relativity is 8304: 8272: 8212: 8081: 7991: 7767: 7543: 7427: 7285: 7218: 7182: 7162: 7115: 7064: 6941:at a rate of about 43 seconds of arc per century. 6925:For example, according to Nordström's theory, the 6895: 6821: 6747: 6666: 6480: 6397: 6370: 6334: 6186: 6068: 6014: 5940: 5889: 5765: 5733: 5592: 5494: 5399: 5353: 5299: 5127:The tidal tensor measured by a static observer is 5107: 4999: 4920: 4859: 4781: 4689: 4624: 4565: 4465: 4369: 4324: 4264: 4154: 4102: 4013: 3920: 3891: 3846: 3807: 3771: 3675: 3495: 3441: 3255: 3194: 3097: 3074: 2914: 2866: 2824: 2757: 2722:{\displaystyle {{S_{a}}^{b}}_{;b}=6\,\pi \,T_{;a}} 2721: 2635:. It immediately follows that Nordström's theory 2620: 2470: 2447: 2287: 2263: 2231: 2211: 2137: 2096: 1975: 1903: 1832: 1778: 1745: 1706: 1647: 1558: 1538: 1460: 1392: 1283: 1183: 1028: 911: 823: 743: 661: 641: 566: 480: 453: 390: 319: 242: 218: 198: 10971:Smeenk, Christopher; Martin, Christopher (2007). 9771:{\displaystyle x\,\partial _{v}+u\,\partial _{x}} 5900:We found earlier that the minimum of V occurs at 4377:for timelike, null, spacelike geodesics, we find 1923:field equations in Nordström's theory are simply 1914: 11745: 2961:and in Einstein's general theory of relativity. 9581:{\displaystyle \partial _{x},\;\;\partial _{y}} 5365:in Nordström gravity than in Einstein gravity. 11398:Degenerate Higher-Order Scalar-Tensor theories 8733:{\displaystyle \delta \phi ={\frac {4\,m}{R}}} 8687:{\displaystyle \delta \phi ={\frac {2\,m}{R}}} 2643:, a differential identity which holds for the 120: 11089: 11024: 11006: 5593:{\displaystyle V=(1+L^{2}/r^{2})/(1+m/r)^{2}} 10970: 10630:transversely to the direction of propagation 10291:{\displaystyle {\vec {e}}_{3}=\partial _{y}} 10241:{\displaystyle {\vec {e}}_{2}=\partial _{x}} 6910:, whereas in general relativity, it rotates 3869:fields. These are readily determined to be 1904:{\displaystyle R=24\pi \,T,\;\;\;C_{abcd}=0} 1461:{\displaystyle g_{ab}=\phi ^{2}\,\eta _{ab}} 11103: 3928:(rotation about an axis through the origin) 2653:first-order covariant differential equation 1309:in which he derived what is now called the 11096: 11082: 11050:Problem Book in Relativity and Gravitation 10973:"Mie's Theories of Matter and Gravitation" 10763: 9885: 9868: 9827: 9685: 9684: 9567: 9566: 9419: 9418: 9417: 9351: 9118: 9117: 9116: 8947: 8446: 8251: 7534: 7489: 6197:which is (to first order) the equation of 6069:{\displaystyle \varepsilon =r-L^{2}/m^{2}} 5718: 5461: 3750: 3731: 3712: 3256:{\displaystyle \eta _{ab}\,dx^{a}\,dx^{b}} 3059: 3058: 2273:conformal perturbation from flat spacetime 2081: 2080: 2079: 1947: 1946: 1945: 1875: 1874: 1873: 649:, where F is some yet unknown function of 454:{\displaystyle {\dot {u}}_{a}=-\phi _{,a}} 93:. Strictly speaking, there were actually 11054:. Princeton: Princeton University Press. 11015: 10818: 10747: 10731: 10573: 10557: 10541: 10523: 10439:{\displaystyle {\vec {X}}={\vec {e}}_{0}} 10172: 10132: 10064: 10024: 9878: 9854: 9837: 9757: 9740: 9706: 9689: 9670: 9653: 9620: 9603: 9449: 9442: 9435: 9370: 9363: 9187: 9148: 9141: 9134: 9074: 9067: 8975: 8964: 8943: 8933: 8908: 8869: 8865: 8798: 8720: 8674: 8377: 8366: 8349: 8114: 7947: 7873: 7723: 7649: 7504: 7411: 7355: 7316: 7276: 7257: 7206: 7143: 7102: 7095: 7048: 6718: 6702: 6308: 6260: 6158: 6128: 6015:{\displaystyle E_{c}=L^{2}/(L^{2}+m^{2})} 5851: 5808: 5682: 5354:{\displaystyle {\vec {X}}={\vec {e}}_{0}} 5256: 5188: 5085: 4986: 4967: 4907: 4846: 4768: 4231: 4190: 4089: 4073: 4044: 4000: 3984: 3955: 3828: 3656: 3614: 3562: 3469: 3417: 3375: 3324: 3239: 3225: 3178: 3164: 3150: 3143: 3041: 3027: 3013: 2777:which vanishes in any vacuum region (but 2705: 2701: 2604: 2600: 2431: 2417: 2403: 2368: 2354: 2340: 2333: 2195: 2181: 2062: 2048: 2034: 1866: 1739: 1684: 1631: 1617: 1603: 1522: 1508: 1444: 1370: 1350: 1336: 1277: 1257: 1243: 1229: 1167: 1153: 1139: 1111: 1055: 1015: 1004: 1000: 898: 844: 775: 713: 694: 553: 384: 358:in the Newtonian field equation with the 346:. Einstein's former calculus professor, 69:Learn how and when to remove this message 9975:five-dimensional Lie group of isometries 9171:a simple solution of the wave equation 5369:Extra-Newtonian precession of periastria 3503:, which brings the metric into the form 2520:curved spacetime Maxwell field equations 2505:this theory can exhibit no light bending 32:This article includes a list of general 11702:Gravitational interaction of antimatter 9477:six-dimensional Lie group of isometries 4690:{\displaystyle r_{c}\approx L^{2}/m-3m} 1990:vacuum solution in Nordström's theory: 642:{\displaystyle FT_{\rm {matter}}=\rho } 391:{\displaystyle \Box \phi =4\pi \,\rho } 199:{\displaystyle \Delta \phi =4\pi \rho } 16:Predecessor to the theory of relativity 11746: 9730:For example, the Killing vector field 6918:and in the second case, a periastrion 4325:{\displaystyle {\dot {\phi }}=L/r^{2}} 153:, were all trying to create competing 11707:Physics in the medieval Islamic world 11630:(2+1)-dimensional topological gravity 11126:Newton's law of universal gravitation 11077: 8523:Acceleration of static test particle 8494: 262:of the test particle is given by the 11452:Asymptotic safety in quantum gravity 9291:smooth function. This represents a 6677:so the periastrion lag per orbit is 3496:{\displaystyle r=\rho \,(1-m/\rho )} 2949:important solutions in this theory: 2930:nonzero for gravitational radiation 1976:{\displaystyle R=0,\;\;\;C_{abcd}=0} 18: 11031:. Oxford: Oxford University Press. 10906:, which in this case happens to be 8223:To first order in the small ratios 4700: 2879:nonzero for gravitational radiation 1301:Meanwhile, a gifted Dutch student, 13: 11204:Gibbons–Hawking–York boundary term 10830: 10827: 10824: 10821: 10585: 10582: 10579: 10576: 10320: 10279: 10229: 10174: 10139: 10066: 10031: 9914: 9759: 9742: 9708: 9691: 9672: 9655: 9622: 9605: 9569: 9554: 9525: 9490: 9475:This Lorentzian manifold admits a 9456: 9423: 9155: 9122: 8848: 8845: 8842: 8829: 8778: 8775: 8772: 8759: 8325: 8287: 8102: 7791: 7567: 7163:{\displaystyle R_{1},\,R_{2}\gg R} 6864: 6861: 6858: 6845: 6790: 6787: 6784: 6771: 6703: 6687: 6543: 6540: 6537: 6522: 6519: 6516: 6501: 6427: 6424: 6421: 6227: 6224: 6221: 5268: 5265: 5262: 5259: 5200: 5197: 5194: 5191: 5021: 4988: 4909: 4848: 4770: 4091: 4046: 4002: 3957: 3909: 3880: 3725: 3706: 3694: 3060: 2530:, say in terms of a power series. 1986:We can immediately write down the 1659:of this curved spacetime is just 1076: 1073: 1070: 1067: 975: 972: 969: 966: 963: 960: 735: 732: 729: 726: 723: 720: 627: 624: 621: 618: 615: 612: 311: 178: 38:it lacks sufficient corresponding 14: 11770: 11319:Modified Newtonian dynamics, MOND 11235:Classical theories of gravitation 10977:The Genesis of General Relativity 10932:Classical theories of gravitation 9973:in general relativity has only a 8744:Rate of precession of periastria 8468:{\displaystyle R/R_{1},\;R/R_{2}} 8273:{\displaystyle R/R_{1},\;R/R_{2}} 7116:{\displaystyle R_{1},\,R,\,R_{2}} 3921:{\displaystyle \partial _{\phi }} 2138:{\displaystyle \phi =\exp(\psi )} 1546:with corresponding wave operator 1305:had written a Ph.D. thesis under 87:Nordström's theory of gravitation 4370:{\displaystyle \epsilon =-1,0,1} 3847:{\displaystyle 4\pi \,r_{0}^{2}} 2526:, so we can also write down the 2244:in the manner familiar from the 23: 11141:History of gravitational theory 10937:Congruence (general relativity) 9031:Vacuum gravitational plane wave 9020:four classic solar system tests 8624:Gravitational red shift factor 7219:{\displaystyle R=r\,\cos \phi } 6378:and removed the subscript from 11447:Causal dynamical triangulation 11136:Poisson's equation for gravity 10964: 10899:{\displaystyle {\vec {e}}_{0}} 10884: 10853: 10835: 10810: 10804: 10784: 10777: 10760: 10757: 10751: 10738: 10710: 10698: 10688: 10681: 10672: 10608: 10590: 10570: 10567: 10561: 10548: 10538: 10532: 10500: 10488: 10478: 10471: 10462: 10424: 10408: 10352: 10331: 10263: 10213: 10169: 10157: 10105: 10061: 10049: 9997: 9889: 9818: 9815: 9812: 9788: 9348: 9345: 9339: 9330: 9274: 9268: 8194: 8172: 8150: 8128: 7978: 7956: 7798: 7788: 7754: 7732: 7574: 7564: 7346: 7339: 7023: 7002: 6944: 6371:{\displaystyle L={\sqrt {mr}}} 6181: 6168: 6155: 6129: 6009: 5983: 5873: 5852: 5848: 5829: 5704: 5683: 5679: 5660: 5581: 5560: 5552: 5518: 5483: 5462: 5458: 5439: 5400:{\displaystyle \theta =\pi /2} 5339: 5323: 5294: 5273: 5226: 5205: 5156: 5149: 5140: 5093: 5053: 5032: 4980: 4974: 4943: 4882: 4804: 4726: 4155:{\displaystyle \theta =\pi /2} 4086: 4080: 4070: 4064: 4041: 4035: 3997: 3991: 3981: 3975: 3952: 3946: 3670: 3647: 3640: 3615: 3598: 3563: 3550: 3529: 3490: 3470: 3431: 3408: 3401: 3376: 3321: 3301: 3147: 3131: 3010: 3001: 2528:general electrovacuum solution 2482:conformal perturbation from a 2400: 2385: 2337: 2327: 2132: 2126: 2031: 2022: 1915:Features of Nordström's theory 362:or wave operator, which gives 288: 1: 10957: 9926:{\displaystyle \partial _{z}} 9537:{\displaystyle \partial _{u}} 9502:{\displaystyle \partial _{v}} 5941:{\displaystyle r_{c}=L^{2}/m} 4625:{\displaystyle r_{c}=L^{2}/m} 4115:Schwarzschild vacuum solution 3892:{\displaystyle \partial _{t}} 3266: 11759:Obsolete theories in physics 11025:Pais, Abraham (1982). "13". 10985:10.1007/978-1-4020-4000-9_35 10952:General Theory of Relativity 10391:obtained by integrating the 10305:(freely falling), since the 4120: 103:metric theory of gravitation 7: 11309:Infinite derivative gravity 10925: 8305:{\displaystyle \Delta t=2m} 2246:Lienard-Wiechert potentials 1746:{\displaystyle R=24\pi \,T} 1566:on the flat background, or 121:Development of the theories 10: 11775: 11497:Unified-field-theoric and 10947:Obsolete physical theories 9280:{\displaystyle \psi =f(u)} 8559:Extra-Coulomb tidal force 5766:{\displaystyle {\dot {r}}} 2840:vanishes in electrovacuum 258:and which states that the 11720: 11652: 11640:Jackiw–Teitelboim gravity 11610: 11496: 11475: 11419:Canonical quantum gravity 11414:Euclidean quantum gravity 11406: 11253: 11227: 11216: 11149: 11118: 11111: 9980:Adopting the frame field 9969:In contrast, the generic 8579:Radius of circular orbit 2837:vanishes in electrovacuum 2484:flat spacetime background 2275:, then to first order in 157:theories of gravitation. 11516:Superfluid vacuum theory 11340:Nonsymmetric gravitation 11189:Post-Newtonian formalism 9971:gravitational plane wave 7170:. We wish to eliminate 2915:{\displaystyle C_{abcd}} 1833:{\displaystyle C_{abcd}} 1792:Riemann curvature tensor 11681:Mechanical explanations 11550:Heterotic string theory 11506:Noncommutative geometry 11425:Wheeler–DeWitt equation 11151:General relativity (GR) 11131:Gauss's law for gravity 11105:Theories of gravitation 10979:. Springer, Dordrecht. 9959:{\displaystyle u=u_{0}} 8650:Angle of light bending 5744:Dividing both sides by 3808:{\displaystyle r=r_{0}} 2547:trace-free Ricci tensor 228:gravitational potential 53:more precise citations. 11522:Logarithmic BEC vacuum 11467:Rainbow gravity theory 11345:Scalar–tensor theories 11119:Newtonian gravity (NG) 10900: 10860: 10615: 10440: 10374: 10292: 10242: 10192: 10084: 9960: 9927: 9896: 9772: 9721: 9635: 9582: 9538: 9503: 9463: 9281: 9243: 9162: 9025:Pound–Rebka experiment 9007: 8916: 8883: 8812: 8734: 8688: 8479:is just about to pass 8469: 8409: 8306: 8274: 8214: 8083: 7993: 7769: 7545: 7429: 7287: 7220: 7184: 7164: 7117: 7066: 6897: 6823: 6749: 6668: 6482: 6399: 6372: 6336: 6199:simple harmonic motion 6188: 6070: 6016: 5942: 5891: 5767: 5735: 5594: 5496: 5401: 5355: 5301: 5109: 5001: 4922: 4861: 4783: 4691: 4626: 4582:stable circular orbits 4567: 4467: 4371: 4326: 4277:Pound–Rebka experiment 4266: 4156: 4104: 4015: 3922: 3893: 3848: 3809: 3773: 3677: 3497: 3443: 3257: 3196: 3099: 3076: 2916: 2868: 2867:{\displaystyle S_{ab}} 2826: 2759: 2758:{\displaystyle S_{ab}} 2723: 2622: 2516:electrovacuum solution 2512:electrovacuum solution 2472: 2449: 2289: 2265: 2233: 2213: 2139: 2098: 1977: 1905: 1834: 1780: 1779:{\displaystyle g_{ab}} 1747: 1708: 1649: 1560: 1540: 1462: 1394: 1311:Fokker–Planck equation 1285: 1185: 1030: 913: 825: 745: 663: 643: 568: 482: 455: 392: 321: 244: 220: 200: 11599:Twistor string theory 11578:Type II string theory 11571:Bosonic string theory 11511:Semiclassical gravity 11476:Unified-field-theoric 11261:Poincaré gauge theory 10901: 10861: 10616: 10441: 10375: 10293: 10243: 10193: 10085: 9961: 9928: 9897: 9773: 9722: 9636: 9583: 9539: 9509:(a null translation, 9504: 9464: 9297:exact vacuum solution 9282: 9244: 9163: 9008: 8917: 8884: 8813: 8735: 8689: 8470: 8410: 8307: 8275: 8215: 8084: 7994: 7770: 7546: 7430: 7288: 7221: 7185: 7183:{\displaystyle \phi } 7165: 7118: 7067: 6898: 6824: 6750: 6669: 6483: 6400: 6398:{\displaystyle r_{c}} 6373: 6337: 6189: 6071: 6017: 5943: 5892: 5768: 5736: 5595: 5497: 5402: 5356: 5302: 5110: 5002: 4923: 4862: 4784: 4692: 4627: 4568: 4468: 4372: 4327: 4267: 4157: 4105: 4016: 3923: 3899:(translation in time) 3894: 3849: 3810: 3774: 3678: 3498: 3444: 3258: 3197: 3105:, the metric becomes 3100: 3098:{\displaystyle \psi } 3077: 2917: 2869: 2827: 2760: 2724: 2623: 2524:conformally invariant 2473: 2471:{\displaystyle \psi } 2450: 2290: 2288:{\displaystyle \psi } 2266: 2264:{\displaystyle \psi } 2248:in electromagnetism. 2234: 2232:{\displaystyle \Box } 2214: 2140: 2099: 1978: 1906: 1835: 1803:Weyl curvature tensor 1781: 1748: 1709: 1650: 1561: 1559:{\displaystyle \Box } 1541: 1470:Cartesian coordinates 1463: 1395: 1286: 1186: 1031: 914: 826: 746: 664: 662:{\displaystyle \phi } 644: 569: 483: 481:{\displaystyle u^{a}} 456: 393: 322: 245: 243:{\displaystyle \rho } 221: 219:{\displaystyle \phi } 201: 89:was a predecessor of 11666:Aristotelian physics 11635:Gauss–Bonnet gravity 11585:Little string theory 11564:Type 0 string theory 11557:Type I string theory 11432:Loop quantum gravity 11359:Scalar–tensor–vector 11332:Tensor–vector–scalar 11287:Gauge theory gravity 11245:Theory of everything 10874: 10666: 10456: 10399: 10316: 10253: 10203: 10095: 9987: 9937: 9910: 9785: 9734: 9647: 9594: 9550: 9521: 9486: 9305: 9256: 9178: 9042: 8927: 8915:{\displaystyle 2\,m} 8902: 8823: 8753: 8702: 8656: 8422: 8322: 8284: 8227: 8099: 8009: 7785: 7561: 7445: 7303: 7233: 7194: 7174: 7127: 7079: 6956: 6839: 6765: 6684: 6498: 6412: 6382: 6349: 6212: 6083: 6026: 5952: 5904: 5780: 5748: 5607: 5509: 5414: 5377: 5314: 5134: 5017: 4933: 4872: 4794: 4716: 4644: 4588: 4483: 4384: 4340: 4286: 4279:. Second, we have 4169: 4132: 4026: 3934: 3905: 3876: 3819: 3786: 3688: 3510: 3457: 3282: 3209: 3112: 3089: 2976: 2924:completely traceless 2890: 2848: 2816: 2739: 2661: 2555: 2462: 2302: 2279: 2255: 2223: 2149: 2111: 1997: 1930: 1851: 1808: 1760: 1724: 1666: 1574: 1550: 1476: 1415: 1317: 1205: 1046: 946: 838: 758: 688: 679:stress–energy tensor 653: 600: 497: 490:velocity four-vector 465: 410: 366: 273: 234: 210: 175: 11754:Theories of gravity 11483:Kaluza–Klein theory 10307:acceleration vector 8506: 8063: 8028: 7925: 7890: 7831: 7701: 7666: 7607: 6491:The discrepancy is 4578:effective potential 3843: 2938:Minkowski spacetime 2882:vanishes in vacuum 2796: 2649:Riemannian manifold 2539:Ricci decomposition 1568:Minkowski spacetime 1406:Lorentzian manifold 1402:geodesic Lagrangian 83:theoretical physics 11728:Gravitational wave 11611:Generalisations / 11499:quantum-mechanical 11407:Quantum-mechanical 11220:general relativity 11194:Linearized gravity 10896: 10856: 10611: 10436: 10395:unit vector field 10370: 10288: 10238: 10188: 10080: 9956: 9923: 9892: 9768: 9717: 9631: 9578: 9534: 9499: 9459: 9277: 9239: 9158: 9003: 8912: 8879: 8808: 8730: 8684: 8502: 8495:Summary of results 8465: 8405: 8302: 8270: 8210: 8079: 8049: 8014: 7989: 7911: 7876: 7810: 7765: 7687: 7652: 7586: 7541: 7425: 7283: 7216: 7180: 7160: 7113: 7062: 6893: 6819: 6745: 6664: 6478: 6395: 6368: 6332: 6184: 6066: 6012: 5938: 5887: 5763: 5731: 5590: 5492: 5397: 5351: 5297: 5105: 4997: 4918: 4857: 4779: 4687: 4622: 4563: 4463: 4367: 4322: 4262: 4152: 4100: 4011: 3918: 3889: 3844: 3829: 3805: 3769: 3673: 3493: 3439: 3253: 3192: 3095: 3072: 2912: 2864: 2822: 2792: 2755: 2719: 2618: 2468: 2445: 2285: 2261: 2229: 2209: 2135: 2094: 1973: 1901: 1830: 1776: 1743: 1704: 1645: 1556: 1536: 1472:with line element 1458: 1390: 1281: 1181: 1026: 909: 821: 741: 659: 639: 564: 478: 451: 388: 317: 256:Newton's force law 240: 216: 196: 115:general relativity 91:general relativity 11741: 11740: 11712:Theory of impetus 11648: 11647: 11620:Liouville gravity 11364:Conformal gravity 11292:Composite gravity 11282:Bimetric theories 11212: 11211: 10994:978-1-4020-4000-9 10887: 10729: 10713: 10701: 10684: 10521: 10520: 10503: 10491: 10474: 10427: 10411: 10355: 10334: 10266: 10216: 10130: 10129: 10108: 10022: 10021: 10000: 9852: 9016: 9015: 8997: 8877: 8854: 8806: 8784: 8728: 8682: 8399: 8203: 8159: 8077: 8042: 7987: 7939: 7904: 7871: 7763: 7715: 7680: 7647: 7527: 7487: 7486: 7409: 7033: 6891: 6870: 6817: 6796: 6743: 6742: 6662: 6661: 6636: 6620: 6618: 6598: 6569: 6568: 6476: 6451: 6366: 6330: 6306: 6286: 6258: 6126: 6095: 5822: 5806: 5792: 5760: 5728: 5653: 5634: 5622: 5427: 5342: 5326: 5254: 5186: 5152: 5096: 5083: 5056: 5035: 4984: 4946: 4905: 4885: 4807: 4729: 4561: 4442: 4399: 4298: 4181: 2934: 2933: 2825:{\displaystyle R} 2802:type of curvature 2598: 2522:. But these are 2271:as controlling a 1797:Some time later, 1702: 1381: 1361: 1227: 895: 855: 800: 587:relativity theory 550: 510: 423: 348:Hermann Minkowski 303: 291: 266:of the potential 79: 78: 71: 11766: 11696: 11694:Entropic gravity 11689: 11613:extensions of GR 11601: 11587: 11580: 11573: 11566: 11559: 11552: 11545: 11538: 11524: 11441: 11434: 11427: 11388:Geometrodynamics 11377: 11353: 11334: 11327: 11276: 11269: 11225: 11224: 11116: 11115: 11098: 11091: 11084: 11075: 11074: 11065: 11053: 11042: 11021: 11019: 10999: 10998: 10968: 10942:Gunnar Nordström 10905: 10903: 10902: 10897: 10895: 10894: 10889: 10888: 10880: 10865: 10863: 10862: 10857: 10834: 10833: 10817: 10813: 10803: 10792: 10791: 10776: 10730: 10722: 10717: 10716: 10715: 10714: 10706: 10703: 10702: 10694: 10686: 10685: 10677: 10620: 10618: 10617: 10612: 10589: 10588: 10531: 10522: 10516: 10512: 10507: 10506: 10505: 10504: 10496: 10493: 10492: 10484: 10476: 10475: 10467: 10448:expansion tensor 10445: 10443: 10442: 10437: 10435: 10434: 10429: 10428: 10420: 10413: 10412: 10404: 10379: 10377: 10376: 10371: 10363: 10362: 10357: 10356: 10348: 10344: 10343: 10342: 10341: 10336: 10335: 10327: 10297: 10295: 10294: 10289: 10287: 10286: 10274: 10273: 10268: 10267: 10259: 10247: 10245: 10244: 10239: 10237: 10236: 10224: 10223: 10218: 10217: 10209: 10197: 10195: 10194: 10189: 10187: 10183: 10182: 10181: 10147: 10146: 10131: 10125: 10121: 10116: 10115: 10110: 10109: 10101: 10089: 10087: 10086: 10081: 10079: 10075: 10074: 10073: 10039: 10038: 10023: 10017: 10013: 10008: 10007: 10002: 10001: 9993: 9965: 9963: 9962: 9957: 9955: 9954: 9932: 9930: 9929: 9924: 9922: 9921: 9901: 9899: 9898: 9893: 9864: 9863: 9853: 9845: 9777: 9775: 9774: 9769: 9767: 9766: 9750: 9749: 9726: 9724: 9723: 9718: 9716: 9715: 9699: 9698: 9680: 9679: 9663: 9662: 9640: 9638: 9637: 9632: 9630: 9629: 9613: 9612: 9587: 9585: 9584: 9579: 9577: 9576: 9562: 9561: 9543: 9541: 9540: 9535: 9533: 9532: 9508: 9506: 9505: 9500: 9498: 9497: 9468: 9466: 9465: 9460: 9413: 9409: 9408: 9407: 9392: 9391: 9320: 9319: 9287:, where f is an 9286: 9284: 9283: 9278: 9248: 9246: 9245: 9240: 9232: 9231: 9216: 9215: 9200: 9199: 9167: 9165: 9164: 9159: 9112: 9111: 9096: 9095: 9057: 9056: 9012: 9010: 9009: 9004: 9002: 8998: 8996: 8995: 8986: 8985: 8984: 8974: 8973: 8959: 8921: 8919: 8918: 8913: 8888: 8886: 8885: 8880: 8878: 8873: 8860: 8855: 8853: 8852: 8851: 8835: 8827: 8817: 8815: 8814: 8809: 8807: 8802: 8793: 8785: 8783: 8782: 8781: 8765: 8757: 8739: 8737: 8736: 8731: 8729: 8724: 8715: 8693: 8691: 8690: 8685: 8683: 8678: 8669: 8507: 8501: 8474: 8472: 8471: 8466: 8464: 8463: 8454: 8442: 8441: 8432: 8414: 8412: 8411: 8406: 8404: 8400: 8398: 8397: 8388: 8387: 8386: 8376: 8375: 8361: 8311: 8309: 8308: 8303: 8279: 8277: 8276: 8271: 8269: 8268: 8259: 8247: 8246: 8237: 8219: 8217: 8216: 8211: 8209: 8205: 8204: 8202: 8201: 8192: 8191: 8182: 8165: 8160: 8158: 8157: 8148: 8147: 8138: 8121: 8088: 8086: 8085: 8080: 8078: 8076: 8075: 8062: 8057: 8048: 8043: 8041: 8040: 8027: 8022: 8013: 7998: 7996: 7995: 7990: 7988: 7986: 7985: 7976: 7975: 7966: 7949: 7940: 7938: 7937: 7924: 7919: 7910: 7905: 7903: 7902: 7889: 7884: 7875: 7872: 7870: 7869: 7860: 7859: 7858: 7842: 7830: 7829: 7828: 7818: 7806: 7805: 7774: 7772: 7771: 7766: 7764: 7762: 7761: 7752: 7751: 7742: 7725: 7716: 7714: 7713: 7700: 7695: 7686: 7681: 7679: 7678: 7665: 7660: 7651: 7648: 7646: 7645: 7636: 7635: 7634: 7618: 7606: 7605: 7604: 7594: 7582: 7581: 7550: 7548: 7547: 7542: 7533: 7529: 7528: 7526: 7525: 7516: 7515: 7506: 7488: 7485: 7484: 7472: 7471: 7462: 7458: 7434: 7432: 7431: 7426: 7424: 7423: 7410: 7408: 7407: 7406: 7394: 7393: 7383: 7382: 7373: 7368: 7367: 7354: 7353: 7329: 7328: 7315: 7314: 7292: 7290: 7289: 7284: 7225: 7223: 7222: 7217: 7189: 7187: 7186: 7181: 7169: 7167: 7166: 7161: 7153: 7152: 7139: 7138: 7122: 7120: 7119: 7114: 7112: 7111: 7091: 7090: 7071: 7069: 7068: 7063: 7061: 7060: 7047: 7046: 7034: 7032: 7031: 7030: 7018: 7000: 6999: 6998: 6983: 6982: 6966: 6912:counterclockwise 6902: 6900: 6899: 6894: 6892: 6887: 6876: 6871: 6869: 6868: 6867: 6851: 6843: 6828: 6826: 6825: 6820: 6818: 6813: 6805: 6797: 6795: 6794: 6793: 6777: 6769: 6754: 6752: 6751: 6746: 6744: 6741: 6740: 6731: 6730: 6721: 6720: 6673: 6671: 6670: 6665: 6663: 6660: 6659: 6650: 6649: 6640: 6639: 6637: 6629: 6621: 6619: 6617: 6616: 6604: 6599: 6597: 6596: 6587: 6586: 6577: 6575: 6570: 6567: 6566: 6554: 6553: 6548: 6547: 6546: 6527: 6526: 6525: 6487: 6485: 6484: 6479: 6477: 6475: 6474: 6465: 6457: 6452: 6450: 6449: 6437: 6432: 6431: 6430: 6404: 6402: 6401: 6396: 6394: 6393: 6377: 6375: 6374: 6369: 6367: 6359: 6341: 6339: 6338: 6333: 6331: 6320: 6319: 6310: 6307: 6305: 6304: 6292: 6287: 6285: 6284: 6272: 6271: 6262: 6259: 6257: 6256: 6247: 6246: 6237: 6232: 6231: 6230: 6193: 6191: 6190: 6185: 6180: 6179: 6154: 6153: 6141: 6140: 6127: 6125: 6124: 6115: 6114: 6105: 6097: 6096: 6088: 6075: 6073: 6072: 6067: 6065: 6064: 6055: 6050: 6049: 6021: 6019: 6018: 6013: 6008: 6007: 5995: 5994: 5982: 5977: 5976: 5964: 5963: 5947: 5945: 5944: 5939: 5934: 5929: 5928: 5916: 5915: 5896: 5894: 5893: 5888: 5886: 5882: 5881: 5880: 5868: 5841: 5840: 5823: 5821: 5810: 5807: 5799: 5794: 5793: 5785: 5772: 5770: 5769: 5764: 5762: 5761: 5753: 5740: 5738: 5737: 5732: 5730: 5729: 5721: 5717: 5713: 5712: 5711: 5699: 5672: 5671: 5654: 5652: 5641: 5636: 5635: 5627: 5624: 5623: 5615: 5599: 5597: 5596: 5591: 5589: 5588: 5576: 5559: 5551: 5550: 5541: 5536: 5535: 5501: 5499: 5498: 5493: 5491: 5490: 5478: 5451: 5450: 5435: 5434: 5429: 5428: 5420: 5406: 5404: 5403: 5398: 5393: 5360: 5358: 5357: 5352: 5350: 5349: 5344: 5343: 5335: 5328: 5327: 5319: 5306: 5304: 5303: 5298: 5272: 5271: 5255: 5253: 5252: 5243: 5242: 5233: 5204: 5203: 5187: 5185: 5184: 5172: 5167: 5166: 5154: 5153: 5145: 5114: 5112: 5111: 5106: 5104: 5103: 5098: 5097: 5089: 5084: 5082: 5081: 5069: 5064: 5063: 5058: 5057: 5049: 5045: 5044: 5043: 5042: 5037: 5036: 5028: 5006: 5004: 5003: 4998: 4996: 4995: 4985: 4983: 4959: 4954: 4953: 4948: 4947: 4939: 4927: 4925: 4924: 4919: 4917: 4916: 4906: 4898: 4893: 4892: 4887: 4886: 4878: 4866: 4864: 4863: 4858: 4856: 4855: 4845: 4841: 4837: 4815: 4814: 4809: 4808: 4800: 4788: 4786: 4785: 4780: 4778: 4777: 4767: 4763: 4759: 4737: 4736: 4731: 4730: 4722: 4701:Static observers 4696: 4694: 4693: 4688: 4674: 4669: 4668: 4656: 4655: 4631: 4629: 4628: 4623: 4618: 4613: 4612: 4600: 4599: 4572: 4570: 4569: 4564: 4562: 4560: 4559: 4554: 4550: 4546: 4526: 4519: 4518: 4509: 4504: 4503: 4493: 4472: 4470: 4469: 4464: 4456: 4455: 4443: 4441: 4440: 4435: 4431: 4427: 4407: 4406: 4401: 4400: 4392: 4388: 4376: 4374: 4373: 4368: 4331: 4329: 4328: 4323: 4321: 4320: 4311: 4300: 4299: 4291: 4271: 4269: 4268: 4263: 4261: 4257: 4253: 4224: 4223: 4218: 4214: 4210: 4183: 4182: 4174: 4161: 4159: 4158: 4153: 4148: 4109: 4107: 4106: 4101: 4099: 4098: 4054: 4053: 4020: 4018: 4017: 4012: 4010: 4009: 3965: 3964: 3927: 3925: 3924: 3919: 3917: 3916: 3898: 3896: 3895: 3890: 3888: 3887: 3853: 3851: 3850: 3845: 3842: 3837: 3814: 3812: 3811: 3806: 3804: 3803: 3778: 3776: 3775: 3770: 3682: 3680: 3679: 3674: 3669: 3668: 3655: 3654: 3630: 3629: 3613: 3612: 3597: 3596: 3581: 3580: 3561: 3560: 3545: 3525: 3524: 3502: 3500: 3499: 3494: 3486: 3448: 3446: 3445: 3440: 3438: 3434: 3430: 3429: 3416: 3415: 3391: 3390: 3374: 3373: 3361: 3360: 3345: 3344: 3317: 3297: 3296: 3262: 3260: 3259: 3254: 3252: 3251: 3238: 3237: 3224: 3223: 3201: 3199: 3198: 3193: 3191: 3190: 3177: 3176: 3163: 3162: 3127: 3126: 3104: 3102: 3101: 3096: 3081: 3079: 3078: 3073: 3054: 3053: 3040: 3039: 3026: 3025: 2991: 2990: 2921: 2919: 2918: 2913: 2911: 2910: 2873: 2871: 2870: 2865: 2863: 2862: 2831: 2829: 2828: 2823: 2797: 2791: 2764: 2762: 2761: 2756: 2754: 2753: 2728: 2726: 2725: 2720: 2718: 2717: 2694: 2693: 2685: 2684: 2683: 2678: 2677: 2676: 2641:Bianchi identity 2627: 2625: 2624: 2619: 2617: 2616: 2599: 2591: 2586: 2585: 2570: 2569: 2477: 2475: 2474: 2469: 2454: 2452: 2451: 2446: 2444: 2443: 2430: 2429: 2416: 2415: 2381: 2380: 2367: 2366: 2353: 2352: 2317: 2316: 2294: 2292: 2291: 2286: 2270: 2268: 2267: 2262: 2238: 2236: 2235: 2230: 2218: 2216: 2215: 2210: 2208: 2207: 2194: 2193: 2180: 2179: 2164: 2163: 2144: 2142: 2141: 2136: 2103: 2101: 2100: 2095: 2075: 2074: 2061: 2060: 2047: 2046: 2012: 2011: 1982: 1980: 1979: 1974: 1966: 1965: 1910: 1908: 1907: 1902: 1894: 1893: 1842:conformally flat 1839: 1837: 1836: 1831: 1829: 1828: 1785: 1783: 1782: 1777: 1775: 1774: 1752: 1750: 1749: 1744: 1713: 1711: 1710: 1705: 1703: 1701: 1700: 1691: 1679: 1654: 1652: 1651: 1646: 1644: 1643: 1630: 1629: 1616: 1615: 1602: 1601: 1589: 1588: 1565: 1563: 1562: 1557: 1545: 1543: 1542: 1537: 1535: 1534: 1521: 1520: 1507: 1506: 1491: 1490: 1467: 1465: 1464: 1459: 1457: 1456: 1443: 1442: 1430: 1429: 1399: 1397: 1396: 1391: 1389: 1388: 1383: 1382: 1374: 1369: 1368: 1363: 1362: 1354: 1349: 1348: 1335: 1334: 1296:action principle 1290: 1288: 1287: 1282: 1270: 1269: 1256: 1255: 1242: 1241: 1228: 1226: 1215: 1190: 1188: 1187: 1182: 1180: 1179: 1166: 1165: 1152: 1151: 1135: 1124: 1123: 1110: 1109: 1094: 1093: 1085: 1081: 1080: 1079: 1035: 1033: 1032: 1027: 1025: 1024: 1014: 1013: 993: 992: 984: 980: 979: 978: 928:Marcel Grossmann 918: 916: 915: 910: 908: 907: 897: 896: 888: 882: 881: 863: 862: 857: 856: 848: 830: 828: 827: 822: 820: 819: 801: 799: 791: 790: 786: 785: 784: 762: 750: 748: 747: 742: 740: 739: 738: 668: 666: 665: 660: 648: 646: 645: 640: 632: 631: 630: 583:electromagnetism 573: 571: 570: 565: 563: 562: 552: 551: 543: 537: 536: 518: 517: 512: 511: 503: 487: 485: 484: 479: 477: 476: 460: 458: 457: 452: 450: 449: 431: 430: 425: 424: 416: 400:Laplace equation 397: 395: 394: 389: 326: 324: 323: 318: 304: 302: 294: 293: 292: 284: 277: 249: 247: 246: 241: 225: 223: 222: 217: 205: 203: 202: 197: 170:Poisson equation 99:Gunnar Nordström 74: 67: 63: 60: 54: 49:this article by 40:inline citations 27: 26: 19: 11774: 11773: 11769: 11768: 11767: 11765: 11764: 11763: 11744: 11743: 11742: 11737: 11716: 11692: 11685: 11656: 11654: 11644: 11625:Lovelock theory 11612: 11606: 11597: 11583: 11576: 11569: 11562: 11555: 11548: 11541: 11534: 11520: 11498: 11492: 11471: 11437: 11430: 11423: 11402: 11393:Induced gravity 11373: 11369:Scalar theories 11349: 11330: 11323: 11314:Massive gravity 11274:Teleparallelism 11272: 11267:Einstein–Cartan 11265: 11249: 11240:Quantum gravity 11219: 11218:Alternatives to 11208: 11174:Exact solutions 11145: 11107: 11102: 11062: 11039: 11003: 11002: 10995: 10969: 10965: 10960: 10928: 10890: 10879: 10878: 10877: 10875: 10872: 10871: 10820: 10819: 10796: 10787: 10783: 10769: 10768: 10764: 10721: 10705: 10704: 10693: 10692: 10691: 10687: 10676: 10675: 10667: 10664: 10663: 10575: 10574: 10524: 10511: 10495: 10494: 10483: 10482: 10481: 10477: 10466: 10465: 10457: 10454: 10453: 10430: 10419: 10418: 10417: 10403: 10402: 10400: 10397: 10396: 10358: 10347: 10346: 10345: 10337: 10326: 10325: 10324: 10323: 10319: 10317: 10314: 10313: 10282: 10278: 10269: 10258: 10257: 10256: 10254: 10251: 10250: 10232: 10228: 10219: 10208: 10207: 10206: 10204: 10201: 10200: 10177: 10173: 10142: 10138: 10137: 10133: 10120: 10111: 10100: 10099: 10098: 10096: 10093: 10092: 10069: 10065: 10034: 10030: 10029: 10025: 10012: 10003: 9992: 9991: 9990: 9988: 9985: 9984: 9950: 9946: 9938: 9935: 9934: 9917: 9913: 9911: 9908: 9907: 9859: 9855: 9844: 9786: 9783: 9782: 9762: 9758: 9745: 9741: 9735: 9732: 9731: 9711: 9707: 9694: 9690: 9675: 9671: 9658: 9654: 9648: 9645: 9644: 9625: 9621: 9608: 9604: 9595: 9592: 9591: 9572: 9568: 9557: 9553: 9551: 9548: 9547: 9528: 9524: 9522: 9519: 9518: 9493: 9489: 9487: 9484: 9483: 9403: 9399: 9387: 9383: 9356: 9352: 9315: 9311: 9306: 9303: 9302: 9257: 9254: 9253: 9224: 9220: 9208: 9204: 9192: 9188: 9179: 9176: 9175: 9107: 9103: 9091: 9087: 9052: 9048: 9043: 9040: 9039: 9033: 8991: 8987: 8980: 8976: 8969: 8965: 8960: 8958: 8954: 8928: 8925: 8924: 8903: 8900: 8899: 8861: 8859: 8841: 8840: 8836: 8828: 8826: 8824: 8821: 8820: 8794: 8792: 8771: 8770: 8766: 8758: 8756: 8754: 8751: 8750: 8716: 8714: 8703: 8700: 8699: 8670: 8668: 8657: 8654: 8653: 8497: 8459: 8455: 8450: 8437: 8433: 8428: 8423: 8420: 8419: 8393: 8389: 8382: 8378: 8371: 8367: 8362: 8360: 8356: 8323: 8320: 8319: 8285: 8282: 8281: 8264: 8260: 8255: 8242: 8238: 8233: 8228: 8225: 8224: 8197: 8193: 8187: 8183: 8178: 8164: 8153: 8149: 8143: 8139: 8134: 8120: 8119: 8115: 8100: 8097: 8096: 8071: 8067: 8058: 8053: 8047: 8036: 8032: 8023: 8018: 8012: 8010: 8007: 8006: 7981: 7977: 7971: 7967: 7962: 7948: 7933: 7929: 7920: 7915: 7909: 7898: 7894: 7885: 7880: 7874: 7865: 7861: 7854: 7850: 7843: 7841: 7824: 7820: 7819: 7814: 7801: 7797: 7786: 7783: 7782: 7757: 7753: 7747: 7743: 7738: 7724: 7709: 7705: 7696: 7691: 7685: 7674: 7670: 7661: 7656: 7650: 7641: 7637: 7630: 7626: 7619: 7617: 7600: 7596: 7595: 7590: 7577: 7573: 7562: 7559: 7558: 7521: 7517: 7511: 7507: 7505: 7494: 7490: 7480: 7476: 7467: 7463: 7457: 7446: 7443: 7442: 7419: 7415: 7402: 7398: 7389: 7385: 7384: 7378: 7374: 7372: 7363: 7359: 7349: 7345: 7324: 7320: 7310: 7306: 7304: 7301: 7300: 7234: 7231: 7230: 7195: 7192: 7191: 7175: 7172: 7171: 7148: 7144: 7134: 7130: 7128: 7125: 7124: 7107: 7103: 7086: 7082: 7080: 7077: 7076: 7056: 7052: 7042: 7038: 7026: 7022: 7014: 7001: 6994: 6990: 6978: 6974: 6967: 6965: 6957: 6954: 6953: 6947: 6877: 6875: 6857: 6856: 6852: 6844: 6842: 6840: 6837: 6836: 6806: 6804: 6783: 6782: 6778: 6770: 6768: 6766: 6763: 6762: 6736: 6732: 6726: 6722: 6719: 6685: 6682: 6681: 6655: 6651: 6645: 6641: 6638: 6628: 6612: 6608: 6603: 6592: 6588: 6582: 6578: 6576: 6574: 6562: 6558: 6552: 6536: 6535: 6531: 6515: 6514: 6510: 6499: 6496: 6495: 6470: 6466: 6461: 6456: 6445: 6441: 6436: 6420: 6419: 6415: 6413: 6410: 6409: 6389: 6385: 6383: 6380: 6379: 6358: 6350: 6347: 6346: 6345:(where we used 6315: 6311: 6309: 6300: 6296: 6291: 6280: 6276: 6267: 6263: 6261: 6252: 6248: 6242: 6238: 6236: 6220: 6219: 6215: 6213: 6210: 6209: 6175: 6171: 6149: 6145: 6136: 6132: 6120: 6116: 6110: 6106: 6104: 6087: 6086: 6084: 6081: 6080: 6060: 6056: 6051: 6045: 6041: 6027: 6024: 6023: 6003: 5999: 5990: 5986: 5978: 5972: 5968: 5959: 5955: 5953: 5950: 5949: 5930: 5924: 5920: 5911: 5907: 5905: 5902: 5901: 5876: 5872: 5864: 5836: 5832: 5828: 5824: 5814: 5809: 5798: 5784: 5783: 5781: 5778: 5777: 5752: 5751: 5749: 5746: 5745: 5720: 5719: 5707: 5703: 5695: 5667: 5663: 5659: 5655: 5645: 5640: 5626: 5625: 5614: 5613: 5608: 5605: 5604: 5584: 5580: 5572: 5555: 5546: 5542: 5537: 5531: 5527: 5510: 5507: 5506: 5486: 5482: 5474: 5446: 5442: 5430: 5419: 5418: 5417: 5415: 5412: 5411: 5389: 5378: 5375: 5374: 5371: 5345: 5334: 5333: 5332: 5318: 5317: 5315: 5312: 5311: 5258: 5257: 5248: 5244: 5238: 5234: 5232: 5190: 5189: 5180: 5176: 5171: 5159: 5155: 5144: 5143: 5135: 5132: 5131: 5099: 5088: 5087: 5086: 5077: 5073: 5068: 5059: 5048: 5047: 5046: 5038: 5027: 5026: 5025: 5024: 5020: 5018: 5015: 5014: 4991: 4987: 4963: 4958: 4949: 4938: 4937: 4936: 4934: 4931: 4930: 4912: 4908: 4897: 4888: 4877: 4876: 4875: 4873: 4870: 4869: 4851: 4847: 4833: 4823: 4819: 4810: 4799: 4798: 4797: 4795: 4792: 4791: 4773: 4769: 4755: 4745: 4741: 4732: 4721: 4720: 4719: 4717: 4714: 4713: 4703: 4670: 4664: 4660: 4651: 4647: 4645: 4642: 4641: 4614: 4608: 4604: 4595: 4591: 4589: 4586: 4585: 4555: 4542: 4532: 4528: 4527: 4514: 4510: 4505: 4499: 4495: 4494: 4492: 4484: 4481: 4480: 4451: 4447: 4436: 4423: 4413: 4409: 4408: 4402: 4391: 4390: 4389: 4387: 4385: 4382: 4381: 4341: 4338: 4337: 4316: 4312: 4307: 4290: 4289: 4287: 4284: 4283: 4249: 4236: 4232: 4219: 4206: 4196: 4192: 4191: 4173: 4172: 4170: 4167: 4166: 4144: 4133: 4130: 4129: 4123: 4094: 4090: 4049: 4045: 4027: 4024: 4023: 4005: 4001: 3960: 3956: 3935: 3932: 3931: 3912: 3908: 3906: 3903: 3902: 3883: 3879: 3877: 3874: 3873: 3838: 3833: 3820: 3817: 3816: 3799: 3795: 3787: 3784: 3783: 3689: 3686: 3685: 3664: 3660: 3650: 3646: 3625: 3621: 3608: 3604: 3592: 3588: 3576: 3572: 3553: 3549: 3541: 3520: 3516: 3511: 3508: 3507: 3482: 3458: 3455: 3454: 3425: 3421: 3411: 3407: 3386: 3382: 3369: 3365: 3356: 3352: 3340: 3336: 3329: 3325: 3313: 3292: 3288: 3283: 3280: 3279: 3269: 3247: 3243: 3233: 3229: 3216: 3212: 3210: 3207: 3206: 3186: 3182: 3172: 3168: 3155: 3151: 3122: 3118: 3113: 3110: 3109: 3090: 3087: 3086: 3049: 3045: 3035: 3031: 3018: 3014: 2986: 2982: 2977: 2974: 2973: 2967: 2927:vanishes always 2897: 2893: 2891: 2888: 2887: 2855: 2851: 2849: 2846: 2845: 2817: 2814: 2813: 2746: 2742: 2740: 2737: 2736: 2710: 2706: 2686: 2679: 2672: 2668: 2667: 2666: 2665: 2664: 2662: 2659: 2658: 2609: 2605: 2590: 2578: 2574: 2562: 2558: 2556: 2553: 2552: 2463: 2460: 2459: 2439: 2435: 2425: 2421: 2408: 2404: 2376: 2372: 2362: 2358: 2345: 2341: 2312: 2308: 2303: 2300: 2299: 2280: 2277: 2276: 2256: 2253: 2252: 2242:Cauchy problems 2224: 2221: 2220: 2203: 2199: 2189: 2185: 2172: 2168: 2159: 2155: 2150: 2147: 2146: 2112: 2109: 2108: 2070: 2066: 2056: 2052: 2039: 2035: 2007: 2003: 1998: 1995: 1994: 1952: 1948: 1931: 1928: 1927: 1917: 1880: 1876: 1852: 1849: 1848: 1815: 1811: 1809: 1806: 1805: 1801:introduced the 1767: 1763: 1761: 1758: 1757: 1725: 1722: 1721: 1696: 1692: 1680: 1678: 1667: 1664: 1663: 1639: 1635: 1625: 1621: 1608: 1604: 1597: 1593: 1584: 1580: 1575: 1572: 1571: 1551: 1548: 1547: 1530: 1526: 1516: 1512: 1499: 1495: 1486: 1482: 1477: 1474: 1473: 1468:. If we adopt 1449: 1445: 1438: 1434: 1422: 1418: 1416: 1413: 1412: 1384: 1373: 1372: 1371: 1364: 1353: 1352: 1351: 1341: 1337: 1330: 1326: 1318: 1315: 1314: 1307:Hendrik Lorentz 1262: 1258: 1248: 1244: 1234: 1230: 1219: 1214: 1206: 1203: 1202: 1172: 1168: 1158: 1154: 1144: 1140: 1131: 1116: 1112: 1102: 1098: 1086: 1066: 1065: 1061: 1057: 1056: 1047: 1044: 1043: 1020: 1016: 1009: 1005: 985: 959: 958: 954: 950: 949: 947: 944: 943: 903: 899: 887: 886: 874: 870: 858: 847: 846: 845: 839: 836: 835: 812: 808: 792: 780: 776: 771: 767: 763: 761: 759: 756: 755: 719: 718: 714: 689: 686: 685: 672: 654: 651: 650: 611: 610: 606: 601: 598: 597: 558: 554: 542: 541: 529: 525: 513: 502: 501: 500: 498: 495: 494: 472: 468: 466: 463: 462: 442: 438: 426: 415: 414: 413: 411: 408: 407: 367: 364: 363: 336:instantaneously 295: 283: 282: 278: 276: 274: 271: 270: 235: 232: 231: 211: 208: 207: 176: 173: 172: 147:Albert Einstein 145:, Germany, and 123: 75: 64: 58: 55: 45:Please help to 44: 28: 24: 17: 12: 11: 5: 11772: 11762: 11761: 11756: 11739: 11738: 11736: 11735: 11730: 11724: 11722: 11721:Related topics 11718: 11717: 11715: 11714: 11709: 11704: 11699: 11698: 11697: 11690: 11678: 11673: 11668: 11662: 11660: 11650: 11649: 11646: 11645: 11643: 11642: 11637: 11632: 11627: 11622: 11616: 11614: 11608: 11607: 11605: 11604: 11603: 11602: 11593:Twistor theory 11590: 11589: 11588: 11581: 11574: 11567: 11560: 11553: 11546: 11539: 11527: 11526: 11525: 11513: 11508: 11502: 11500: 11494: 11493: 11491: 11490: 11485: 11479: 11477: 11473: 11472: 11470: 11469: 11464: 11459: 11454: 11449: 11444: 11443: 11442: 11435: 11428: 11416: 11410: 11408: 11404: 11403: 11401: 11400: 11395: 11390: 11385: 11380: 11379: 11378: 11366: 11361: 11356: 11355: 11354: 11342: 11337: 11336: 11335: 11328: 11316: 11311: 11306: 11294: 11289: 11284: 11279: 11278: 11277: 11270: 11257: 11255: 11251: 11250: 11248: 11247: 11242: 11237: 11231: 11229: 11222: 11214: 11213: 11210: 11209: 11207: 11206: 11201: 11196: 11191: 11186: 11181: 11176: 11171: 11166: 11161: 11155: 11153: 11147: 11146: 11144: 11143: 11138: 11133: 11128: 11122: 11120: 11113: 11109: 11108: 11101: 11100: 11093: 11086: 11078: 11072: 11071: 11060: 11043: 11037: 11022: 11001: 11000: 10993: 10962: 10961: 10959: 10956: 10955: 10954: 10949: 10944: 10939: 10934: 10927: 10924: 10893: 10886: 10883: 10867: 10866: 10855: 10852: 10849: 10846: 10843: 10840: 10837: 10832: 10829: 10826: 10823: 10816: 10812: 10809: 10806: 10802: 10799: 10795: 10790: 10786: 10782: 10779: 10775: 10772: 10767: 10762: 10759: 10756: 10753: 10750: 10746: 10743: 10740: 10737: 10734: 10728: 10725: 10720: 10712: 10709: 10700: 10697: 10690: 10683: 10680: 10674: 10671: 10622: 10621: 10610: 10607: 10604: 10601: 10598: 10595: 10592: 10587: 10584: 10581: 10578: 10572: 10569: 10566: 10563: 10560: 10556: 10553: 10550: 10547: 10544: 10540: 10537: 10534: 10530: 10527: 10519: 10515: 10510: 10502: 10499: 10490: 10487: 10480: 10473: 10470: 10464: 10461: 10433: 10426: 10423: 10416: 10410: 10407: 10381: 10380: 10369: 10366: 10361: 10354: 10351: 10340: 10333: 10330: 10322: 10299: 10298: 10285: 10281: 10277: 10272: 10265: 10262: 10248: 10235: 10231: 10227: 10222: 10215: 10212: 10198: 10186: 10180: 10176: 10171: 10168: 10165: 10162: 10159: 10156: 10153: 10150: 10145: 10141: 10136: 10128: 10124: 10119: 10114: 10107: 10104: 10090: 10078: 10072: 10068: 10063: 10060: 10057: 10054: 10051: 10048: 10045: 10042: 10037: 10033: 10028: 10020: 10016: 10011: 10006: 9999: 9996: 9953: 9949: 9945: 9942: 9920: 9916: 9903: 9902: 9891: 9888: 9884: 9881: 9877: 9874: 9871: 9867: 9862: 9858: 9851: 9848: 9843: 9840: 9836: 9833: 9830: 9826: 9823: 9820: 9817: 9814: 9811: 9808: 9805: 9802: 9799: 9796: 9793: 9790: 9765: 9761: 9756: 9753: 9748: 9744: 9739: 9728: 9727: 9714: 9710: 9705: 9702: 9697: 9693: 9688: 9683: 9678: 9674: 9669: 9666: 9661: 9657: 9652: 9642: 9628: 9624: 9619: 9616: 9611: 9607: 9602: 9599: 9589: 9575: 9571: 9565: 9560: 9556: 9545: 9531: 9527: 9496: 9492: 9470: 9469: 9458: 9455: 9452: 9448: 9445: 9441: 9438: 9434: 9431: 9428: 9425: 9422: 9416: 9412: 9406: 9402: 9398: 9395: 9390: 9386: 9382: 9379: 9376: 9373: 9369: 9366: 9362: 9359: 9355: 9350: 9347: 9344: 9341: 9338: 9335: 9332: 9329: 9326: 9323: 9318: 9314: 9310: 9276: 9273: 9270: 9267: 9264: 9261: 9250: 9249: 9238: 9235: 9230: 9227: 9223: 9219: 9214: 9211: 9207: 9203: 9198: 9195: 9191: 9186: 9183: 9169: 9168: 9157: 9154: 9151: 9147: 9144: 9140: 9137: 9133: 9130: 9127: 9124: 9121: 9115: 9110: 9106: 9102: 9099: 9094: 9090: 9086: 9083: 9080: 9077: 9073: 9070: 9066: 9063: 9060: 9055: 9051: 9047: 9032: 9029: 9014: 9013: 9001: 8994: 8990: 8983: 8979: 8972: 8968: 8963: 8957: 8953: 8950: 8946: 8942: 8939: 8936: 8932: 8922: 8911: 8907: 8897: 8894: 8890: 8889: 8876: 8872: 8868: 8864: 8858: 8850: 8847: 8844: 8839: 8834: 8831: 8818: 8805: 8801: 8797: 8791: 8788: 8780: 8777: 8774: 8769: 8764: 8761: 8748: 8745: 8741: 8740: 8727: 8723: 8719: 8713: 8710: 8707: 8697: 8694: 8681: 8677: 8673: 8667: 8664: 8661: 8651: 8647: 8646: 8637: 8628: 8625: 8621: 8620: 8604: 8592: 8580: 8576: 8575: 8572: 8563: 8560: 8556: 8555: 8540: 8532: 8524: 8520: 8519: 8516: 8513: 8510: 8496: 8493: 8462: 8458: 8453: 8449: 8445: 8440: 8436: 8431: 8427: 8416: 8415: 8403: 8396: 8392: 8385: 8381: 8374: 8370: 8365: 8359: 8355: 8352: 8348: 8345: 8342: 8339: 8336: 8333: 8330: 8327: 8301: 8298: 8295: 8292: 8289: 8267: 8263: 8258: 8254: 8250: 8245: 8241: 8236: 8232: 8221: 8220: 8208: 8200: 8196: 8190: 8186: 8181: 8177: 8174: 8171: 8168: 8163: 8156: 8152: 8146: 8142: 8137: 8133: 8130: 8127: 8124: 8118: 8113: 8110: 8107: 8104: 8090: 8089: 8074: 8070: 8066: 8061: 8056: 8052: 8046: 8039: 8035: 8031: 8026: 8021: 8017: 8000: 7999: 7984: 7980: 7974: 7970: 7965: 7961: 7958: 7955: 7952: 7946: 7943: 7936: 7932: 7928: 7923: 7918: 7914: 7908: 7901: 7897: 7893: 7888: 7883: 7879: 7868: 7864: 7857: 7853: 7849: 7846: 7840: 7837: 7834: 7827: 7823: 7817: 7813: 7809: 7804: 7800: 7796: 7793: 7790: 7776: 7775: 7760: 7756: 7750: 7746: 7741: 7737: 7734: 7731: 7728: 7722: 7719: 7712: 7708: 7704: 7699: 7694: 7690: 7684: 7677: 7673: 7669: 7664: 7659: 7655: 7644: 7640: 7633: 7629: 7625: 7622: 7616: 7613: 7610: 7603: 7599: 7593: 7589: 7585: 7580: 7576: 7572: 7569: 7566: 7552: 7551: 7540: 7537: 7532: 7524: 7520: 7514: 7510: 7503: 7500: 7497: 7493: 7483: 7479: 7475: 7470: 7466: 7461: 7456: 7453: 7450: 7436: 7435: 7422: 7418: 7414: 7405: 7401: 7397: 7392: 7388: 7381: 7377: 7371: 7366: 7362: 7358: 7352: 7348: 7344: 7341: 7338: 7335: 7332: 7327: 7323: 7319: 7313: 7309: 7294: 7293: 7282: 7279: 7275: 7272: 7269: 7266: 7263: 7260: 7256: 7253: 7250: 7247: 7244: 7241: 7238: 7215: 7212: 7209: 7205: 7202: 7199: 7179: 7159: 7156: 7151: 7147: 7142: 7137: 7133: 7110: 7106: 7101: 7098: 7094: 7089: 7085: 7073: 7072: 7059: 7055: 7051: 7045: 7041: 7037: 7029: 7025: 7021: 7017: 7013: 7010: 7007: 7004: 6997: 6993: 6989: 6986: 6981: 6977: 6973: 6970: 6964: 6961: 6946: 6943: 6904: 6903: 6890: 6886: 6883: 6880: 6874: 6866: 6863: 6860: 6855: 6850: 6847: 6830: 6829: 6816: 6812: 6809: 6803: 6800: 6792: 6789: 6786: 6781: 6776: 6773: 6756: 6755: 6739: 6735: 6729: 6725: 6717: 6714: 6711: 6708: 6705: 6701: 6698: 6695: 6692: 6689: 6675: 6674: 6658: 6654: 6648: 6644: 6635: 6632: 6627: 6624: 6615: 6611: 6607: 6602: 6595: 6591: 6585: 6581: 6573: 6565: 6561: 6557: 6551: 6545: 6542: 6539: 6534: 6530: 6524: 6521: 6518: 6513: 6509: 6506: 6503: 6489: 6488: 6473: 6469: 6464: 6460: 6455: 6448: 6444: 6440: 6435: 6429: 6426: 6423: 6418: 6392: 6388: 6365: 6362: 6357: 6354: 6343: 6342: 6329: 6326: 6323: 6318: 6314: 6303: 6299: 6295: 6290: 6283: 6279: 6275: 6270: 6266: 6255: 6251: 6245: 6241: 6235: 6229: 6226: 6223: 6218: 6195: 6194: 6183: 6178: 6174: 6170: 6167: 6164: 6161: 6157: 6152: 6148: 6144: 6139: 6135: 6131: 6123: 6119: 6113: 6109: 6103: 6100: 6094: 6091: 6063: 6059: 6054: 6048: 6044: 6040: 6037: 6034: 6031: 6011: 6006: 6002: 5998: 5993: 5989: 5985: 5981: 5975: 5971: 5967: 5962: 5958: 5937: 5933: 5927: 5923: 5919: 5914: 5910: 5898: 5897: 5885: 5879: 5875: 5871: 5867: 5863: 5860: 5857: 5854: 5850: 5847: 5844: 5839: 5835: 5831: 5827: 5820: 5817: 5813: 5805: 5802: 5797: 5791: 5788: 5759: 5756: 5742: 5741: 5727: 5724: 5716: 5710: 5706: 5702: 5698: 5694: 5691: 5688: 5685: 5681: 5678: 5675: 5670: 5666: 5662: 5658: 5651: 5648: 5644: 5639: 5633: 5630: 5621: 5618: 5612: 5587: 5583: 5579: 5575: 5571: 5568: 5565: 5562: 5558: 5554: 5549: 5545: 5540: 5534: 5530: 5526: 5523: 5520: 5517: 5514: 5503: 5502: 5489: 5485: 5481: 5477: 5473: 5470: 5467: 5464: 5460: 5457: 5454: 5449: 5445: 5441: 5438: 5433: 5426: 5423: 5396: 5392: 5388: 5385: 5382: 5370: 5367: 5348: 5341: 5338: 5331: 5325: 5322: 5310:where we take 5308: 5307: 5296: 5293: 5290: 5287: 5284: 5281: 5278: 5275: 5270: 5267: 5264: 5261: 5251: 5247: 5241: 5237: 5231: 5228: 5225: 5222: 5219: 5216: 5213: 5210: 5207: 5202: 5199: 5196: 5193: 5183: 5179: 5175: 5170: 5165: 5162: 5158: 5151: 5148: 5142: 5139: 5116: 5115: 5102: 5095: 5092: 5080: 5076: 5072: 5067: 5062: 5055: 5052: 5041: 5034: 5031: 5023: 5008: 5007: 4994: 4990: 4982: 4979: 4976: 4973: 4970: 4966: 4962: 4957: 4952: 4945: 4942: 4928: 4915: 4911: 4904: 4901: 4896: 4891: 4884: 4881: 4867: 4854: 4850: 4844: 4840: 4836: 4832: 4829: 4826: 4822: 4818: 4813: 4806: 4803: 4789: 4776: 4772: 4766: 4762: 4758: 4754: 4751: 4748: 4744: 4740: 4735: 4728: 4725: 4702: 4699: 4686: 4683: 4680: 4677: 4673: 4667: 4663: 4659: 4654: 4650: 4621: 4617: 4611: 4607: 4603: 4598: 4594: 4574: 4573: 4558: 4553: 4549: 4545: 4541: 4538: 4535: 4531: 4525: 4522: 4517: 4513: 4508: 4502: 4498: 4491: 4488: 4474: 4473: 4462: 4459: 4454: 4450: 4446: 4439: 4434: 4430: 4426: 4422: 4419: 4416: 4412: 4405: 4398: 4395: 4366: 4363: 4360: 4357: 4354: 4351: 4348: 4345: 4333: 4332: 4319: 4315: 4310: 4306: 4303: 4297: 4294: 4273: 4272: 4260: 4256: 4252: 4248: 4245: 4242: 4239: 4235: 4230: 4227: 4222: 4217: 4213: 4209: 4205: 4202: 4199: 4195: 4189: 4186: 4180: 4177: 4151: 4147: 4143: 4140: 4137: 4122: 4119: 4111: 4110: 4097: 4093: 4088: 4085: 4082: 4079: 4076: 4072: 4069: 4066: 4063: 4060: 4057: 4052: 4048: 4043: 4040: 4037: 4034: 4031: 4021: 4008: 4004: 3999: 3996: 3993: 3990: 3987: 3983: 3980: 3977: 3974: 3971: 3968: 3963: 3959: 3954: 3951: 3948: 3945: 3942: 3939: 3929: 3915: 3911: 3900: 3886: 3882: 3867:Killing vector 3841: 3836: 3832: 3827: 3824: 3802: 3798: 3794: 3791: 3780: 3779: 3768: 3765: 3762: 3759: 3756: 3753: 3749: 3746: 3743: 3740: 3737: 3734: 3730: 3727: 3724: 3721: 3718: 3715: 3711: 3708: 3705: 3702: 3699: 3696: 3693: 3683: 3672: 3667: 3663: 3659: 3653: 3649: 3645: 3642: 3639: 3636: 3633: 3628: 3624: 3620: 3617: 3611: 3607: 3603: 3600: 3595: 3591: 3587: 3584: 3579: 3575: 3571: 3568: 3565: 3559: 3556: 3552: 3548: 3544: 3540: 3537: 3534: 3531: 3528: 3523: 3519: 3515: 3492: 3489: 3485: 3481: 3478: 3475: 3472: 3468: 3465: 3462: 3450: 3449: 3437: 3433: 3428: 3424: 3420: 3414: 3410: 3406: 3403: 3400: 3397: 3394: 3389: 3385: 3381: 3378: 3372: 3368: 3364: 3359: 3355: 3351: 3348: 3343: 3339: 3335: 3332: 3328: 3323: 3320: 3316: 3312: 3309: 3306: 3303: 3300: 3295: 3291: 3287: 3273:exact solution 3268: 3265: 3250: 3246: 3242: 3236: 3232: 3228: 3222: 3219: 3215: 3203: 3202: 3189: 3185: 3181: 3175: 3171: 3167: 3161: 3158: 3154: 3149: 3146: 3142: 3139: 3136: 3133: 3130: 3125: 3121: 3117: 3094: 3083: 3082: 3071: 3068: 3065: 3062: 3057: 3052: 3048: 3044: 3038: 3034: 3030: 3024: 3021: 3017: 3012: 3009: 3006: 3003: 3000: 2997: 2994: 2989: 2985: 2981: 2966: 2963: 2958: 2957: 2954: 2932: 2931: 2928: 2925: 2922: 2909: 2906: 2903: 2900: 2896: 2884: 2883: 2880: 2877: 2876:once traceless 2874: 2861: 2858: 2854: 2842: 2841: 2838: 2835: 2832: 2821: 2810: 2809: 2806: 2803: 2800: 2783:Weyl curvature 2752: 2749: 2745: 2730: 2729: 2716: 2713: 2709: 2704: 2700: 2697: 2692: 2689: 2682: 2675: 2671: 2647:in any (semi)- 2645:Riemann tensor 2629: 2628: 2615: 2612: 2608: 2603: 2597: 2594: 2589: 2584: 2581: 2577: 2573: 2568: 2565: 2561: 2467: 2456: 2455: 2442: 2438: 2434: 2428: 2424: 2420: 2414: 2411: 2407: 2402: 2399: 2396: 2393: 2390: 2387: 2384: 2379: 2375: 2371: 2365: 2361: 2357: 2351: 2348: 2344: 2339: 2336: 2332: 2329: 2326: 2323: 2320: 2315: 2311: 2307: 2284: 2260: 2228: 2206: 2202: 2198: 2192: 2188: 2184: 2178: 2175: 2171: 2167: 2162: 2158: 2154: 2134: 2131: 2128: 2125: 2122: 2119: 2116: 2105: 2104: 2093: 2090: 2087: 2084: 2078: 2073: 2069: 2065: 2059: 2055: 2051: 2045: 2042: 2038: 2033: 2030: 2027: 2024: 2021: 2018: 2015: 2010: 2006: 2002: 1984: 1983: 1972: 1969: 1964: 1961: 1958: 1955: 1951: 1944: 1941: 1938: 1935: 1916: 1913: 1912: 1911: 1900: 1897: 1892: 1889: 1886: 1883: 1879: 1872: 1869: 1865: 1862: 1859: 1856: 1827: 1824: 1821: 1818: 1814: 1773: 1770: 1766: 1754: 1753: 1742: 1738: 1735: 1732: 1729: 1715: 1714: 1699: 1695: 1690: 1687: 1683: 1677: 1674: 1671: 1642: 1638: 1634: 1628: 1624: 1620: 1614: 1611: 1607: 1600: 1596: 1592: 1587: 1583: 1579: 1555: 1533: 1529: 1525: 1519: 1515: 1511: 1505: 1502: 1498: 1494: 1489: 1485: 1481: 1455: 1452: 1448: 1441: 1437: 1433: 1428: 1425: 1421: 1387: 1380: 1377: 1367: 1360: 1357: 1347: 1344: 1340: 1333: 1329: 1325: 1322: 1303:Adriaan Fokker 1292: 1291: 1280: 1276: 1273: 1268: 1265: 1261: 1254: 1251: 1247: 1240: 1237: 1233: 1225: 1222: 1218: 1213: 1210: 1192: 1191: 1178: 1175: 1171: 1164: 1161: 1157: 1150: 1147: 1143: 1138: 1134: 1130: 1127: 1122: 1119: 1115: 1108: 1105: 1101: 1097: 1092: 1089: 1084: 1078: 1075: 1072: 1069: 1064: 1060: 1054: 1051: 1037: 1036: 1023: 1019: 1012: 1008: 1003: 999: 996: 991: 988: 983: 977: 974: 971: 968: 965: 962: 957: 953: 906: 902: 894: 891: 885: 880: 877: 873: 869: 866: 861: 854: 851: 843: 832: 831: 818: 815: 811: 807: 804: 798: 795: 789: 783: 779: 774: 770: 766: 737: 734: 731: 728: 725: 722: 717: 712: 709: 706: 703: 700: 697: 693: 670: 658: 638: 635: 629: 626: 623: 620: 617: 614: 609: 605: 561: 557: 549: 546: 540: 535: 532: 528: 524: 521: 516: 509: 506: 475: 471: 448: 445: 441: 437: 434: 429: 422: 419: 387: 383: 380: 377: 374: 371: 344:speed of light 328: 327: 316: 313: 310: 307: 301: 298: 290: 287: 281: 239: 215: 195: 192: 189: 186: 183: 180: 166:field equation 122: 119: 77: 76: 31: 29: 22: 15: 9: 6: 4: 3: 2: 11771: 11760: 11757: 11755: 11752: 11751: 11749: 11734: 11731: 11729: 11726: 11725: 11723: 11719: 11713: 11710: 11708: 11705: 11703: 11700: 11695: 11691: 11688: 11687:Fatio–Le Sage 11684: 11683: 11682: 11679: 11677: 11674: 11672: 11669: 11667: 11664: 11663: 11661: 11659: 11653:Pre-Newtonian 11651: 11641: 11638: 11636: 11633: 11631: 11628: 11626: 11623: 11621: 11618: 11617: 11615: 11609: 11600: 11596: 11595: 11594: 11591: 11586: 11582: 11579: 11575: 11572: 11568: 11565: 11561: 11558: 11554: 11551: 11547: 11544: 11540: 11537: 11533: 11532: 11531: 11530:String theory 11528: 11523: 11519: 11518: 11517: 11514: 11512: 11509: 11507: 11504: 11503: 11501: 11495: 11489: 11486: 11484: 11481: 11480: 11478: 11474: 11468: 11465: 11463: 11460: 11458: 11455: 11453: 11450: 11448: 11445: 11440: 11436: 11433: 11429: 11426: 11422: 11421: 11420: 11417: 11415: 11412: 11411: 11409: 11405: 11399: 11396: 11394: 11391: 11389: 11386: 11384: 11381: 11376: 11372: 11371: 11370: 11367: 11365: 11362: 11360: 11357: 11352: 11348: 11347: 11346: 11343: 11341: 11338: 11333: 11329: 11326: 11322: 11321: 11320: 11317: 11315: 11312: 11310: 11307: 11305: 11303: 11299: 11295: 11293: 11290: 11288: 11285: 11283: 11280: 11275: 11271: 11268: 11264: 11263: 11262: 11259: 11258: 11256: 11252: 11246: 11243: 11241: 11238: 11236: 11233: 11232: 11230: 11226: 11223: 11221: 11215: 11205: 11202: 11200: 11199:ADM formalism 11197: 11195: 11192: 11190: 11187: 11185: 11182: 11180: 11177: 11175: 11172: 11170: 11167: 11165: 11162: 11160: 11157: 11156: 11154: 11152: 11148: 11142: 11139: 11137: 11134: 11132: 11129: 11127: 11124: 11123: 11121: 11117: 11114: 11110: 11106: 11099: 11094: 11092: 11087: 11085: 11080: 11079: 11076: 11069: 11063: 11061:0-691-08162-X 11057: 11052: 11051: 11044: 11040: 11038:0-19-280672-6 11034: 11030: 11029: 11023: 11018: 11017:gr-qc/0405030 11013: 11009: 11008:Ravndal, Finn 11005: 11004: 10996: 10990: 10986: 10982: 10978: 10974: 10967: 10963: 10953: 10950: 10948: 10945: 10943: 10940: 10938: 10935: 10933: 10930: 10929: 10923: 10920: 10916: 10911: 10909: 10891: 10881: 10850: 10847: 10844: 10841: 10838: 10814: 10807: 10800: 10797: 10793: 10788: 10780: 10773: 10770: 10765: 10754: 10748: 10744: 10741: 10735: 10732: 10726: 10723: 10718: 10707: 10695: 10678: 10669: 10662: 10661: 10660: 10659: 10655: 10651: 10647: 10646:tensor theory 10643: 10642:scalar theory 10639: 10635: 10631: 10627: 10626:isotropically 10605: 10602: 10599: 10596: 10593: 10564: 10558: 10554: 10551: 10545: 10542: 10535: 10528: 10525: 10517: 10513: 10508: 10497: 10485: 10468: 10459: 10452: 10451: 10450: 10449: 10431: 10421: 10414: 10405: 10394: 10390: 10386: 10367: 10364: 10359: 10349: 10338: 10328: 10312: 10311: 10310: 10308: 10304: 10283: 10275: 10270: 10260: 10249: 10233: 10225: 10220: 10210: 10199: 10184: 10178: 10166: 10163: 10160: 10154: 10151: 10148: 10143: 10134: 10126: 10122: 10117: 10112: 10102: 10091: 10076: 10070: 10058: 10055: 10052: 10046: 10043: 10040: 10035: 10026: 10018: 10014: 10009: 10004: 9994: 9983: 9982: 9981: 9978: 9976: 9972: 9967: 9951: 9947: 9943: 9940: 9918: 9886: 9882: 9879: 9875: 9872: 9869: 9865: 9860: 9856: 9849: 9846: 9841: 9838: 9834: 9831: 9828: 9824: 9821: 9809: 9806: 9803: 9800: 9797: 9794: 9791: 9781: 9780: 9779: 9763: 9754: 9751: 9746: 9737: 9712: 9703: 9700: 9695: 9686: 9681: 9676: 9667: 9664: 9659: 9650: 9643: 9626: 9617: 9614: 9609: 9600: 9597: 9590: 9573: 9563: 9558: 9546: 9529: 9516: 9512: 9494: 9482: 9481: 9480: 9478: 9473: 9453: 9450: 9446: 9443: 9439: 9436: 9432: 9429: 9426: 9420: 9414: 9410: 9404: 9400: 9396: 9393: 9388: 9384: 9380: 9377: 9374: 9371: 9367: 9364: 9360: 9357: 9353: 9342: 9336: 9333: 9327: 9324: 9321: 9316: 9312: 9308: 9301: 9300: 9299: 9298: 9294: 9290: 9271: 9265: 9262: 9259: 9236: 9233: 9228: 9225: 9221: 9217: 9212: 9209: 9205: 9201: 9196: 9193: 9189: 9184: 9181: 9174: 9173: 9172: 9152: 9149: 9145: 9142: 9138: 9135: 9131: 9128: 9125: 9119: 9113: 9108: 9104: 9100: 9097: 9092: 9088: 9084: 9081: 9078: 9075: 9071: 9068: 9064: 9061: 9058: 9053: 9049: 9045: 9038: 9037: 9036: 9028: 9026: 9021: 8999: 8992: 8988: 8981: 8977: 8970: 8966: 8961: 8955: 8951: 8948: 8944: 8940: 8937: 8934: 8930: 8923: 8909: 8905: 8898: 8895: 8892: 8891: 8874: 8870: 8866: 8862: 8856: 8837: 8832: 8819: 8803: 8799: 8795: 8789: 8786: 8767: 8762: 8749: 8746: 8743: 8742: 8725: 8721: 8717: 8711: 8708: 8705: 8698: 8695: 8679: 8675: 8671: 8665: 8662: 8659: 8652: 8649: 8648: 8645: 8642: 8638: 8636: 8633: 8629: 8626: 8623: 8622: 8619: 8615: 8612: 8608: 8605: 8603: 8600: 8596: 8593: 8591: 8588: 8584: 8581: 8578: 8577: 8573: 8570: 8567: 8564: 8561: 8558: 8557: 8554: 8551: 8547: 8544: 8541: 8539: 8536: 8533: 8531: 8528: 8525: 8522: 8521: 8517: 8514: 8511: 8509: 8508: 8505: 8500: 8492: 8490: 8486: 8482: 8478: 8460: 8456: 8451: 8447: 8443: 8438: 8434: 8429: 8425: 8401: 8394: 8390: 8383: 8379: 8372: 8368: 8363: 8357: 8353: 8350: 8346: 8343: 8340: 8337: 8334: 8331: 8328: 8318: 8317: 8316: 8313: 8299: 8296: 8293: 8290: 8280:this is just 8265: 8261: 8256: 8252: 8248: 8243: 8239: 8234: 8230: 8206: 8198: 8188: 8184: 8179: 8175: 8169: 8166: 8161: 8154: 8144: 8140: 8135: 8131: 8125: 8122: 8116: 8111: 8108: 8105: 8095: 8094: 8093: 8072: 8068: 8064: 8059: 8054: 8050: 8044: 8037: 8033: 8029: 8024: 8019: 8015: 8005: 8004: 8003: 7982: 7972: 7968: 7963: 7959: 7953: 7950: 7944: 7941: 7934: 7930: 7926: 7921: 7916: 7912: 7906: 7899: 7895: 7891: 7886: 7881: 7877: 7866: 7862: 7855: 7851: 7847: 7844: 7838: 7835: 7832: 7825: 7821: 7815: 7811: 7807: 7802: 7794: 7781: 7780: 7779: 7778:and likewise 7758: 7748: 7744: 7739: 7735: 7729: 7726: 7720: 7717: 7710: 7706: 7702: 7697: 7692: 7688: 7682: 7675: 7671: 7667: 7662: 7657: 7653: 7642: 7638: 7631: 7627: 7623: 7620: 7614: 7611: 7608: 7601: 7597: 7591: 7587: 7583: 7578: 7570: 7557: 7556: 7555: 7538: 7535: 7530: 7522: 7518: 7512: 7508: 7501: 7498: 7495: 7491: 7481: 7477: 7473: 7468: 7464: 7459: 7454: 7451: 7448: 7441: 7440: 7439: 7420: 7416: 7412: 7403: 7399: 7395: 7390: 7386: 7379: 7375: 7369: 7364: 7360: 7356: 7350: 7342: 7336: 7333: 7330: 7325: 7321: 7317: 7311: 7307: 7299: 7298: 7297: 7280: 7277: 7273: 7270: 7267: 7264: 7261: 7258: 7254: 7251: 7248: 7245: 7242: 7239: 7236: 7229: 7228: 7227: 7213: 7210: 7207: 7203: 7200: 7197: 7177: 7157: 7154: 7149: 7145: 7140: 7135: 7131: 7108: 7104: 7099: 7096: 7092: 7087: 7083: 7057: 7053: 7049: 7043: 7039: 7035: 7027: 7019: 7015: 7011: 7008: 7005: 6995: 6991: 6987: 6984: 6979: 6975: 6971: 6968: 6962: 6959: 6952: 6951: 6950: 6942: 6940: 6936: 6932: 6928: 6923: 6921: 6917: 6913: 6909: 6888: 6884: 6881: 6878: 6872: 6853: 6848: 6835: 6834: 6833: 6814: 6810: 6807: 6801: 6798: 6779: 6774: 6761: 6760: 6759: 6737: 6733: 6727: 6723: 6715: 6712: 6709: 6706: 6699: 6696: 6693: 6690: 6680: 6679: 6678: 6656: 6652: 6646: 6642: 6633: 6630: 6625: 6622: 6613: 6609: 6605: 6600: 6593: 6589: 6583: 6579: 6571: 6563: 6559: 6555: 6549: 6532: 6528: 6511: 6507: 6504: 6494: 6493: 6492: 6471: 6467: 6462: 6458: 6453: 6446: 6442: 6438: 6433: 6416: 6408: 6407: 6406: 6390: 6386: 6363: 6360: 6355: 6352: 6327: 6324: 6321: 6316: 6312: 6301: 6297: 6293: 6288: 6281: 6277: 6273: 6268: 6264: 6253: 6249: 6243: 6239: 6233: 6216: 6208: 6207: 6206: 6202: 6200: 6176: 6172: 6165: 6162: 6159: 6150: 6146: 6142: 6137: 6133: 6121: 6117: 6111: 6107: 6101: 6098: 6092: 6089: 6079: 6078: 6077: 6061: 6057: 6052: 6046: 6042: 6038: 6035: 6032: 6029: 6004: 6000: 5996: 5991: 5987: 5979: 5973: 5969: 5965: 5960: 5956: 5935: 5931: 5925: 5921: 5917: 5912: 5908: 5883: 5877: 5869: 5865: 5861: 5858: 5855: 5845: 5842: 5837: 5833: 5825: 5818: 5815: 5811: 5803: 5800: 5795: 5789: 5786: 5776: 5775: 5774: 5757: 5754: 5725: 5722: 5714: 5708: 5700: 5696: 5692: 5689: 5686: 5676: 5673: 5668: 5664: 5656: 5649: 5646: 5642: 5637: 5631: 5628: 5619: 5616: 5610: 5603: 5602: 5601: 5585: 5577: 5573: 5569: 5566: 5563: 5556: 5547: 5543: 5538: 5532: 5528: 5524: 5521: 5515: 5512: 5487: 5479: 5475: 5471: 5468: 5465: 5455: 5452: 5447: 5443: 5436: 5431: 5424: 5421: 5410: 5409: 5408: 5394: 5390: 5386: 5383: 5380: 5366: 5364: 5346: 5336: 5329: 5320: 5291: 5288: 5285: 5282: 5279: 5276: 5249: 5245: 5239: 5235: 5229: 5223: 5220: 5217: 5214: 5211: 5208: 5181: 5177: 5173: 5168: 5163: 5160: 5146: 5137: 5130: 5129: 5128: 5125: 5122: 5100: 5090: 5078: 5074: 5070: 5065: 5060: 5050: 5039: 5029: 5013: 5012: 5011: 4992: 4977: 4971: 4968: 4964: 4960: 4955: 4950: 4940: 4929: 4913: 4902: 4899: 4894: 4889: 4879: 4868: 4852: 4842: 4838: 4834: 4830: 4827: 4824: 4820: 4816: 4811: 4801: 4790: 4774: 4764: 4760: 4756: 4752: 4749: 4746: 4742: 4738: 4733: 4723: 4712: 4711: 4710: 4709: 4698: 4684: 4681: 4678: 4675: 4671: 4665: 4661: 4657: 4652: 4648: 4639: 4635: 4619: 4615: 4609: 4605: 4601: 4596: 4592: 4583: 4579: 4576:is a kind of 4556: 4551: 4547: 4543: 4539: 4536: 4533: 4529: 4523: 4520: 4515: 4511: 4506: 4500: 4496: 4489: 4486: 4479: 4478: 4477: 4460: 4457: 4452: 4448: 4444: 4437: 4432: 4428: 4424: 4420: 4417: 4414: 4410: 4403: 4396: 4393: 4380: 4379: 4378: 4364: 4361: 4358: 4355: 4352: 4349: 4346: 4343: 4317: 4313: 4308: 4304: 4301: 4295: 4292: 4282: 4281: 4280: 4278: 4258: 4254: 4250: 4246: 4243: 4240: 4237: 4233: 4228: 4225: 4220: 4215: 4211: 4207: 4203: 4200: 4197: 4193: 4187: 4184: 4178: 4175: 4165: 4164: 4163: 4149: 4145: 4141: 4138: 4135: 4126: 4118: 4116: 4095: 4083: 4077: 4074: 4067: 4061: 4058: 4055: 4050: 4038: 4032: 4029: 4022: 4006: 3994: 3988: 3985: 3978: 3972: 3969: 3966: 3961: 3949: 3943: 3940: 3937: 3930: 3913: 3901: 3884: 3872: 3871: 3870: 3868: 3864: 3860: 3855: 3839: 3834: 3830: 3825: 3822: 3800: 3796: 3792: 3789: 3766: 3763: 3760: 3757: 3754: 3751: 3747: 3744: 3741: 3738: 3735: 3732: 3728: 3722: 3719: 3716: 3713: 3709: 3703: 3700: 3697: 3691: 3684: 3665: 3661: 3657: 3651: 3643: 3637: 3634: 3631: 3626: 3622: 3618: 3609: 3605: 3601: 3593: 3589: 3585: 3582: 3577: 3573: 3569: 3566: 3557: 3554: 3546: 3542: 3538: 3535: 3532: 3526: 3521: 3517: 3513: 3506: 3505: 3504: 3487: 3483: 3479: 3476: 3473: 3466: 3463: 3460: 3435: 3426: 3422: 3418: 3412: 3404: 3398: 3395: 3392: 3387: 3383: 3379: 3370: 3366: 3362: 3357: 3353: 3349: 3346: 3341: 3337: 3333: 3330: 3326: 3318: 3314: 3310: 3307: 3304: 3298: 3293: 3289: 3285: 3278: 3277: 3276: 3274: 3264: 3248: 3244: 3240: 3234: 3230: 3226: 3220: 3217: 3213: 3187: 3183: 3179: 3173: 3169: 3165: 3159: 3156: 3152: 3144: 3140: 3137: 3134: 3128: 3123: 3119: 3115: 3108: 3107: 3106: 3092: 3069: 3066: 3063: 3055: 3050: 3046: 3042: 3036: 3032: 3028: 3022: 3019: 3015: 3007: 3004: 2998: 2995: 2992: 2987: 2983: 2979: 2972: 2971: 2970: 2962: 2955: 2952: 2951: 2950: 2946: 2943: 2942:together with 2939: 2929: 2926: 2923: 2907: 2904: 2901: 2898: 2894: 2886: 2885: 2881: 2878: 2875: 2859: 2856: 2852: 2844: 2843: 2839: 2836: 2833: 2819: 2812: 2811: 2807: 2804: 2801: 2799: 2798: 2795: 2790: 2787: 2784: 2780: 2776: 2771: 2769: 2750: 2747: 2743: 2733: 2714: 2711: 2707: 2702: 2698: 2695: 2690: 2687: 2680: 2673: 2669: 2657: 2656: 2655: 2654: 2650: 2646: 2642: 2638: 2634: 2613: 2610: 2606: 2601: 2595: 2592: 2587: 2582: 2579: 2575: 2571: 2566: 2563: 2559: 2551: 2550: 2549: 2548: 2544: 2540: 2535: 2531: 2529: 2525: 2521: 2517: 2513: 2508: 2506: 2501: 2497: 2493: 2487: 2485: 2481: 2465: 2440: 2436: 2432: 2426: 2422: 2418: 2412: 2409: 2405: 2397: 2394: 2391: 2388: 2382: 2377: 2373: 2369: 2363: 2359: 2355: 2349: 2346: 2342: 2334: 2330: 2324: 2321: 2318: 2313: 2309: 2305: 2298: 2297: 2296: 2282: 2274: 2258: 2249: 2247: 2243: 2226: 2204: 2200: 2196: 2190: 2186: 2182: 2176: 2173: 2169: 2165: 2160: 2156: 2152: 2129: 2123: 2120: 2117: 2114: 2091: 2088: 2085: 2082: 2076: 2071: 2067: 2063: 2057: 2053: 2049: 2043: 2040: 2036: 2028: 2025: 2019: 2016: 2013: 2008: 2004: 2000: 1993: 1992: 1991: 1989: 1970: 1967: 1962: 1959: 1956: 1953: 1949: 1942: 1939: 1936: 1933: 1926: 1925: 1924: 1922: 1898: 1895: 1890: 1887: 1884: 1881: 1877: 1870: 1867: 1863: 1860: 1857: 1854: 1847: 1846: 1845: 1843: 1825: 1822: 1819: 1816: 1812: 1804: 1800: 1795: 1793: 1789: 1771: 1768: 1764: 1740: 1736: 1733: 1730: 1727: 1720: 1719: 1718: 1697: 1693: 1688: 1685: 1681: 1675: 1672: 1669: 1662: 1661: 1660: 1658: 1640: 1636: 1632: 1626: 1622: 1618: 1612: 1609: 1605: 1598: 1594: 1590: 1585: 1581: 1577: 1569: 1553: 1531: 1527: 1523: 1517: 1513: 1509: 1503: 1500: 1496: 1492: 1487: 1483: 1479: 1471: 1453: 1450: 1446: 1439: 1435: 1431: 1426: 1423: 1419: 1411: 1410:metric tensor 1407: 1404:for a curved 1403: 1385: 1378: 1375: 1365: 1358: 1355: 1345: 1342: 1338: 1331: 1327: 1323: 1320: 1312: 1308: 1304: 1299: 1297: 1278: 1274: 1271: 1266: 1263: 1259: 1252: 1249: 1245: 1238: 1235: 1231: 1223: 1220: 1216: 1211: 1208: 1201: 1200: 1199: 1197: 1176: 1173: 1169: 1162: 1159: 1155: 1148: 1145: 1141: 1136: 1132: 1128: 1125: 1120: 1117: 1113: 1106: 1103: 1099: 1095: 1090: 1087: 1082: 1062: 1058: 1052: 1049: 1042: 1041: 1040: 1021: 1017: 1010: 1006: 1001: 997: 994: 989: 986: 981: 955: 951: 942: 941: 940: 938: 937:perfect fluid 934: 929: 925: 920: 904: 900: 892: 889: 883: 878: 875: 871: 867: 864: 859: 852: 849: 841: 816: 813: 809: 805: 802: 796: 793: 787: 781: 777: 772: 768: 764: 754: 753: 752: 715: 710: 707: 704: 701: 698: 695: 691: 682: 680: 676: 669:, and where T 656: 636: 633: 607: 603: 594: 592: 588: 584: 580: 575: 559: 555: 547: 544: 538: 533: 530: 526: 522: 519: 514: 507: 504: 491: 473: 469: 446: 443: 439: 435: 432: 427: 420: 417: 405: 404:wave equation 401: 385: 381: 378: 375: 372: 369: 361: 360:D'Alembertian 357: 352: 349: 345: 341: 340:test particle 337: 333: 314: 308: 305: 299: 296: 285: 279: 269: 268: 267: 265: 261: 257: 253: 252:test particle 237: 229: 213: 193: 190: 187: 184: 181: 171: 167: 163: 158: 156: 152: 148: 144: 140: 136: 132: 128: 118: 116: 110: 108: 104: 100: 96: 92: 88: 84: 73: 70: 62: 52: 48: 42: 41: 35: 30: 21: 20: 11655:theories and 11488:Supergravity 11374: 11301: 11297: 11159:Introduction 11068:problem 13.2 11067: 11049: 11026: 10976: 10966: 10918: 10914: 10912: 10908:irrotational 10868: 10658:tidal tensor 10657: 10653: 10649: 10645: 10641: 10638:spin-2 waves 10637: 10633: 10629: 10625: 10623: 10447: 10382: 10302: 10300: 9979: 9974: 9968: 9904: 9729: 9510: 9476: 9474: 9471: 9296: 9288: 9251: 9170: 9034: 9019: 9017: 8643: 8640: 8634: 8631: 8617: 8613: 8610: 8606: 8601: 8598: 8594: 8589: 8586: 8582: 8571:diag(-1,1,1) 8568: 8565: 8552: 8549: 8545: 8542: 8537: 8534: 8529: 8526: 8503: 8498: 8489:microseconds 8480: 8417: 8314: 8222: 8091: 8001: 7777: 7553: 7437: 7295: 7074: 6948: 6938: 6934: 6924: 6919: 6915: 6911: 6907: 6905: 6831: 6757: 6676: 6490: 6344: 6203: 6196: 5899: 5743: 5504: 5372: 5362: 5309: 5126: 5120: 5117: 5009: 4704: 4637: 4636:but not the 4633: 4581: 4577: 4575: 4475: 4336:Third, with 4334: 4274: 4127: 4124: 4112: 3856: 3781: 3451: 3272: 3270: 3204: 3084: 2968: 2959: 2947: 2941: 2935: 2793: 2788: 2782: 2778: 2775:Ricci tensor 2774: 2772: 2767: 2734: 2731: 2652: 2636: 2630: 2543:Ricci scalar 2536: 2532: 2527: 2523: 2515: 2509: 2504: 2499: 2495: 2491: 2488: 2483: 2479: 2457: 2272: 2250: 2106: 1987: 1985: 1920: 1918: 1841: 1799:Hermann Weyl 1796: 1788:Ricci tensor 1755: 1716: 1657:Ricci scalar 1300: 1293: 1195: 1193: 1038: 935:(that is, a 932: 921: 833: 683: 595: 590: 576: 353: 335: 329: 260:acceleration 162:field theory 159: 155:relativistic 124: 111: 94: 86: 80: 65: 56: 37: 11457:Causal sets 11351:Brans–Dicke 11169:Mathematics 10915:propagation 10634:spin-0 wave 10389:world lines 9515:wave vector 8893:Time delay 8483:the Sun, a 6945:Light delay 6405:), whereas 4708:frame field 3863:Lie algebra 2633:Weyl tensor 1655:, then the 332:proper time 131:Max Abraham 51:introducing 11748:Categories 11671:CGHS model 11658:toy models 10958:References 10919:generation 10654:transverse 10650:transverse 10385:congruence 9511:"opposing" 9293:plane wave 8616:− 3 6076:, we find 4128:If we set 3267:The metric 579:Lagrangian 143:Greifswald 139:Gustav Mie 59:March 2024 34:references 11676:RST model 11462:DGP model 11439:Spin foam 11383:Whitehead 11375:Nordström 11304:) gravity 11254:Classical 11228:Paradigms 11179:Resources 10885:→ 10794:− 10742:− 10736:⁡ 10711:^ 10699:^ 10682:→ 10552:− 10546:⁡ 10501:^ 10489:^ 10472:→ 10460:θ 10425:→ 10409:→ 10353:→ 10332:→ 10321:∇ 10309:vanishes 10280:∂ 10264:→ 10230:∂ 10214:→ 10175:∂ 10161:− 10155:⁡ 10149:− 10140:∂ 10106:→ 10067:∂ 10053:− 10047:⁡ 10032:∂ 9998:→ 9915:∂ 9880:λ 9857:λ 9839:λ 9816:⟶ 9760:∂ 9743:∂ 9709:∂ 9692:∂ 9673:∂ 9656:∂ 9623:∂ 9606:∂ 9598:− 9570:∂ 9555:∂ 9526:∂ 9491:∂ 9457:∞ 9424:∞ 9421:− 9358:− 9328:⁡ 9289:arbitrary 9260:ψ 9222:ψ 9206:ψ 9190:ψ 9182:− 9156:∞ 9123:∞ 9120:− 9062:− 8952:⁡ 8867:π 8838:ω 8833:ϕ 8830:Δ 8796:π 8790:− 8768:ω 8763:ϕ 8760:Δ 8709:ϕ 8706:δ 8663:ϕ 8660:δ 8518:Einstein 8515:Nordström 8354:⁡ 8326:Δ 8288:Δ 8170:− 8126:− 8103:Δ 8065:− 8030:− 7954:− 7927:− 7892:− 7839:≈ 7812:∫ 7792:Δ 7730:− 7703:− 7668:− 7615:≈ 7588:∫ 7568:Δ 7474:− 7455:≈ 7396:− 7343:ϕ 7337:⁡ 7322:ϕ 7274:ϕ 7271:⁡ 7262:ϕ 7255:ϕ 7252:⁡ 7243:− 7214:ϕ 7211:⁡ 7190:, so put 7178:ϕ 7155:≫ 7054:ϕ 6969:− 6927:perihelia 6908:clockwise 6882:π 6873:≈ 6854:ω 6849:ϕ 6846:Δ 6808:π 6802:− 6799:≈ 6780:ω 6775:ϕ 6772:Δ 6716:π 6713:− 6710:≈ 6707:ω 6704:Δ 6700:π 6691:ϕ 6688:Δ 6626:− 6623:≈ 6572:− 6533:ω 6529:− 6512:ω 6505:ω 6502:Δ 6417:ω 6234:≈ 6217:ω 6173:ε 6160:ε 6102:− 6093:¨ 6090:ε 6039:− 6030:ε 5843:− 5790:¨ 5758:˙ 5726:˙ 5674:− 5632:¨ 5620:˙ 5453:− 5425:˙ 5387:π 5381:θ 5340:→ 5324:→ 5277:− 5209:− 5150:→ 5094:→ 5054:→ 5033:→ 5022:∇ 4993:ϕ 4989:∂ 4978:θ 4972:⁡ 4944:→ 4914:θ 4910:∂ 4883:→ 4849:∂ 4805:→ 4771:∂ 4727:→ 4679:− 4658:≈ 4524:ϵ 4521:− 4458:− 4397:˙ 4350:− 4344:ϵ 4296:˙ 4293:ϕ 4226:≈ 4179:˙ 4142:π 4136:θ 4121:Geodesics 4096:ϕ 4092:∂ 4084:ϕ 4078:⁡ 4068:θ 4062:⁡ 4051:θ 4047:∂ 4039:ϕ 4033:⁡ 4007:ϕ 4003:∂ 3995:ϕ 3989:⁡ 3979:θ 3973:⁡ 3962:θ 3958:∂ 3950:ϕ 3944:⁡ 3938:− 3914:ϕ 3910:∂ 3881:∂ 3859:Lie group 3826:π 3767:π 3761:ϕ 3755:π 3752:− 3745:π 3739:θ 3726:∞ 3707:∞ 3695:∞ 3692:− 3662:ϕ 3644:θ 3638:⁡ 3623:θ 3567:− 3555:− 3488:ρ 3477:− 3467:ρ 3423:ϕ 3405:θ 3399:⁡ 3384:θ 3367:ρ 3354:ρ 3331:− 3319:ρ 3308:− 3214:η 3153:η 3145:ψ 3093:ψ 3064:ψ 3061:Δ 3016:η 3008:ψ 2999:⁡ 2808:Einstein 2805:Nordström 2703:π 2588:− 2496:traceless 2466:ψ 2406:η 2398:ψ 2383:≈ 2343:η 2335:ψ 2325:⁡ 2283:ψ 2259:ψ 2227:◻ 2170:η 2157:σ 2130:ψ 2124:⁡ 2115:ϕ 2086:ψ 2083:◻ 2037:η 2029:ψ 2020:⁡ 1864:π 1737:π 1694:ϕ 1689:ϕ 1686:◻ 1676:− 1606:η 1595:ϕ 1554:◻ 1497:η 1484:σ 1447:η 1436:ϕ 1400:, is the 1379:˙ 1359:˙ 1339:η 1328:ϕ 1279:ϕ 1275:ρ 1272:− 1260:ϕ 1246:ϕ 1232:η 1224:π 1196:conserved 1170:ϕ 1156:ϕ 1142:η 1126:− 1114:ϕ 1100:ϕ 1053:π 1002:ρ 998:ϕ 893:˙ 890:ϕ 884:− 872:ϕ 868:− 853:˙ 842:ϕ 810:ϕ 806:− 773:ϕ 711:π 705:− 699:ϕ 696:◻ 692:ϕ 657:ϕ 637:ρ 591:nonlinear 548:˙ 545:ϕ 539:− 527:ϕ 523:− 508:˙ 440:ϕ 436:− 421:˙ 386:ρ 382:π 373:ϕ 370:◻ 356:Laplacian 315:ϕ 312:∇ 309:− 289:→ 238:ρ 214:ϕ 194:ρ 191:π 182:ϕ 179:Δ 107:spacetime 11733:Graviton 11543:F-theory 11536:M-theory 11112:Standard 10926:See also 10801:″ 10774:′ 10529:′ 10393:timelike 10303:inertial 5407:we have 5363:stronger 3815:is just 2631:and the 2295:we have 264:gradient 206:, where 127:Helsinki 11164:History 10656:. The 6939:advance 6933:should 6931:Mercury 6920:advance 4634:angular 1988:general 677:of the 673:is the 488:is the 402:to the 226:is the 168:is the 47:improve 11058: 11035: 10991: 10446:, the 9517:field 8512:Newton 8481:behind 5948:where 5773:gives 5505:where 5121:second 4638:radial 4476:where 3205:where 2834:scalar 2545:, the 2107:where 1921:vacuum 924:Vienna 671:matter 151:Prague 36:, but 11325:AQUAL 11184:Tests 11012:arXiv 8485:radar 8477:Venus 7296:Thus 7123:with 2498:, so 2492:trace 2480:small 1408:with 675:trace 135:Milan 11066:See 11056:ISBN 11033:ISBN 10989:ISBN 10628:and 9513:the 9454:< 9427:< 9153:< 9126:< 8639:1 + 8630:1 + 3764:< 3758:< 3742:< 3736:< 3723:< 3717:< 3704:< 3698:< 2145:and 933:dust 230:and 10981:doi 10733:exp 10543:exp 10387:of 10152:exp 10044:exp 9325:exp 9252:is 8949:log 8351:log 7334:cot 7268:cos 7249:sin 7208:cos 6935:lag 6929:of 6916:lag 4969:sin 4584:at 4075:cos 4059:cot 4030:sin 3986:sin 3970:cot 3941:cos 3865:of 3635:sin 3396:sin 3275:as 2996:exp 2779:not 2322:exp 2121:exp 2017:exp 834:or 149:in 141:in 133:in 95:two 81:In 11750:: 10987:. 9966:. 8609:= 8597:= 8585:= 8574:0 8548:+ 8312:. 6201:. 3854:. 2770:. 2507:. 2486:. 1861:24 1734:24 919:. 574:. 137:, 129:, 109:. 85:, 11302:R 11300:( 11298:f 11097:e 11090:t 11083:v 11070:. 11064:. 11041:. 11020:. 11014:: 10997:. 10983:: 10892:0 10882:e 10854:) 10851:1 10848:, 10845:1 10842:, 10839:0 10836:( 10831:g 10828:a 10825:i 10822:d 10815:) 10811:) 10808:u 10805:( 10798:f 10789:2 10785:) 10781:u 10778:( 10771:f 10766:( 10761:) 10758:) 10755:u 10752:( 10749:f 10745:4 10739:( 10727:2 10724:1 10719:= 10708:q 10696:p 10689:] 10679:X 10673:[ 10670:E 10609:) 10606:1 10603:, 10600:1 10597:, 10594:0 10591:( 10586:g 10583:a 10580:i 10577:d 10571:) 10568:) 10565:u 10562:( 10559:f 10555:2 10549:( 10539:) 10536:u 10533:( 10526:f 10518:2 10514:1 10509:= 10498:q 10486:p 10479:] 10469:X 10463:[ 10432:0 10422:e 10415:= 10406:X 10368:0 10365:= 10360:0 10350:e 10339:0 10329:e 10284:y 10276:= 10271:3 10261:e 10234:x 10226:= 10221:2 10211:e 10185:) 10179:u 10170:) 10167:f 10164:2 10158:( 10144:v 10135:( 10127:2 10123:1 10118:= 10113:1 10103:e 10077:) 10071:u 10062:) 10059:f 10056:2 10050:( 10041:+ 10036:v 10027:( 10019:2 10015:1 10010:= 10005:0 9995:e 9952:0 9948:u 9944:= 9941:u 9919:z 9890:) 9887:y 9883:, 9876:u 9873:+ 9870:x 9866:, 9861:2 9850:2 9847:u 9842:+ 9835:x 9832:+ 9829:v 9825:, 9822:u 9819:( 9813:) 9810:y 9807:, 9804:x 9801:, 9798:v 9795:, 9792:u 9789:( 9764:x 9755:u 9752:+ 9747:v 9738:x 9713:y 9704:u 9701:+ 9696:v 9687:y 9682:, 9677:x 9668:u 9665:+ 9660:v 9651:x 9627:y 9618:x 9615:+ 9610:x 9601:y 9574:y 9564:, 9559:x 9544:) 9530:u 9495:v 9451:y 9447:, 9444:x 9440:, 9437:v 9433:, 9430:u 9415:, 9411:) 9405:2 9401:y 9397:d 9394:+ 9389:2 9385:x 9381:d 9378:+ 9375:v 9372:d 9368:u 9365:d 9361:2 9354:( 9349:) 9346:) 9343:u 9340:( 9337:f 9334:2 9331:( 9322:= 9317:2 9313:s 9309:d 9275:) 9272:u 9269:( 9266:f 9263:= 9237:0 9234:= 9229:y 9226:y 9218:+ 9213:x 9210:x 9202:+ 9197:v 9194:u 9185:2 9150:y 9146:, 9143:x 9139:, 9136:v 9132:, 9129:u 9114:, 9109:2 9105:y 9101:d 9098:+ 9093:2 9089:x 9085:d 9082:+ 9079:v 9076:d 9072:u 9069:d 9065:2 9059:= 9054:2 9050:s 9046:d 9000:) 8993:2 8989:R 8982:2 8978:R 8971:1 8967:R 8962:4 8956:( 8945:m 8941:2 8938:+ 8935:m 8931:2 8910:m 8906:2 8896:0 8875:R 8871:m 8863:6 8857:= 8849:b 8846:r 8843:o 8804:R 8800:m 8787:= 8779:b 8776:r 8773:o 8747:0 8726:R 8722:m 8718:4 8712:= 8696:0 8680:R 8676:m 8672:2 8666:= 8644:r 8641:m 8635:r 8632:m 8627:1 8618:m 8614:m 8611:L 8607:R 8602:m 8599:L 8595:R 8590:m 8587:L 8583:R 8569:r 8566:m 8562:0 8553:r 8550:m 8546:r 8543:m 8538:r 8535:m 8530:r 8527:m 8461:2 8457:R 8452:/ 8448:R 8444:, 8439:1 8435:R 8430:/ 8426:R 8402:) 8395:2 8391:R 8384:2 8380:R 8373:1 8369:R 8364:4 8358:( 8347:m 8344:2 8341:+ 8338:m 8335:2 8332:= 8329:t 8300:m 8297:2 8294:= 8291:t 8266:2 8262:R 8257:/ 8253:R 8249:, 8244:1 8240:R 8235:/ 8231:R 8207:) 8199:2 8195:) 8189:2 8185:R 8180:/ 8176:R 8173:( 8167:1 8162:+ 8155:2 8151:) 8145:1 8141:R 8136:/ 8132:R 8129:( 8123:1 8117:( 8112:m 8109:= 8106:t 8073:2 8069:R 8060:2 8055:2 8051:R 8045:+ 8038:2 8034:R 8025:2 8020:1 8016:R 7983:2 7979:) 7973:2 7969:R 7964:/ 7960:R 7957:( 7951:1 7945:m 7942:+ 7935:2 7931:R 7922:2 7917:2 7913:R 7907:= 7900:2 7896:R 7887:2 7882:2 7878:R 7867:2 7863:R 7856:2 7852:R 7848:+ 7845:m 7836:t 7833:d 7826:2 7822:R 7816:R 7808:= 7803:2 7799:) 7795:t 7789:( 7759:2 7755:) 7749:1 7745:R 7740:/ 7736:R 7733:( 7727:1 7721:m 7718:+ 7711:2 7707:R 7698:2 7693:1 7689:R 7683:= 7676:2 7672:R 7663:2 7658:1 7654:R 7643:1 7639:R 7632:1 7628:R 7624:+ 7621:m 7612:t 7609:d 7602:1 7598:R 7592:R 7584:= 7579:1 7575:) 7571:t 7565:( 7539:r 7536:d 7531:) 7523:2 7519:r 7513:2 7509:R 7502:m 7499:+ 7496:r 7492:( 7482:2 7478:R 7469:2 7465:r 7460:1 7452:t 7449:d 7421:2 7417:r 7413:d 7404:2 7400:R 7391:2 7387:r 7380:2 7376:R 7370:= 7365:2 7361:r 7357:d 7351:2 7347:) 7340:( 7331:= 7326:2 7318:d 7312:2 7308:r 7281:r 7278:d 7265:+ 7259:d 7246:r 7240:= 7237:0 7204:r 7201:= 7198:R 7158:R 7150:2 7146:R 7141:, 7136:1 7132:R 7109:2 7105:R 7100:, 7097:R 7093:, 7088:1 7084:R 7058:2 7050:d 7044:2 7040:r 7036:+ 7028:2 7024:) 7020:r 7016:/ 7012:m 7009:+ 7006:1 7003:( 6996:2 6992:r 6988:d 6985:+ 6980:2 6976:t 6972:d 6963:= 6960:0 6889:r 6885:m 6879:6 6865:b 6862:r 6859:o 6815:r 6811:m 6791:b 6788:r 6785:o 6738:5 6734:r 6728:3 6724:m 6697:2 6694:= 6657:5 6653:r 6647:3 6643:m 6634:2 6631:1 6614:3 6610:r 6606:m 6601:+ 6594:4 6590:r 6584:2 6580:m 6564:3 6560:r 6556:m 6550:= 6544:m 6541:h 6538:s 6523:b 6520:r 6517:o 6508:= 6472:3 6468:r 6463:/ 6459:m 6454:= 6447:2 6443:r 6439:L 6434:= 6428:b 6425:r 6422:o 6391:c 6387:r 6364:r 6361:m 6356:= 6353:L 6328:r 6325:m 6322:+ 6317:2 6313:m 6302:2 6298:r 6294:1 6289:= 6282:2 6278:L 6274:+ 6269:2 6265:m 6254:4 6250:L 6244:2 6240:m 6228:m 6225:h 6222:s 6182:) 6177:2 6169:( 6166:O 6163:+ 6156:) 6151:2 6147:L 6143:+ 6138:2 6134:m 6130:( 6122:8 6118:L 6112:4 6108:m 6099:= 6062:2 6058:m 6053:/ 6047:2 6043:L 6036:r 6033:= 6010:) 6005:2 6001:m 5997:+ 5992:2 5988:L 5984:( 5980:/ 5974:2 5970:L 5966:= 5961:c 5957:E 5936:m 5932:/ 5926:2 5922:L 5918:= 5913:c 5909:r 5884:) 5878:4 5874:) 5870:r 5866:/ 5862:m 5859:+ 5856:1 5853:( 5849:) 5846:V 5838:2 5834:E 5830:( 5826:( 5819:r 5816:d 5812:d 5804:2 5801:1 5796:= 5787:r 5755:r 5723:r 5715:) 5709:4 5705:) 5701:r 5697:/ 5693:m 5690:+ 5687:1 5684:( 5680:) 5677:V 5669:2 5665:E 5661:( 5657:( 5650:r 5647:d 5643:d 5638:= 5629:r 5617:r 5611:2 5586:2 5582:) 5578:r 5574:/ 5570:m 5567:+ 5564:1 5561:( 5557:/ 5553:) 5548:2 5544:r 5539:/ 5533:2 5529:L 5525:+ 5522:1 5519:( 5516:= 5513:V 5488:4 5484:) 5480:r 5476:/ 5472:m 5469:+ 5466:1 5463:( 5459:) 5456:V 5448:2 5444:E 5440:( 5437:= 5432:2 5422:r 5395:2 5391:/ 5384:= 5347:0 5337:e 5330:= 5321:X 5295:) 5292:1 5289:, 5286:1 5283:, 5280:1 5274:( 5269:g 5266:a 5263:i 5260:d 5250:4 5246:r 5240:2 5236:m 5230:+ 5227:) 5224:1 5221:, 5218:1 5215:, 5212:2 5206:( 5201:g 5198:a 5195:i 5192:d 5182:3 5178:r 5174:m 5169:= 5164:b 5161:a 5157:] 5147:X 5141:[ 5138:E 5101:1 5091:e 5079:2 5075:r 5071:m 5066:= 5061:0 5051:e 5040:0 5030:e 4981:) 4975:( 4965:r 4961:1 4956:= 4951:3 4941:e 4903:r 4900:1 4895:= 4890:2 4880:e 4853:r 4843:) 4839:r 4835:/ 4831:m 4828:+ 4825:1 4821:( 4817:= 4812:1 4802:e 4775:t 4765:) 4761:r 4757:/ 4753:m 4750:+ 4747:1 4743:( 4739:= 4734:0 4724:e 4685:m 4682:3 4676:m 4672:/ 4666:2 4662:L 4653:c 4649:r 4620:m 4616:/ 4610:2 4606:L 4602:= 4597:c 4593:r 4557:2 4552:) 4548:r 4544:/ 4540:m 4537:+ 4534:1 4530:( 4516:2 4512:r 4507:/ 4501:2 4497:L 4490:= 4487:V 4461:V 4453:2 4449:E 4445:= 4438:4 4433:) 4429:r 4425:/ 4421:m 4418:+ 4415:1 4411:( 4404:2 4394:r 4365:1 4362:, 4359:0 4356:, 4353:1 4347:= 4318:2 4314:r 4309:/ 4305:L 4302:= 4259:) 4255:r 4251:/ 4247:m 4244:2 4241:+ 4238:1 4234:( 4229:E 4221:2 4216:) 4212:r 4208:/ 4204:m 4201:+ 4198:1 4194:( 4188:E 4185:= 4176:t 4150:2 4146:/ 4139:= 4087:) 4081:( 4071:) 4065:( 4056:+ 4042:) 4036:( 3998:) 3992:( 3982:) 3976:( 3967:+ 3953:) 3947:( 3885:t 3840:2 3835:0 3831:r 3823:4 3801:0 3797:r 3793:= 3790:r 3748:, 3733:0 3729:, 3720:r 3714:0 3710:, 3701:t 3671:) 3666:2 3658:d 3652:2 3648:) 3641:( 3632:+ 3627:2 3619:d 3616:( 3610:2 3606:r 3602:+ 3599:) 3594:2 3590:r 3586:d 3583:+ 3578:2 3574:t 3570:d 3564:( 3558:2 3551:) 3547:r 3543:/ 3539:m 3536:+ 3533:1 3530:( 3527:= 3522:2 3518:s 3514:d 3491:) 3484:/ 3480:m 3474:1 3471:( 3464:= 3461:r 3436:) 3432:) 3427:2 3419:d 3413:2 3409:) 3402:( 3393:+ 3388:2 3380:d 3377:( 3371:2 3363:+ 3358:2 3350:d 3347:+ 3342:2 3338:t 3334:d 3327:( 3322:) 3315:/ 3311:m 3305:1 3302:( 3299:= 3294:2 3290:s 3286:d 3249:b 3245:x 3241:d 3235:a 3231:x 3227:d 3221:b 3218:a 3188:b 3184:x 3180:d 3174:a 3170:x 3166:d 3160:b 3157:a 3148:) 3141:2 3138:+ 3135:1 3132:( 3129:= 3124:2 3120:s 3116:d 3070:0 3067:= 3056:, 3051:b 3047:x 3043:d 3037:a 3033:x 3029:d 3023:b 3020:a 3011:) 3005:2 3002:( 2993:= 2988:2 2984:s 2980:d 2908:d 2905:c 2902:b 2899:a 2895:C 2860:b 2857:a 2853:S 2820:R 2751:b 2748:a 2744:S 2715:a 2712:; 2708:T 2699:6 2696:= 2691:b 2688:; 2681:b 2674:a 2670:S 2614:b 2611:a 2607:g 2602:R 2596:4 2593:1 2583:b 2580:a 2576:R 2572:= 2567:b 2564:a 2560:S 2441:b 2437:x 2433:d 2427:a 2423:x 2419:d 2413:b 2410:a 2401:) 2395:2 2392:+ 2389:1 2386:( 2378:b 2374:x 2370:d 2364:a 2360:x 2356:d 2350:b 2347:a 2338:) 2331:2 2328:( 2319:= 2314:2 2310:s 2306:d 2205:b 2201:x 2197:d 2191:a 2187:x 2183:d 2177:b 2174:a 2166:= 2161:2 2153:d 2133:) 2127:( 2118:= 2092:0 2089:= 2077:, 2072:b 2068:x 2064:d 2058:a 2054:x 2050:d 2044:b 2041:a 2032:) 2026:2 2023:( 2014:= 2009:2 2005:s 2001:d 1971:0 1968:= 1963:d 1960:c 1957:b 1954:a 1950:C 1943:, 1940:0 1937:= 1934:R 1899:0 1896:= 1891:d 1888:c 1885:b 1882:a 1878:C 1871:, 1868:T 1858:= 1855:R 1826:d 1823:c 1820:b 1817:a 1813:C 1772:b 1769:a 1765:g 1741:T 1731:= 1728:R 1698:3 1682:6 1673:= 1670:R 1641:b 1637:x 1633:d 1627:a 1623:x 1619:d 1613:b 1610:a 1599:2 1591:= 1586:2 1582:s 1578:d 1532:b 1528:x 1524:d 1518:a 1514:x 1510:d 1504:b 1501:a 1493:= 1488:2 1480:d 1454:b 1451:a 1440:2 1432:= 1427:b 1424:a 1420:g 1386:b 1376:u 1366:a 1356:u 1346:b 1343:a 1332:2 1324:= 1321:L 1267:b 1264:, 1253:a 1250:, 1239:b 1236:a 1221:8 1217:1 1212:= 1209:L 1177:m 1174:, 1163:m 1160:, 1149:b 1146:a 1137:2 1133:/ 1129:1 1121:b 1118:, 1107:a 1104:, 1096:= 1091:b 1088:a 1083:) 1077:v 1074:a 1071:r 1068:g 1063:T 1059:( 1050:4 1022:b 1018:u 1011:a 1007:u 995:= 990:b 987:a 982:) 976:r 973:e 970:t 967:t 964:a 961:m 956:T 952:( 905:a 901:u 879:a 876:, 865:= 860:a 850:u 817:a 814:, 803:= 797:s 794:d 788:) 782:a 778:u 769:( 765:d 736:r 733:e 730:t 727:t 724:a 721:m 716:T 708:4 702:= 634:= 628:r 625:e 622:t 619:t 616:a 613:m 608:T 604:F 560:a 556:u 534:a 531:, 520:= 515:a 505:u 474:a 470:u 447:a 444:, 433:= 428:a 418:u 379:4 376:= 306:= 300:t 297:d 286:u 280:d 188:4 185:= 72:) 66:( 61:) 57:( 43:.

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Knowledge

Nordström's theory of gravitation

Index