8375:
8348:
8387:
238:. In the system of Planck units, a similar set of base quantities and associated units may be selected, in terms of which other quantities and coherent units may be expressed. The Planck unit of length has become known as the Planck length, and the Planck unit of time is known as the Planck time, but this nomenclature has not been established as extending to all quantities.
3119:". It is possible that the Planck length is the shortest physically measurable distance, since any attempt to investigate the possible existence of shorter distances, by performing higher-energy collisions, would result in black hole production. Higher-energy collisions, rather than splitting matter into finer pieces, would simply produce bigger black holes.
802:... die Möglichkeit gegeben ist, Einheiten für Länge, Masse, Zeit und Temperatur aufzustellen, welche, unabhängig von speciellen Körpern oder Substanzen, ihre Bedeutung für alle Zeiten und für alle, auch ausserirdische und aussermenschliche Culturen nothwendig behalten und welche daher als »natürliche Maasseinheiten« bezeichnet werden können.
1355:, there is no official entity that establishes a definition of a Planck unit system. Some authors define the base Planck units to be those of mass, length and time, regarding an additional unit for temperature to be redundant. Other tabulations add, in addition to a unit for temperature, a unit for electric charge, so that the
551:
6814:
The point at which our physical theories run into most serious difficulties is that where matter reaches a temperature of approximately 10 degrees, also known as Planck's temperature. The extreme density of radiation emitted at this temperature creates a disproportionately intense field of gravity.
2549:
When Planck proposed his units, the goal was only that of establishing a universal ("natural") way of measuring objects, without giving any special meaning to quantities that measured one single unit. However, in 1959, C. A. Mead showed that distances that measured of the order of one Planck length,
1619:
In any system of measurement, units for many physical quantities can be derived from base units. Table 2 offers a sample of derived Planck units, some of which are seldom used. As with the base units, their use is mostly confined to theoretical physics because most of them are too large or too small
809:... it is possible to set up units for length, mass, time and temperature, which are independent of special bodies or substances, necessarily retaining their meaning for all times and for all civilizations, including extraterrestrial and non-human ones, which can be called "natural units of measure".
6347:
Bird, D. J.; Corbato, S. C.; Dai, H. Y.; Elbert, J. W.; Green, K. D.; Huang, M. A.; Kieda, D. B.; Ko, S.; Larsen, C. G.; Loh, E. C.; Luo, M. Z.; Salamon, M. H.; Smith, J. D.; Sokolsky, P.; Sommers, P.; Tang, J. K. K.; Thomas, S. B. (March 1995). "Detection of a cosmic ray with measured energy well
5782:
Like all orders of magnitude estimates, this procedure is not rigorous since it extrapolates the concepts of black hole and of
Compton wavelength to a new regime in which both concepts would probably lose their accepted meanings and would, strictly speaking, cease being valid. However, this is how
2521:
While it is true that the electrostatic repulsive force between two protons (alone in free space) greatly exceeds the gravitational attractive force between the same two protons, this is not about the relative strengths of the two fundamental forces. From the point of view of Planck units, this is
2448:
too large or too small to be of practical use, so that Planck units as a system are typically only relevant to theoretical physics. In some cases, a Planck unit may suggest a limit to a range of a physical quantity where present-day theories of physics apply. For example, our understanding of the
2637:
At the Planck length scale, the strength of gravity is expected to become comparable with the other forces, and it has been theorized that all the fundamental forces are unified at that scale, but the exact mechanism of this unification remains unknown. The Planck scale is therefore the point at
241:
All Planck units are derived from the dimensional universal physical constants that define the system, and in a convention in which these units are omitted (i.e. treated as having the dimensionless value 1), these constants are then eliminated from equations of physics in which they appear. For
3510:
As already stated above, Planck units are derived by "normalizing" the numerical values of certain fundamental constants to 1. These normalizations are neither the only ones possible nor necessarily the best. Moreover, the choice of what factors to normalize, among the factors appearing in the
3353:
It is the gravitational attractive force of two bodies of 1 Planck mass each that are held 1 Planck length apart. One convention for the Planck charge is to choose it so that the electrostatic repulsion of two objects with Planck charge and mass that are held 1 Planck length apart balances the
572:
since any ratio of two like-dimensioned quantities is a dimensionless quantity. If, by a shorthand convention, it is understood that each physical quantity is the corresponding ratio with a coherent Planck unit (or "expressed in Planck units"), the ratios above may be expressed simply with the
6778:
However, for most experimental physicists, approaching energies comparable to Planck energy is little more than a distant fantasy. The most powerful accelerators on Earth miss Planck energy of 15 orders of magnitude, while ultra high energy cosmic rays are still 9 orders of magnitude short of
2330:
401:
5047:
Just as in our discussion of the Fermi theory, the nonrenormalizability of quantum gravity tells us that at the Planck energy scale ... new physics must appear. Fermi's theory cried out, and the new physics turned out to be the electroweak theory. Einstein's theory is now crying
3114:
in units of Planck length squared. Since the 1950s, it has been conjectured that quantum fluctuations of the spacetime metric might make the familiar notion of distance inapplicable below the Planck length. This is sometimes expressed by saying that "spacetime becomes a
2181:
3349:
1842:
1481:
3415:
Physical quantities that have different dimensions (such as time and length) cannot be equated even if they are numerically equal (e.g., 1 second is not the same as 1 metre). In theoretical physics, however, this scruple may be set aside, by a process called
2512:
We see that the question is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in natural (Planck) units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number 1/13
2054:
411:
3655:) into the nondimensionalized form of the law of universal gravitation, consistent with the modern rationalized formulation of Coulomb's law in terms of the vacuum permittivity. In fact, alternative normalizations frequently preserve the factor of
3190:. No current physical theory can describe timescales shorter than the Planck time, such as the earliest events after the Big Bang. Some conjectures state that the structure of time need not remain smooth on intervals comparable to the Planck time.
1940:
3743:
In 1899, Newton's law of universal gravitation was still seen as exact, rather than as a convenient approximation holding for "small" velocities and masses (the approximate nature of Newton's law was shown following the development of
2425:
1328:
2207:
251:
3248:
3064:
648:
1388:
5690:
1229:
1072:
3420:. The effective result is that many fundamental equations of physics, which often include some of the constants used to define Planck units, become equations where these constants are replaced by a 1.
2472:: very large in comparison with subatomic particles, and within the mass range of living organisms. Similarly, the related units of energy and of momentum are in the range of some everyday phenomena.
1704:
1147:
2657:
at very high energies using the usual framework of quantum field theory. At lesser energy levels it is usually ignored, while for energies approaching or exceeding the Planck scale, a new theory of
2080:
1730:
6974:
784:
introduced what became later known as the Planck constant. At the end of the paper, he proposed the base units that were later named in his honor. The Planck units are based on the quantum of
1585:
546:{\displaystyle {\frac {F}{F_{\text{P}}}}={\frac {\left({\dfrac {m_{1}}{m_{\text{P}}}}\right)\left({\dfrac {m_{2}}{m_{\text{P}}}}\right)}{\left({\dfrac {r}{l_{\text{P}}}}\right)^{2}}}}
3357:
Some authors have argued that the Planck force is on the order of the maximum force that can occur between two bodies. However, the validity of these conjectures has been disputed.
2642:, where current calculations and approaches begin to break down, and a means to take account of its impact is necessary. On these grounds, it has been speculated that it may be an
1962:
7249:
943:
3233:
posit that, in addition to the speed of light, an energy scale is also invariant for all inertial observers. Typically, this energy scale is chosen to be the Planck energy.
730:, etc., to 1" if the correspondence of the quantities is thought of as equality. For this reason, Planck or other natural units should be employed with care. Referring to "
1383:
900:
214:
Variants of the basic idea of Planck units exist, such as alternate choices of normalization that give other numeric values to one or more of the four constants above.
2968:
After the measurement of the cosmological constant (Λ) in 1998, estimated at 10 in Planck units, it was noted that this is suggestively close to the reciprocal of the
1605:
103:
refers to quantities of space, time, energy and other units that are similar in magnitude to corresponding Planck units. This region may be characterized by particle
5911:
Studies in the
History of General Relativity: Based on the proceedings of the 2nd International Conference on the History of General Relativity, Luminy, France, 1988
963:
1868:
5678:
3103:, though whether those concepts are in fact simultaneously applicable is open to debate. (The same heuristic argument simultaneously motivates the Planck mass.)
3148:
2562:
An analysis of the effect of gravitation on hypothetical experiments indicates that it is impossible to measure the position of a particle with error less than
983:
871:
851:
831:
3205:
is approximately equal to the energy released in the combustion of the fuel in an automobile fuel tank (57.2 L at 34.2 MJ/L of chemical energy). The
2728:, or approximately 10 seconds. There is no currently available physical theory to describe such short times, and it is not clear in what sense the concept of
2356:
6016:
5909:
3013:
135:(approximately the energy-equivalent of the Planck mass, the Planck time and the Planck length, respectively). At the Planck scale, the predictions of the
4895:
4849:
2325:{\displaystyle \rho _{\text{P}}={\frac {m_{\text{P}}}{l_{\text{P}}^{3}}}={\frac {\hbar t_{\text{P}}}{l_{\text{P}}^{5}}}={\frac {c^{5}}{\hbar G^{2}}}}
745:
wrote that, "Mathematically it is an acceptable trick which saves labour. Physically it represents a loss of information and can lead to confusion."
7049:
5208:
1262:
7294:
7015:
4958:
4928:
4669:
4636:
4606:
4576:
4333:
396:{\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}=\left({\frac {F_{\text{P}}l_{\text{P}}^{2}}{m_{\text{P}}^{2}}}\right){\frac {m_{1}m_{2}}{r^{2}}}}
4470:
7244:
2504:
and Planck time are conceptually linked at a fundamental physical level. Consequently, natural units help physicists to reframe questions.
6913:
5403:
4761:
3763:
or a small integer multiple thereof. Hence, a choice to be made when designing a system of natural units is which, if any, instances of 4
7816:
7563:
7288:
6107:
5989:
7936:
7841:
3674:
in the nondimensionalized form of
Coulomb's law as well, so that the nondimensionalized Maxwell's equations for electromagnetism and
5356:
8077:
4427:
1180:
1023:
243:
6325:
5883:
4527:
3399:
at the Planck temperature might contain Planck-scale black holes, constantly being formed from thermal radiation and decaying via
2457:, i.e., when the universe was less than one Planck time old. Describing the universe during the Planck epoch requires a theory of
7836:
7553:
7373:
3577:
2696:
2566:, where 𝐺 is the gravitational constant in natural units. A similar limitation applies to the precise synchronization of clocks.
8042:
5239:
2176:{\displaystyle F_{\text{P}}={\frac {E_{\text{P}}}{l_{\text{P}}}}={\frac {\hbar }{l_{\text{P}}t_{\text{P}}}}={\frac {c^{4}}{G}}}
1652:
1105:
5594:
3344:{\displaystyle F_{\text{P}}={\frac {m_{\text{P}}c}{t_{\text{P}}}}={\frac {c^{4}}{G}}\approx \mathrm {1.2103\times 10^{44}~N} }
7872:
7867:
7821:
7472:
7462:
7363:
6984:
6949:
6807:
6553:
6149:
6101:
5769:
5580:
5478:
5397:
5074:
5032:
4843:
4712:
2909:
2892:
2866:
5272:
1837:{\displaystyle l_{\text{P}}^{3}=\left({\frac {\hbar G}{c^{3}}}\right)^{\frac {3}{2}}={\sqrt {\frac {(\hbar G)^{3}}{c^{9}}}}}
7877:
7447:
7385:
7042:
6255:
Wendel, Garrett; Martínez, Luis; Bojowald, Martin (19 June 2020). "Physical
Implications of a Fundamental Period of Time".
5465:
1534:. Hence, the uncertainties in the SI values of the Planck units derive almost entirely from uncertainty in the SI value of
68:(described further below). Expressing one of these physical constants in terms of Planck units yields a numerical value of
6522:
1476:{\displaystyle q_{\text{P}}={\sqrt {4\pi \epsilon _{0}\hbar c}}\approx 1.875546\times 10^{-18}{\text{ C}}\approx 11.7\ e.}
7811:
7598:
2649:
While physicists have a fairly good understanding of the other fundamental interactions of forces on the quantum level,
7851:
7826:
7679:
7427:
5504:
5014:
4889:
4755:
578:
8351:
7735:
6907:
5919:
5440:
4309:
2621:
become significant. At this scale, present descriptions and theories of sub-atomic particle interactions in terms of
1548:
4792:
3395:; a quantum theory of gravity would be required to model the extreme energies attained. Hypothetically, a system in
2454:
8067:
7964:
7705:
7669:
7578:
7316:
7254:
4881:
4747:
3511:
fundamental equations of physics, is not evident, and the values of the Planck units are sensitive to this choice.
8374:
7791:
7776:
7730:
7608:
7583:
7568:
7528:
7452:
7035:
6065:
5933:
2834:
2643:
8257:
8125:
7786:
7771:
7761:
7751:
7689:
7659:
7629:
7593:
7518:
7437:
7412:
7407:
7402:
7395:
7099:
6579:
4872:
34:
3569:
around a point charge will be distributed uniformly over that sphere. From this, it follows that a factor of 4
2049:{\displaystyle E_{\text{P}}=m_{\text{P}}c^{2}={\frac {\hbar }{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{G}}}}
8140:
8052:
8014:
7929:
7781:
7756:
7720:
7684:
7664:
7654:
7639:
7634:
7624:
7533:
7457:
7358:
7089:
7079:
6743:
4254:
1352:
1004:
227:
4815:
8145:
7846:
7766:
7725:
7715:
7644:
7548:
7487:
7482:
7442:
7432:
7422:
7417:
7321:
7223:
7218:
7137:
7132:
7094:
6834:(17 July 2018). "Scrambling Time and Causal Structure of the Photon Sphere of a Schwarzschild Black Hole".
6182:
5908:(1992). "First Steps of Quantum Gravity and the Planck Values". In Eisenstaedt, Jean; Kox, Anne J. (eds.).
3584:
would change if space were higher-dimensional; the correct expressions can be deduced from the geometry of
3424:
3206:
7181:
5433:
The
Constants of Nature: From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe
2550:
or, similarly, times that measured of the order of Planck time, did carry special implications related to
2484:
arbitrariness, but do still involve some arbitrary choices in terms of the defining constants. Unlike the
8223:
8024:
7674:
7649:
7588:
7538:
7523:
7513:
7477:
7368:
7331:
7300:
7171:
5718:
5621:
4075:
3599:
Hence a substantial body of physical theory developed since Planck's 1899 paper suggests normalizing not
3388:
reaches the Planck length. There are no known physical models able to describe temperatures greater than
3084:
2599:
5198:
5136:
Mead, Chester Alden (10 August 1964). "Possible
Connection Between Gravitation and Fundamental Length".
2854:
8365:
8120:
7710:
7558:
7492:
7390:
7336:
7326:
5020:
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produced by point objects have spherical symmetry, and so the electric flux through a sphere of radius
3096:
2752:, where gravitation is separated from the unified force of the Standard Model, in turn followed by the
2690:
222:
Any system of measurement may be assigned a mutually independent set of base quantities and associated
152:
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921:
6548:. International series of monographs on physics. Vol. 155. Oxford University Press. p. 5.
3245:
in the Planck system if the Planck units of time, length, and mass are considered to be base units.
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2639:
2435:
2340:
569:
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1361:
8277:
8087:
8019:
7979:
7166:
7117:
6860:
6734:
Afshordi, Niayesh (1 March 2012). "Where will
Einstein fail? Leasing for Gravity and cosmology".
6257:
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4209:
4158:
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223:
194:
55:
4820:. Fundamental Theories of Physics. Vol. 119. Dordrecht: Kluwer Academic. pp. 347–352.
8407:
7959:
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both take the same form as those for electromagnetism in SI, which do not have any factors of 4
2514:
1385:
is added to the list of constants used by Planck. Thus, this charge unit is typically given by
761:, noting that electric charge is quantized, derived units of length, time, and mass, now named
184:
49:
6895:
5387:
4741:
2812:
8185:
8097:
7994:
7984:
7058:
6935:
4352:
3593:
3417:
3127:
3106:
The Planck length is a distance scale of interest in speculations about quantum gravity. The
2977:
2916:
2555:
1935:{\displaystyle m_{\text{P}}c={\frac {\hbar }{l_{\text{P}}}}={\sqrt {\frac {\hbar c^{3}}{G}}}}
1590:
85:
2546:
is far less than the unit mass in a system that treats both forces as having the same form.
948:
8218:
6868:
6761:
6706:
6643:
6588:
6498:
6430:
6369:
6275:
6201:
6033:
5965:
5851:
5809:
5737:
5640:
5523:
5352:
5312:
5248:
5147:
5108:
4989:
4506:
4411:
4361:
4258:
4148:
3100:
2745:
2670:
2622:
2535:
2468:
Several quantities are not "extreme" in magnitude, such as the Planck mass, which is about
793:
573:
symbols of physical quantity, without being scaled explicitly by their corresponding unit:
557:
140:
6085:
Quantum Field Theory II: Introductions to
Quantum Gravity, Supersymmetry and String Theory
5951:
5024:
8:
8412:
7831:
7351:
7282:
6472:
6173:
6024:
5344:
5064:
4079:
3871:
3562:
3558:
3519:
3396:
2969:
2793:
2778:
2445:
722:, but not as a direct equality of quantities. This may seem to be "setting the constants
6872:
6765:
6710:
6647:
6592:
6502:
6434:
6373:
6279:
6205:
6037:
5969:
5855:
5813:
5741:
5644:
5527:
5316:
5252:
5151:
5112:
4993:
4415:
4399:
4365:
4262:
780:
would be numerically equal to 1. In 1899, one year before the advent of quantum theory,
8272:
8200:
8155:
7346:
7122:
6835:
6751:
6696:
6687:
6659:
6633:
6604:
6514:
6488:
6454:
6420:
6385:
6359:
6317:
6299:
6265:
6232:
6191:
6177:
6155:
6129:
6057:
5875:
5761:
5727:
5656:
5630:
5539:
5513:
5264:
5177:
4821:
4540:
4500:
4377:
4327:
4299:
4205:
4192:
3745:
3542:
3209:
3133:
2753:
2702:
2580:
2538:
quantities. Rather, the disparity of magnitude of force is a manifestation of that the
2523:
2462:
2420:{\displaystyle a_{\text{P}}={\frac {c}{t_{\text{P}}}}={\sqrt {\frac {c^{7}}{\hbar G}}}}
968:
856:
836:
816:
789:
207:
163:
144:
60:
26:
6408:
6045:
8317:
8312:
8092:
8057:
6990:
6980:
6955:
6945:
6903:
6803:
6769:
6663:
6608:
6577:
de
Sabbata, Venzo; Sivaram, C. (1993). "On limiting field strengths in gravitation".
6559:
6549:
6458:
6446:
6389:
6303:
6291:
6237:
6219:
6159:
6145:
6097:
6083:
6049:
5981:
5925:
5915:
5879:
5867:
5765:
5753:
5586:
5576:
5484:
5474:
5436:
5393:
5070:
5038:
5028:
4885:
4839:
4751:
4718:
4708:
4419:
4381:
4315:
4305:
4287:
4163:
4107:. Also, the Bekenstein–Hawking formula for the entropy of a black hole simplifies to
3400:
3385:
2766:
Table 3 lists properties of the observable universe today expressed in Planck units.
2737:
2654:
2551:
2539:
774:
38:
6518:
6061:
5660:
5543:
5502:
Barrow, John D.; Shaw, Douglas J. (2011). "The value of the cosmological constant".
5268:
4661:
3407:
its temperature by creating larger black holes, whose
Hawking temperature is lower.
2732:
is meaningful for values smaller than the Planck time. It is generally assumed that
8135:
7378:
7011:
6876:
6714:
6651:
6596:
6506:
6438:
6377:
6287:
6283:
6227:
6209:
6137:
6089:
6088:. Graduate Texts in Physics. Cham: Springer International Publishing. p. 187.
6041:
5973:
5859:
5817:
5745:
5648:
5564:
5531:
5320:
5256:
5155:
5116:
4831:
4369:
4266:
3883:
3853:
2858:
2576:
1356:
785:
231:
22:
5713:
4628:
4598:
4568:
4483:
pp. 478–80 contain the first appearance of the Planck base units, and of the
3767:
appearing in the equations of physics are to be eliminated via the normalization.
1620:
for empirical or practical use and there are large uncertainties in their values.
8322:
7945:
7543:
6816:
6012:
5947:
5905:
5800:
5568:
5460:
5234:
5230:
5138:
4920:
4700:
4484:
4187:
2940:
2748:. Immeasurably hot and dense, the state of the Planck epoch was succeeded by the
2733:
2658:
2626:
2614:
2531:
2481:
2458:
148:
93:
77:
6318:"HiRes – The High Resolution Fly's Eye Ultra High Energy Cosmic Ray Observatory"
4950:
4463:
Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin
8391:
8379:
8292:
7893:
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6141:
5456:
5428:
4395:
4347:
3891:
3546:
3456:
2741:
742:
174:
166:
that, by definition, have a numeric value 1 when expressed in these units are:
136:
43:
6994:
6880:
6655:
6622:
Gibbons, G. W. (2002). "The Maximum Tension Principle in General Relativity".
6476:
6093:
5535:
5325:
5298:
3696:. When applied additionally to gravitation and Planck units, these are called
151:
are expected to dominate. One example is represented by the conditions in the
8401:
8267:
8177:
7341:
7213:
7153:
6959:
6855:
6773:
6563:
6450:
6223:
5985:
5956:
5871:
5757:
5590:
5294:
5159:
5099:
5090:
5060:
5042:
4423:
4245:
4240:
4183:
3988:
3879:
3123:
3111:
2662:
2505:
2501:
2191:
1946:
1323:{\displaystyle T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{Gk_{\text{B}}^{2}}}}}
754:
73:
6719:
6682:
5929:
4722:
4536:
3095:. It can be motivated in various ways, such as considering a particle whose
8195:
8102:
8009:
7999:
7974:
7898:
7186:
7161:
6510:
6295:
6241:
6053:
5835:
5821:
5674:
5303:
5203:
4529:
Natural Systems of Units. To the Centenary Anniversary of the Planck System
3887:
3554:
3116:
2594:
2584:
2493:
2348:
1714:
1542:
762:
6939:
4835:
4319:
8262:
7497:
7228:
7127:
7027:
6678:
6404:
6364:
5010:
4784:
4143:
3596:
when relating the gravitational potential to the distribution of matter.
2827:
2718:
2678:
2543:
1852:
1254:
915:
6348:
beyond the expected spectral cutoff due to cosmic microwave radiation".
6214:
4537:
Proceedings Of The XXII Workshop On High Energy Physics And Field Theory
8327:
8112:
8047:
7969:
6976:
Curvature of Space and Time, with an Introduction to Geometric Analysis
6831:
6638:
6600:
5863:
5370:
5260:
5182:
4737:
4454:
4373:
4291:
3550:
2849:
2638:
which the effects of quantum gravity can no longer be ignored in other
781:
89:
69:
8386:
6425:
5977:
5749:
5652:
5121:
5094:
4826:
4270:
3126:
are modeled to be on the order of the Planck length. In theories with
2736:
dominate physical interactions at this time scale. At this scale, the
796:. Planck underlined the universality of the new unit system, writing:
8282:
8150:
8062:
8034:
5839:
5560:
5173:
3878:. These equations have the same form as Maxwell's equations (and the
3059:{\displaystyle \ell _{\mathrm {P} }={\sqrt {\frac {\hbar G}{c^{3}}}}}
3588:.) Likewise for Newton's law of universal gravitation: a factor of 4
2625:
break down and become inadequate, due to the impact of the apparent
8190:
7989:
6840:
6701:
6381:
6270:
5732:
5374:
3585:
2714:
2666:
2469:
2450:
1860:
226:, from which all other quantities and units may be derived. In the
156:
7914:
7022:
The Planck scale: relativity meets quantum mechanics meets gravity
6896:"A Model of Topological Quantization of the Electromagnetic Field"
6756:
6493:
6196:
5635:
5518:
3241:
The Planck unit of force may be thought of as the derived unit of
989:
Table 1: Modern values for Planck's original choice of quantities
813:
Planck considered only the units based on the universal constants
788:, now usually known as the Planck constant, which appeared in the
7903:
2650:
2618:
2199:
5488:
4350:(1980). "The application of dimensional analysis to cosmology".
3580:. (Both the numerical factor and the power of the dependence on
7573:
5686:
5619:
Adler, Ronald J. (2010). "Six easy roads to the Planck scale".
3523:
3186:
3176:
3092:
3007:
2976:) squared. Barrow and Shaw proposed a modified theory in which
2960:
2489:
1954:
1722:
1015:
903:
128:
104:
81:
6481:
Symmetry, Integrability and Geometry: Methods and Applications
3212:
had a measured energy of about 50 J, equivalent to about
3242:
3172:
2980:
is a field evolving in such a way that its value remains Λ ~
2485:
2072:
2064:
1224:{\displaystyle t_{\text{P}}={\sqrt {\frac {\hbar G}{c^{5}}}}}
1067:{\displaystyle l_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}}
918:. His definitions differ from the modern ones by a factor of
235:
6819:
would be necessary, but such a theory has yet to be written.
92:, they are relevant in research on unified theories such as
6979:. Providence, Rhode Island: American Mathematical Society.
3538:
3168:
2729:
2527:
1644:
1172:
1097:
911:
907:
116:
3522:
because in three-dimensional space, the surface area of a
3384:. At this temperature, the wavelength of light emitted by
3130:, the Planck length calculated from the observed value of
3083:(the two digits enclosed by parentheses are the estimated
5679:"What Is The Smallest Possible Distance In The Universe?"
4874:
Quantum Field Theory I: Basics in Mathematics and Physics
3150:
can be smaller than the true, fundamental Planck length.
1699:{\displaystyle l_{\text{P}}^{2}={\frac {\hbar G}{c^{3}}}}
1142:{\displaystyle m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}}
2444:
Some Planck units, such as of time and length, are many
1515:
in SI units respectively have relative uncertainties of
6132:(December 2021). "From the S Matrix to String Theory".
4955:
The NIST Reference on Constants, Units, and Uncertainty
4925:
The NIST Reference on Constants, Units, and Uncertainty
4666:
The NIST Reference on Constants, Units, and Uncertainty
4633:
The NIST Reference on Constants, Units, and Uncertainty
4603:
The NIST Reference on Constants, Units, and Uncertainty
4573:
The NIST Reference on Constants, Units, and Uncertainty
3735:
There are several possible alternative normalizations.
3505:
3087:
associated with the reported numerical value) or about
2756:, which ended after about 10 seconds (or about 10
2497:
1636:
6254:
4951:"2022 CODATA Value: Newtonian constant of gravitation"
1551:
8363:
5565:"Higher-Dimensional Algebra and Planck-Scale Physics"
4776:
3682:. When this is applied to electromagnetic constants,
3251:
3136:
3016:
2359:
2210:
2083:
1965:
1871:
1733:
1655:
1593:
1391:
1364:
1265:
1183:
1108:
1026:
971:
951:
924:
879:
859:
839:
819:
581:
513:
477:
443:
414:
254:
6346:
6178:"Minimal Length Scale Scenarios for Quantum Gravity"
564:
system of quantities, but the second equation, with
4817:
The Landscape of Theoretical Physics: A Global View
4404:
Quarterly Journal of the Royal Astronomical Society
4286:
3576:will appear in the denominator of Coulomb's law in
2646:at which a black hole could be formed by collapse.
6409:"Conservation laws in "doubly special relativity""
3755:in Newton's law. In theories emerging after 1899,
3703:The rationalized Planck units are defined so that
3343:
3142:
3058:
2419:
2324:
2175:
2048:
1934:
1836:
1698:
1599:
1579:
1475:
1377:
1322:
1223:
1141:
1066:
977:
957:
937:
894:
865:
845:
825:
642:
545:
395:
88:. Originally proposed in 1899 by German physicist
7014:, including the Planck units, as reported by the
6683:"Counterexamples to the maximum force conjecture"
6576:
5338:
5336:
4980:Ray, T. P. (1981). "Stoney's Fundamental Units".
4921:"2022 CODATA Value: vacuum electric permittivity"
4082:, for gravitation. Planck units modified so that
3759:nearly always appears in formulae multiplied by 4
2661:is necessary. Approaches to this problem include
643:{\displaystyle F'={\frac {m_{1}'m_{2}'}{r'^{2}}}}
8399:
5575:. Cambridge University Press. pp. 172–195.
5095:"Scaling Mount Planck I: A View from the Bottom"
3175:to travel a distance of 1 Planck length in
1624:Table 2: Coherent derived units of Planck units
1580:{\textstyle {\sqrt {{1}/{\alpha }}}\approx 11.7}
5783:one gains intuition into a new physical regime.
4656:
4654:
3410:
1545:, Planck base units are all larger by a factor
234:include length with the associated unit of the
37:defined exclusively in terms of four universal
7016:National Institute of Standards and Technology
6934:
5423:
5421:
5379:
5333:
5223:
4913:
4563:
4561:
2653:is problematic, and cannot be integrated with
7930:
7043:
6941:General Relativity and the Einstein Equations
6898:. In M. Ferrero; Alwyn van der Merwe (eds.).
6736:Bulletin of the Astronomical Society of India
5199:"Can experiment access Planck-scale physics?"
4449:
4447:
4445:
4282:
4280:
3976:from the Bekenstein–Hawking formula (for the
3617:) to 1. Doing so would introduce a factor of
765:in his honor. Stoney chose his units so that
6676:
6471:
6172:
6010:
5840:"Gravitational fields and quantum mechanics"
5614:
5612:
5573:Physics Meets Philosophy at the Planck Scale
5455:
5386:Kolb, Edward W.; Turner, Michael S. (1994).
5287:
5146:(3B). American Physical Society: B849–B862.
4651:
3179:, which is a time interval of approximately
2461:that would incorporate quantum effects into
800:
6402:
5555:
5553:
5418:
5053:
4943:
4621:
4591:
4558:
3748:in 1915). Hence Planck normalized to 1 the
7937:
7923:
7057:
7050:
7036:
5501:
5385:
4442:
4332:: CS1 maint: location missing publisher (
4277:
3942:in free space to 1 (normally expressed as
3108:Bekenstein–Hawking entropy of a black hole
2770:Table 3: Today's universe in Planck units
2632:
76:, defined using fundamental properties of
6839:
6755:
6718:
6700:
6637:
6492:
6424:
6363:
6269:
6231:
6213:
6195:
6134:Geoffrey Chew: Architect of the Bootstrap
6081:
5731:
5634:
5609:
5517:
5324:
5181:
5120:
4825:
4243:(2005). "On Absolute Units, I: Choices".
3738:
2598:(the Planck energy, corresponding to the
2542:is approximately the unit charge but the
748:
685:being the dimensionless ratio quantities
6733:
5914:. Boston: Birkhäuser. pp. 364–379.
5550:
4735:
4699:
4097:, because the Planck mass is divided by
2987:
2984:throughout the history of the universe.
159:, approximately 13.8 billion years ago.
16:Units defined only by physical constants
6972:
6900:Fundamental Problems in Quantum Physics
6621:
6128:
5904:
5794:
5711:
5293:
5229:
5196:
5089:
5059:
4870:
4662:"2022 CODATA Value: Planck temperature"
4525:
4239:
3403:. Adding energy to such a system might
3236:
2697:Time-variation of fundamental constants
8400:
6894:Rañada, Antonio F. (31 October 1995).
6893:
6854:
6802:. W. H. Freeman Company. p. 117.
6797:
6543:
5946:
5693:from the original on 18 September 2021
5673:
5427:
5240:Communications in Mathematical Physics
5172:
4813:
4459:"Über irreversible Strahlungsvorgänge"
4453:
4394:
4346:
689:the standard quantities, written e.g.
7918:
7031:
6916:from the original on 1 September 2020
6248:
5834:
5772:from the original on 30 December 2017
5714:"Three new roads to the Planck scale"
5618:
5359:from the original on 28 November 2018
5342:
5211:from the original on 30 November 2020
5005:
5003:
4852:from the original on 5 September 2021
4695:
4693:
4691:
4689:
4687:
4521:
4519:
4517:
4473:from the original on 17 November 2020
3700:and are seen in high-energy physics.
3360:
2910:cosmic microwave background radiation
945:, because the modern definitions use
244:Newton's law of universal gravitation
84:) rather than properties of a chosen
6830:
6136:. World Scientific. pp. 72–83.
5559:
5466:The Anthropic Cosmological Principle
5135:
4430:from the original on 20 January 2022
4235:
4233:
4169:
3870:(GEM) equations, which hold in weak
3506:Alternative choices of normalization
2629:of gravity within current theories.
2465:. Such a theory does not yet exist.
7944:
6944:. Oxford: Oxford University Press.
6328:from the original on 15 August 2009
5009:
4979:
4807:
3354:Newtonian attraction between them.
2500:system for historical reasons, the
13:
7080:International System of Units (SI)
7012:Value of the fundamental constants
6525:from the original on 19 March 2022
6110:from the original on 24 March 2022
5992:from the original on 22 March 2022
5886:from the original on 24 March 2022
5712:Faraoni, Valerio (November 2017).
5505:General Relativity and Gravitation
5235:"Particle Creation by Black Holes"
5197:Bingham, Robert (4 October 2006).
5016:Quantum Field Theory in a Nutshell
5000:
4684:
4569:"2022 CODATA Value: Planck length"
4539:. pp. 287–296. Archived from
4514:
3537:. This, along with the concept of
3337:
3023:
886:
14:
8429:
8352:Template:Quantum mechanics topics
7005:
6858:(1983). "Kaluza-Klein Monopole".
6046:10.1038/scientificamerican0505-48
5406:from the original on 6 March 2021
5176:(2002). "Quest For Unification".
4901:from the original on 19 June 2020
4795:from the original on 3 March 2021
4782:
4764:from the original on 6 March 2021
4230:
1804:
1423:
8385:
8373:
8347:
8346:
8068:Topological quantum field theory
5597:from the original on 8 July 2021
5275:from the original on 5 July 2014
4629:"2022 CODATA Value: Planck time"
4599:"2022 CODATA Value: Planck mass"
4499:in this paper correspond to the
4212:propagates at the same speed as
3193:
2992:
1614:
8053:Jackiw–Teitelboim gravity
6966:
6928:
6887:
6848:
6824:
6791:
6727:
6670:
6615:
6570:
6537:
6465:
6396:
6340:
6310:
6166:
6122:
6075:
6004:
5940:
5898:
5828:
5788:
5705:
5667:
5495:
5449:
5373:" at the very beginning of the
5190:
5166:
5129:
5083:
4973:
4864:
4729:
4705:A First Course in String Theory
4199:
4176:
2807:4.35 × 10 s or 1.38 × 10 years
2684:
2570:
2475:
938:{\displaystyle {\sqrt {2\pi }}}
902:to arrive at natural units for
217:
147:are not expected to apply, and
8258:Causal dynamical triangulation
8126:Black hole information paradox
6580:Foundations of Physics Letters
6288:10.1103/PhysRevLett.124.241301
6082:Manoukian, Edouard B. (2016).
4707:. Cambridge University Press.
4388:
4340:
3689:, this unit system is called "
3153:
3110:is one-fourth the area of its
2721:was equal to the Planck time,
1811:
1801:
35:system of units of measurement
1:
8224:Wheeler–DeWitt equation
7090:US customary units (USCS/USC)
7024:from 'Einstein Light' at UNSW
6744:Astronomical Society of India
5952:"What's bad about this habit"
5369:Discusses "Planck time" and "
5069:. New York: Alfred A. Knopf.
4400:"Natural Units Before Planck"
4255:American Institute of Physics
4223:
2713:is the earliest stage of the
2524:comparing apples with oranges
1353:International System of Units
757:was introduced in 1874, when
228:International System of Units
80:(specifically, properties of
8146:Cosmic censorship hypothesis
7965:Batalin–Vilkovisky formalism
6183:Living Reviews in Relativity
5392:. Basic Books. p. 447.
4190:do so, as does the textbook
3778:to 1 (and therefore setting
3411:Nondimensionalized equations
3207:ultra-high-energy cosmic ray
1504:are exact and the values of
1378:{\displaystyle k_{\text{e}}}
653:This last equation (without
7:
6815:To go even farther back, a
5719:American Journal of Physics
5622:American Journal of Physics
4137:
2997:The Planck length, denoted
1485:In SI units, the values of
895:{\displaystyle k_{\rm {B}}}
10:
8434:
8242:Hartle–Hawking state
8141:Bousso's holographic bound
8121:Black hole complementarity
6443:10.1103/PhysRevD.68.045001
6142:10.1142/9789811219832_0013
5021:Princeton University Press
4982:Irish Astronomical Journal
4871:Zeidler, Eberhard (2006).
4487:, which Planck denoted by
3586:higher-dimensional spheres
3229:Proposals for theories of
3097:reduced Compton wavelength
2734:quantum effects of gravity
2694:
2691:Chronology of the universe
2688:
155:of our universe after the
149:quantum effects of gravity
8338:
8305:
8250:
8234:Euclidean quantum gravity
8232:
8211:Canonical quantum gravity
8209:
8176:
8169:
8161:Gravitational singularity
8131:Black-hole thermodynamics
8111:
8076:
8033:
8015:Ryu–Takayanagi conjecture
7952:
7886:
7860:
7804:
7744:
7698:
7617:
7506:
7309:
7275:
7268:
7237:
7206:
7199:
7152:
7143:English Engineering Units
7110:
7072:
7065:
6902:. Springer. p. 271.
6881:10.1103/PhysRevLett.51.87
6817:quantum theory of gravity
6477:"The Soccer-Ball Problem"
6351:The Astrophysical Journal
6094:10.1007/978-3-319-33852-1
5798:(January 1955). "Geons".
5536:10.1007/s10714-011-1199-1
5326:10.4249/scholarpedia.7117
4789:The Physics Hypertextbook
4743:Encyclopedia of Distances
4214:electromagnetic radiation
4154:Doubly special relativity
4063:). This would eliminate 8
3997:) which is simplified to
3698:rationalized Planck units
3231:doubly special relativity
2583:, the Planck scale is an
2564:𝛥𝑥 ≳ √𝐺 = 1.6 × 10 cm
2480:Planck units have little
1351:Unlike the case with the
568:absent, is relating only
8288:Superfluid vacuum theory
8043:2+1D topological gravity
6748:Astrophysics Data System
5571:; Huggett, Nick (eds.).
5160:10.1103/PhysRev.135.B849
4072:Einstein field equations
3928:characteristic impedance
3541:, are the basis for the
3425:energy–momentum relation
3117:foam at the Planck scale
3091:times the diameter of a
2857:(sometimes known as the
2746:unified with gravitation
2640:fundamental interactions
570:dimensionless quantities
558:dimensionally consistent
8278:Noncommutative geometry
8025:Weinberg–Witten theorem
8020:Trans-Planckian problem
7182:Quantum chromodynamical
6800:The Hour of Our Delight
6798:Reeves, Hubert (1991).
6720:10.3390/universe7110403
6656:10.1023/A:1022370717626
5471:Oxford University Press
5345:"Birth of the Universe"
4526:Tomilin, K. A. (1999).
4210:gravitational radiation
4159:Trans-Planckian problem
4076:Einstein–Hilbert action
4040:(and therefore setting
3978:entropy of a black hole
3940:gravitational radiation
3807:Gauss's law for gravity
3676:gravitoelectromagnetism
3365:The Planck temperature
2826:8.7 × 10 m or 9.2 × 10
2750:grand unification epoch
2675:noncommutative geometry
2644:approximate lower limit
2633:Relationship to gravity
2453:does not extend to the
1609:fine-structure constant
1600:{\displaystyle \alpha }
759:George Johnstone Stoney
195:reduced Planck constant
72:. They are a system of
7960:AdS/CFT correspondence
7295:centimetre–gram–second
7059:Systems of measurement
6936:Choquet-Bruhat, Yvonne
6625:Foundations of Physics
6544:Kiefer, Claus (2012).
6511:10.3842/SIGMA.2014.074
5822:10.1103/PhysRev.97.511
4814:Pavšic, Matej (2001).
3876:locally flat spacetime
3868:gravitoelectromagnetic
3750:gravitational constant
3739:Gravitational constant
3345:
3144:
3128:large extra dimensions
3060:
2848:3 × 10 kg or 1.5 × 10
2568:
2519:
2421:
2326:
2177:
2050:
1936:
1838:
1700:
1601:
1581:
1477:
1379:
1324:
1225:
1143:
1068:
979:
959:
958:{\displaystyle \hbar }
939:
896:
867:
847:
827:
811:
806:
801:
749:History and definition
644:
547:
397:
185:gravitational constant
8186:Bosonic string theory
8098:Semiclassical gravity
7995:Holographic principle
7985:Gravitational anomaly
7817:Biblical and Talmudic
7283:metre–kilogram–second
6973:Stavrov, Iva (2020).
6017:"Quantum Black Holes"
4836:10.1007/0-306-47136-1
4785:"Blackbody Radiation"
4353:Space Science Reviews
3980:in terms of its mass
3592:naturally appears in
3423:Examples include the
3418:nondimensionalization
3346:
3145:
3099:is comparable to its
3061:
2988:Analysis of the units
2917:Cosmological constant
2852:(only counting stars)
2695:Further information:
2627:non-renormalizability
2560:
2556:uncertainty principle
2510:
2422:
2327:
2178:
2051:
1937:
1839:
1701:
1602:
1582:
1478:
1380:
1325:
1226:
1144:
1069:
980:
960:
940:
897:
868:
848:
828:
807:
798:
687:corresponding to
645:
560:and equally valid in
548:
406:can be expressed as:
398:
8219:Loop quantum gravity
8078:Quantum field theory
6473:Hossenfelder, Sabine
6174:Hossenfelder, Sabine
6071:on 14 February 2019.
5353:University of Oregon
4736:Deza, Michel Marie;
4149:Dimensional analysis
4095:reduced Planck units
3987:and the area of its
3872:gravitational fields
3563:electrostatic fields
3553:operator applied to
3249:
3237:Planck unit of force
3134:
3101:Schwarzschild radius
3014:
2671:loop quantum gravity
2623:quantum field theory
2602:of the Planck mass,
2508:puts it succinctly:
2357:
2208:
2081:
1963:
1869:
1731:
1653:
1591:
1549:
1389:
1362:
1263:
1181:
1106:
1024:
969:
949:
922:
877:
857:
837:
817:
794:black-body radiation
579:
412:
252:
141:quantum field theory
119:intervals of around
8088:Bunch–Davies vacuum
8080:in curved spacetime
7980:Faddeev–Popov ghost
6873:1983PhRvL..51...87S
6766:2012BASI...40....1A
6711:2021Univ....7..403J
6681:(3 February 2021).
6648:2002FoPh...32.1891G
6593:1993FoPhL...6..561D
6503:2014SIGMA..10..074H
6435:2003PhRvD..68d5001J
6374:1995ApJ...441..144B
6280:2020PhRvL.124x1301W
6215:10.12942/lrr-2013-2
6206:2013LRR....16....2H
6038:2005SciAm.292e..48C
6025:Scientific American
6013:Giddings, Steven B.
5970:2009PhT....62e...8M
5856:1958NCim....7..215R
5814:1955PhRv...97..511W
5742:2017AmJPh..85..865F
5645:2010AmJPh..78..925A
5528:2011GReGr..43.2555B
5317:2008SchpJ...3.7117R
5253:1975CMaPh..43..199H
5152:1964PhRv..135..849M
5113:2001PhT....54f..12W
5066:The Road to Reality
5019:(second ed.).
4994:1981IrAJ...15..152R
4546:on 12 December 2020
4416:1983QJRAS..24...24B
4366:1980SSRv...27..109W
4263:2005PhT....58j..12W
4080:Friedmann equations
3520:theoretical physics
3401:Hawking evaporation
3397:thermal equilibrium
2970:age of the universe
2959:≈ 10 s ≈ 10 (km/s)/
2908:temperature of the
2779:observable universe
2771:
2446:orders of magnitude
2287:
2250:
1748:
1670:
1625:
1315:
1251:Planck temperature
990:
621:
608:
556:Both equations are
350:
334:
230:, for example, the
164:universal constants
8273:Group field theory
8201:Superstring theory
8156:Firewall (physics)
7289:metre–tonne–second
7085:UK imperial system
6601:10.1007/BF00662806
6322:www.cosmic-ray.org
6011:Carr, Bernard J.;
5864:10.1007/BF02744199
5838:(1 January 1958).
5683:Starts with a Bang
5435:. Pantheon Books.
5389:The Early Universe
5343:Schombert, James.
5261:10.1007/BF02345020
4374:10.1007/bf00212237
4288:Misner, Charles W.
4206:General relativity
4182:For example, both
3746:general relativity
3594:Poisson's equation
3543:inverse-square law
3361:Planck temperature
3341:
3198:The Planck energy
3140:
3056:
2783:Approximate number
2769:
2754:inflationary epoch
2703:Big Bang cosmology
2581:physical cosmology
2463:general relativity
2417:
2322:
2273:
2236:
2173:
2046:
1932:
1834:
1734:
1696:
1656:
1623:
1597:
1577:
1473:
1375:
1320:
1301:
1221:
1139:
1064:
988:
975:
955:
935:
892:
863:
843:
823:
790:Wien approximation
640:
609:
596:
543:
529:
500:
466:
393:
336:
320:
232:SI base quantities
208:Boltzmann constant
145:general relativity
39:physical constants
27:physical cosmology
8361:
8360:
8318:Eternal inflation
8313:Quantum cosmology
8301:
8300:
8093:Hawking radiation
8058:Liouville gravity
7912:
7911:
7800:
7799:
7264:
7263:
7255:Foot–pound–second
7195:
7194:
7172:Heaviside–Lorentz
6986:978-1-4704-6313-7
6951:978-0-19-155226-7
6809:978-0-7167-2220-5
6632:(12): 1891–1901.
6555:978-0-191-62885-6
6413:Physical Review D
6407:(4 August 2003).
6176:(December 2013).
6151:978-981-12-1982-5
6103:978-3-319-33851-4
5978:10.1063/1.3141952
5936:on 25 April 2019.
5750:10.1119/1.4994804
5653:10.1119/1.3439650
5582:978-0-521-66280-2
5512:(10): 2555–2560.
5480:978-0-19-282147-8
5399:978-0-201-62674-2
5349:HC 441: Cosmology
5299:"Quantum gravity"
5122:10.1063/1.1387576
5076:978-0-679-45443-4
5034:978-0-691-14034-6
4845:978-0-7923-7006-2
4714:978-0-521-83143-7
4271:10.1063/1.2138392
4170:Explanatory notes
4164:Zero-point energy
3842:in Planck units).
3578:rationalized form
3518:is ubiquitous in
3386:thermal radiation
3336:
3312:
3292:
3289:
3275:
3259:
3143:{\displaystyle G}
3054:
3053:
2966:
2965:
2744:is assumed to be
2679:causal set theory
2655:quantum mechanics
2600:energy equivalent
2492:, which exist as
2442:
2441:
2415:
2414:
2388:
2385:
2367:
2320:
2288:
2280:
2269:
2251:
2243:
2233:
2218:
2171:
2151:
2147:
2137:
2119:
2116:
2106:
2091:
2044:
2043:
2017:
2014:
1986:
1973:
1930:
1929:
1903:
1900:
1879:
1832:
1831:
1791:
1777:
1741:
1694:
1663:
1569:
1466:
1456:
1429:
1399:
1372:
1349:
1348:
1318:
1317:
1308:
1273:
1219:
1218:
1191:
1137:
1136:
1116:
1062:
1061:
1034:
978:{\displaystyle h}
933:
866:{\displaystyle c}
846:{\displaystyle h}
826:{\displaystyle G}
638:
541:
528:
525:
499:
496:
465:
462:
430:
427:
391:
351:
343:
327:
317:
298:
8425:
8390:
8389:
8378:
8377:
8369:
8350:
8349:
8174:
8173:
8136:Bekenstein bound
7953:Central concepts
7939:
7932:
7925:
7916:
7915:
7379:Mesures usuelles
7273:
7272:
7204:
7203:
7070:
7069:
7052:
7045:
7038:
7029:
7028:
6999:
6998:
6970:
6964:
6963:
6932:
6926:
6925:
6923:
6921:
6891:
6885:
6884:
6861:Phys. Rev. Lett.
6852:
6846:
6845:
6843:
6828:
6822:
6821:
6795:
6789:
6788:
6759:
6731:
6725:
6724:
6722:
6704:
6674:
6668:
6667:
6641:
6619:
6613:
6612:
6574:
6568:
6567:
6541:
6535:
6534:
6532:
6530:
6496:
6469:
6463:
6462:
6428:
6400:
6394:
6393:
6367:
6365:astro-ph/9410067
6344:
6338:
6337:
6335:
6333:
6314:
6308:
6307:
6273:
6258:Phys. Rev. Lett.
6252:
6246:
6245:
6235:
6217:
6199:
6170:
6164:
6163:
6130:Schwarz, John H.
6126:
6120:
6119:
6117:
6115:
6079:
6073:
6072:
6070:
6064:. Archived from
6021:
6008:
6002:
6001:
5999:
5997:
5948:Mermin, N. David
5944:
5938:
5937:
5932:. Archived from
5906:Gorelik, Gennady
5902:
5896:
5895:
5893:
5891:
5844:Il Nuovo Cimento
5832:
5826:
5825:
5792:
5786:
5785:
5779:
5777:
5735:
5709:
5703:
5702:
5700:
5698:
5677:(26 June 2019).
5671:
5665:
5664:
5638:
5616:
5607:
5606:
5604:
5602:
5569:Callender, Craig
5557:
5548:
5547:
5521:
5499:
5493:
5492:
5469:(1st ed.).
5461:Tipler, Frank J.
5453:
5447:
5446:
5425:
5416:
5415:
5413:
5411:
5383:
5377:
5368:
5366:
5364:
5340:
5331:
5330:
5328:
5291:
5285:
5284:
5282:
5280:
5227:
5221:
5220:
5218:
5216:
5194:
5188:
5187:
5185:
5170:
5164:
5163:
5133:
5127:
5126:
5124:
5087:
5081:
5080:
5057:
5051:
5050:
5007:
4998:
4997:
4977:
4971:
4970:
4968:
4966:
4947:
4941:
4940:
4938:
4936:
4917:
4911:
4910:
4908:
4906:
4900:
4879:
4868:
4862:
4861:
4859:
4857:
4829:
4811:
4805:
4804:
4802:
4800:
4780:
4774:
4773:
4771:
4769:
4733:
4727:
4726:
4701:Zwiebach, Barton
4697:
4682:
4681:
4679:
4677:
4658:
4649:
4648:
4646:
4644:
4625:
4619:
4618:
4616:
4614:
4595:
4589:
4588:
4586:
4584:
4565:
4556:
4555:
4553:
4551:
4545:
4534:
4523:
4512:
4511:in this article.
4482:
4480:
4478:
4451:
4440:
4439:
4437:
4435:
4398:(1 March 1983).
4392:
4386:
4385:
4344:
4338:
4337:
4331:
4323:
4296:Wheeler, John A.
4284:
4275:
4274:
4237:
4217:
4203:
4197:
4180:
4132:
4125:
4106:
4105:
4104:
4092:
4087:
4066:
4062:
4060:
4059:
4058:
4053:
4050:
4039:
4034:
4023:
4008:
3972:
3965:
3963:
3962:
3957:
3954:
3950:
3915:
3913:
3912:
3908:
3903:
3900:
3884:electromagnetism
3862:
3854:Poisson equation
3848:
3841:
3837:
3823:
3802:
3801:
3799:
3798:
3797:
3792:
3789:
3774:
3766:
3762:
3731:
3695:
3681:
3673:
3671:
3670:
3669:
3664:
3661:
3654:
3652:
3651:
3650:
3645:
3642:
3635:
3633:
3632:
3631:
3626:
3623:
3613:
3606:
3591:
3572:
3533:
3517:
3501:
3480:, which becomes
3479:
3454:
3441:, which becomes
3440:
3383:
3381:
3377:
3350:
3348:
3347:
3342:
3340:
3334:
3333:
3332:
3313:
3308:
3307:
3298:
3293:
3291:
3290:
3287:
3281:
3277:
3276:
3273:
3266:
3261:
3260:
3257:
3225:
3217:
3210:observed in 1991
3189:
3184:
3166:
3158:The Planck time
3149:
3147:
3146:
3141:
3090:
3082:
3079:
3077:
3073:
3065:
3063:
3062:
3057:
3055:
3052:
3051:
3042:
3034:
3033:
3028:
3027:
3026:
3005:
2956:
2955:
2932:
2931:
2859:Eddington number
2785:of Planck units
2772:
2768:
2612:
2610:
2607:
2597:
2592:
2577:particle physics
2565:
2438:
2433:
2426:
2424:
2423:
2418:
2416:
2413:
2405:
2404:
2395:
2394:
2389:
2387:
2386:
2383:
2374:
2369:
2368:
2365:
2343:
2338:
2331:
2329:
2328:
2323:
2321:
2319:
2318:
2317:
2304:
2303:
2294:
2289:
2286:
2281:
2278:
2272:
2271:
2270:
2267:
2257:
2252:
2249:
2244:
2241:
2235:
2234:
2231:
2225:
2220:
2219:
2216:
2194:
2189:
2182:
2180:
2179:
2174:
2172:
2167:
2166:
2157:
2152:
2150:
2149:
2148:
2145:
2139:
2138:
2135:
2125:
2120:
2118:
2117:
2114:
2108:
2107:
2104:
2098:
2093:
2092:
2089:
2067:
2062:
2055:
2053:
2052:
2047:
2045:
2039:
2038:
2037:
2024:
2023:
2018:
2016:
2015:
2012:
2003:
1998:
1997:
1988:
1987:
1984:
1975:
1974:
1971:
1949:
1941:
1939:
1938:
1933:
1931:
1925:
1924:
1923:
1910:
1909:
1904:
1902:
1901:
1898:
1889:
1881:
1880:
1877:
1855:
1850:
1843:
1841:
1840:
1835:
1833:
1830:
1829:
1820:
1819:
1818:
1799:
1798:
1793:
1792:
1784:
1782:
1778:
1776:
1775:
1766:
1758:
1747:
1742:
1739:
1717:
1712:
1705:
1703:
1702:
1697:
1695:
1693:
1692:
1683:
1675:
1669:
1664:
1661:
1629:Derived unit of
1626:
1622:
1606:
1604:
1603:
1598:
1586:
1584:
1583:
1578:
1570:
1568:
1563:
1558:
1553:
1533:
1531:
1525:
1522:
1520:
1482:
1480:
1479:
1474:
1464:
1457:
1454:
1452:
1451:
1430:
1422:
1421:
1406:
1401:
1400:
1397:
1384:
1382:
1381:
1376:
1374:
1373:
1370:
1357:Coulomb constant
1345:
1342:
1340:
1336:
1329:
1327:
1326:
1321:
1319:
1316:
1314:
1309:
1306:
1296:
1295:
1294:
1281:
1280:
1275:
1274:
1271:
1246:
1243:
1241:
1237:
1230:
1228:
1227:
1222:
1220:
1217:
1216:
1207:
1199:
1198:
1193:
1192:
1189:
1164:
1161:
1159:
1155:
1148:
1146:
1145:
1140:
1138:
1132:
1124:
1123:
1118:
1117:
1114:
1089:
1086:
1084:
1080:
1073:
1071:
1070:
1065:
1063:
1060:
1059:
1050:
1042:
1041:
1036:
1035:
1032:
991:
987:
984:
982:
981:
976:
964:
962:
961:
956:
944:
942:
941:
936:
934:
926:
901:
899:
898:
893:
891:
890:
889:
872:
870:
869:
864:
852:
850:
849:
844:
832:
830:
829:
824:
804:
740:
721:
709:
701:
696:
684:
663:
657:) is valid with
649:
647:
646:
641:
639:
637:
636:
635:
622:
617:
604:
594:
589:
552:
550:
549:
544:
542:
540:
539:
534:
530:
527:
526:
523:
514:
506:
505:
501:
498:
497:
494:
488:
487:
478:
471:
467:
464:
463:
460:
454:
453:
444:
436:
431:
429:
428:
425:
416:
402:
400:
399:
394:
392:
390:
389:
380:
379:
378:
369:
368:
358:
356:
352:
349:
344:
341:
335:
333:
328:
325:
319:
318:
315:
308:
299:
297:
296:
287:
286:
285:
276:
275:
265:
153:first 10 seconds
134:
126:
124:
114:
110:
86:prototype object
23:particle physics
8433:
8432:
8428:
8427:
8426:
8424:
8423:
8422:
8418:Quantum gravity
8398:
8397:
8396:
8384:
8372:
8364:
8362:
8357:
8356:
8334:
8323:FRW/CFT duality
8297:
8246:
8228:
8205:
8165:
8107:
8079:
8072:
8029:
7948:
7946:Quantum gravity
7943:
7913:
7908:
7882:
7856:
7796:
7740:
7694:
7613:
7502:
7305:
7260:
7233:
7191:
7148:
7106:
7061:
7056:
7008:
7003:
7002:
6987:
6971:
6967:
6952:
6933:
6929:
6919:
6917:
6910:
6892:
6888:
6853:
6849:
6829:
6825:
6810:
6796:
6792:
6785:
6732:
6728:
6675:
6671:
6620:
6616:
6575:
6571:
6556:
6546:Quantum Gravity
6542:
6538:
6528:
6526:
6475:(9 July 2014).
6470:
6466:
6401:
6397:
6345:
6341:
6331:
6329:
6316:
6315:
6311:
6253:
6249:
6171:
6167:
6152:
6127:
6123:
6113:
6111:
6104:
6080:
6076:
6068:
6019:
6009:
6005:
5995:
5993:
5945:
5941:
5922:
5903:
5899:
5889:
5887:
5833:
5829:
5801:Physical Review
5793:
5789:
5775:
5773:
5726:(11): 865–869.
5710:
5706:
5696:
5694:
5672:
5668:
5617:
5610:
5600:
5598:
5583:
5558:
5551:
5500:
5496:
5481:
5457:Barrow, John D.
5454:
5450:
5443:
5429:Barrow, John D.
5426:
5419:
5409:
5407:
5400:
5384:
5380:
5362:
5360:
5341:
5334:
5292:
5288:
5278:
5276:
5228:
5224:
5214:
5212:
5195:
5191:
5171:
5167:
5139:Physical Review
5134:
5130:
5088:
5084:
5077:
5058:
5054:
5035:
5008:
5001:
4978:
4974:
4964:
4962:
4949:
4948:
4944:
4934:
4932:
4919:
4918:
4914:
4904:
4902:
4898:
4892:
4884:. p. 953.
4877:
4869:
4865:
4855:
4853:
4846:
4812:
4808:
4798:
4796:
4781:
4777:
4767:
4765:
4758:
4750:. p. 602.
4734:
4730:
4715:
4698:
4685:
4675:
4673:
4660:
4659:
4652:
4642:
4640:
4627:
4626:
4622:
4612:
4610:
4597:
4596:
4592:
4582:
4580:
4567:
4566:
4559:
4549:
4547:
4543:
4532:
4524:
4515:
4485:Planck constant
4476:
4474:
4452:
4443:
4433:
4431:
4393:
4389:
4345:
4341:
4325:
4324:
4312:
4285:
4278:
4238:
4231:
4226:
4221:
4220:
4204:
4200:
4188:Barton Zwiebach
4181:
4177:
4172:
4140:
4131:
4123:
4121:
4114:
4108:
4102:
4100:
4098:
4085:
4083:
4064:
4056:
4054:
4051:
4048:
4047:
4045:
4032:
4030:
4021:
4014:
4006:
4004:
3998:
3996:
3986:
3970:
3958:
3955:
3948:
3946:
3945:
3943:
3937:
3926:Normalizes the
3922:
3906:
3904:
3901:
3898:
3897:
3895:
3860:
3846:
3835:
3833:
3825:
3818:
3810:
3795:
3793:
3790:
3787:
3786:
3784:
3779:
3772:
3764:
3760:
3741:
3729:
3722:
3704:
3693:
3688:
3679:
3667:
3665:
3662:
3659:
3658:
3656:
3648:
3646:
3643:
3640:
3639:
3637:
3629:
3627:
3624:
3621:
3620:
3618:
3611:
3604:
3589:
3570:
3557:. For example,
3531:
3515:
3508:
3491:
3481:
3469:
3459:
3442:
3427:
3413:
3394:
3379:
3375:
3373:
3371:
3363:
3328:
3324:
3317:
3303:
3299:
3297:
3286:
3282:
3272:
3268:
3267:
3265:
3256:
3252:
3250:
3247:
3246:
3239:
3224:
3215:
3213:
3204:
3196:
3182:
3180:
3165:
3162:
3159:
3156:
3135:
3132:
3131:
3122:The strings of
3088:
3080:
3075:
3071:
3069:
3068:It is equal to
3047:
3043:
3035:
3032:
3022:
3021:
3017:
3015:
3012:
3011:
3006:, is a unit of
3004:
3001:
2998:
2995:
2990:
2954:
2951:
2950:
2949:
2941:Hubble constant
2930:
2927:
2926:
2925:
2907:
2903:
2884:
2877:
2853:
2845:
2823:
2804:
2784:
2776:
2762:
2727:
2699:
2693:
2687:
2659:quantum gravity
2635:
2615:quantum effects
2608:
2605:
2603:
2590:
2588:
2573:
2563:
2536:incommensurable
2532:electric charge
2482:anthropocentric
2478:
2459:quantum gravity
2431:
2429:
2406:
2400:
2396:
2393:
2382:
2378:
2373:
2364:
2360:
2358:
2355:
2354:
2336:
2334:
2313:
2309:
2305:
2299:
2295:
2293:
2282:
2277:
2266:
2262:
2258:
2256:
2245:
2240:
2230:
2226:
2224:
2215:
2211:
2209:
2206:
2205:
2187:
2185:
2162:
2158:
2156:
2144:
2140:
2134:
2130:
2129:
2124:
2113:
2109:
2103:
2099:
2097:
2088:
2084:
2082:
2079:
2078:
2060:
2058:
2033:
2029:
2025:
2022:
2011:
2007:
2002:
1993:
1989:
1983:
1979:
1970:
1966:
1964:
1961:
1960:
1944:
1919:
1915:
1911:
1908:
1897:
1893:
1888:
1876:
1872:
1870:
1867:
1866:
1848:
1846:
1825:
1821:
1814:
1810:
1800:
1797:
1783:
1771:
1767:
1759:
1757:
1753:
1752:
1743:
1738:
1732:
1729:
1728:
1710:
1708:
1688:
1684:
1676:
1674:
1665:
1660:
1654:
1651:
1650:
1617:
1592:
1589:
1588:
1564:
1559:
1554:
1552:
1550:
1547:
1546:
1529:
1527:
1523:
1518:
1516:
1510:
1503:
1453:
1444:
1440:
1417:
1413:
1405:
1396:
1392:
1390:
1387:
1386:
1369:
1365:
1363:
1360:
1359:
1343:
1338:
1334:
1332:
1310:
1305:
1297:
1290:
1286:
1282:
1279:
1270:
1266:
1264:
1261:
1260:
1244:
1239:
1235:
1233:
1212:
1208:
1200:
1197:
1188:
1184:
1182:
1179:
1178:
1162:
1157:
1153:
1151:
1125:
1122:
1113:
1109:
1107:
1104:
1103:
1087:
1082:
1078:
1076:
1055:
1051:
1043:
1040:
1031:
1027:
1025:
1022:
1021:
970:
967:
966:
950:
947:
946:
925:
923:
920:
919:
885:
884:
880:
878:
875:
874:
858:
855:
854:
838:
835:
834:
818:
815:
814:
775:electron charge
753:The concept of
751:
731:
720:
707:
703:
694:
690:
682:
677:
670:
661:
651:
631:
627:
623:
613:
600:
595:
593:
582:
580:
577:
576:
554:
535:
522:
518:
512:
508:
507:
493:
489:
483:
479:
476:
472:
459:
455:
449:
445:
442:
438:
437:
435:
424:
420:
415:
413:
410:
409:
404:
385:
381:
374:
370:
364:
360:
359:
357:
345:
340:
329:
324:
314:
310:
309:
307:
303:
292:
288:
281:
277:
271:
267:
266:
264:
253:
250:
249:
220:
205:
132:
122:
120:
112:
108:
94:quantum gravity
66:
17:
12:
11:
5:
8431:
8421:
8420:
8415:
8410:
8395:
8394:
8382:
8359:
8358:
8355:
8354:
8340:
8339:
8336:
8335:
8333:
8332:
8331:
8330:
8325:
8320:
8309:
8307:
8303:
8302:
8299:
8298:
8296:
8295:
8293:Twistor theory
8290:
8285:
8280:
8275:
8270:
8265:
8260:
8254:
8252:
8248:
8247:
8245:
8244:
8238:
8236:
8230:
8229:
8227:
8226:
8221:
8215:
8213:
8207:
8206:
8204:
8203:
8198:
8193:
8188:
8182:
8180:
8171:
8167:
8166:
8164:
8163:
8158:
8153:
8148:
8143:
8138:
8133:
8128:
8123:
8117:
8115:
8109:
8108:
8106:
8105:
8100:
8095:
8090:
8084:
8082:
8074:
8073:
8071:
8070:
8065:
8060:
8055:
8050:
8045:
8039:
8037:
8031:
8030:
8028:
8027:
8022:
8017:
8012:
8007:
8002:
7997:
7992:
7987:
7982:
7977:
7972:
7967:
7962:
7956:
7954:
7950:
7949:
7942:
7941:
7934:
7927:
7919:
7910:
7909:
7907:
7906:
7901:
7896:
7894:Absolute scale
7890:
7888:
7884:
7883:
7881:
7880:
7875:
7870:
7864:
7862:
7858:
7857:
7855:
7854:
7849:
7844:
7839:
7834:
7829:
7824:
7819:
7814:
7808:
7806:
7802:
7801:
7798:
7797:
7795:
7794:
7789:
7784:
7779:
7774:
7769:
7764:
7759:
7754:
7748:
7746:
7742:
7741:
7739:
7738:
7733:
7728:
7723:
7718:
7713:
7708:
7702:
7700:
7696:
7695:
7693:
7692:
7687:
7682:
7677:
7672:
7667:
7662:
7657:
7652:
7647:
7642:
7637:
7632:
7627:
7621:
7619:
7615:
7614:
7612:
7611:
7606:
7601:
7596:
7591:
7586:
7581:
7576:
7571:
7566:
7561:
7556:
7551:
7546:
7541:
7536:
7531:
7526:
7521:
7516:
7510:
7508:
7504:
7503:
7501:
7500:
7495:
7490:
7485:
7480:
7475:
7470:
7465:
7460:
7455:
7450:
7445:
7440:
7435:
7430:
7425:
7420:
7415:
7410:
7405:
7400:
7399:
7398:
7388:
7383:
7382:
7381:
7376:
7366:
7361:
7356:
7355:
7354:
7349:
7339:
7334:
7329:
7324:
7319:
7313:
7311:
7307:
7306:
7304:
7303:
7298:
7292:
7286:
7279:
7277:
7270:
7266:
7265:
7262:
7261:
7259:
7258:
7252:
7247:
7241:
7239:
7235:
7234:
7232:
7231:
7226:
7221:
7216:
7210:
7208:
7201:
7197:
7196:
7193:
7192:
7190:
7189:
7184:
7179:
7174:
7169:
7164:
7158:
7156:
7150:
7149:
7147:
7146:
7140:
7135:
7130:
7125:
7120:
7114:
7112:
7108:
7107:
7105:
7104:
7103:
7102:
7092:
7087:
7082:
7076:
7074:
7067:
7063:
7062:
7055:
7054:
7047:
7040:
7032:
7026:
7025:
7019:
7007:
7006:External links
7004:
7001:
7000:
6985:
6965:
6950:
6927:
6908:
6886:
6856:Sorkin, Rafael
6847:
6832:Shor, Peter W.
6823:
6808:
6790:
6783:
6726:
6677:Jowsey, Aden;
6669:
6639:hep-th/0210109
6614:
6587:(6): 561–570.
6569:
6554:
6536:
6464:
6403:Judes, Simon;
6395:
6382:10.1086/175344
6339:
6309:
6264:(24): 241301.
6247:
6165:
6150:
6121:
6102:
6074:
6003:
5939:
5920:
5897:
5850:(2): 215–221.
5827:
5808:(2): 511–536.
5796:Wheeler, J. A.
5787:
5704:
5666:
5629:(9): 925–932.
5608:
5581:
5549:
5494:
5479:
5448:
5441:
5417:
5398:
5378:
5332:
5295:Rovelli, Carlo
5286:
5247:(3): 199–220.
5231:Hawking, S. W.
5222:
5189:
5183:hep-ph/0207124
5165:
5128:
5091:Wilczek, Frank
5082:
5075:
5061:Penrose, Roger
5052:
5033:
4999:
4972:
4942:
4912:
4891:978-3540347620
4890:
4863:
4844:
4806:
4783:Elert, Glenn.
4775:
4757:978-3662528433
4756:
4728:
4713:
4683:
4650:
4620:
4590:
4557:
4513:
4441:
4387:
4339:
4310:
4292:Thorne, Kip S.
4276:
4241:Wilczek, Frank
4228:
4227:
4225:
4222:
4219:
4218:
4208:predicts that
4198:
4174:
4173:
4171:
4168:
4167:
4166:
4161:
4156:
4151:
4146:
4139:
4136:
4135:
4134:
4129:
4119:
4112:
4027:
4026:
4025:
4019:
4012:
4002:
3994:
3984:
3967:
3933:
3924:
3920:
3892:charge density
3857:
3843:
3829:
3814:
3740:
3737:
3727:
3720:
3686:
3507:
3504:
3487:
3465:
3457:Dirac equation
3412:
3409:
3392:
3369:
3362:
3359:
3339:
3331:
3327:
3323:
3320:
3316:
3311:
3306:
3302:
3296:
3285:
3280:
3271:
3264:
3255:
3238:
3235:
3222:
3202:
3195:
3192:
3163:
3160:
3155:
3152:
3139:
3085:standard error
3050:
3046:
3041:
3038:
3031:
3025:
3020:
3002:
2999:
2994:
2991:
2989:
2986:
2964:
2963:
2957:
2952:
2943:
2937:
2936:
2933:
2928:
2919:
2913:
2912:
2904:
2901:
2895:
2889:
2888:
2887:9.9 × 10 kg⋅m
2885:
2882:
2875:
2869:
2863:
2862:
2846:
2843:
2837:
2831:
2830:
2824:
2821:
2815:
2809:
2808:
2805:
2802:
2796:
2790:
2789:
2786:
2781:
2760:
2742:Standard Model
2725:
2689:Main article:
2686:
2683:
2634:
2631:
2572:
2569:
2477:
2474:
2440:
2439:
2427:
2412:
2409:
2403:
2399:
2392:
2381:
2377:
2372:
2363:
2352:
2345:
2344:
2332:
2316:
2312:
2308:
2302:
2298:
2292:
2285:
2276:
2265:
2261:
2255:
2248:
2239:
2229:
2223:
2214:
2203:
2196:
2195:
2183:
2170:
2165:
2161:
2155:
2143:
2133:
2128:
2123:
2112:
2102:
2096:
2087:
2076:
2069:
2068:
2056:
2042:
2036:
2032:
2028:
2021:
2010:
2006:
2001:
1996:
1992:
1982:
1978:
1969:
1958:
1951:
1950:
1942:
1928:
1922:
1918:
1914:
1907:
1896:
1892:
1887:
1884:
1875:
1864:
1857:
1856:
1844:
1828:
1824:
1817:
1813:
1809:
1806:
1803:
1796:
1790:
1787:
1781:
1774:
1770:
1765:
1762:
1756:
1751:
1746:
1737:
1726:
1719:
1718:
1706:
1691:
1687:
1682:
1679:
1673:
1668:
1659:
1648:
1641:
1640:
1633:
1630:
1616:
1613:
1596:
1576:
1573:
1567:
1562:
1557:
1508:
1501:
1472:
1469:
1463:
1460:
1450:
1447:
1443:
1439:
1436:
1433:
1428:
1425:
1420:
1416:
1412:
1409:
1404:
1395:
1368:
1347:
1346:
1330:
1313:
1304:
1300:
1293:
1289:
1285:
1278:
1269:
1258:
1252:
1248:
1247:
1231:
1215:
1211:
1206:
1203:
1196:
1187:
1176:
1170:
1166:
1165:
1149:
1135:
1131:
1128:
1121:
1112:
1101:
1095:
1091:
1090:
1074:
1058:
1054:
1049:
1046:
1039:
1030:
1019:
1013:
1012:Planck length
1009:
1008:
1001:
998:
995:
974:
954:
932:
929:
888:
883:
862:
842:
822:
750:
747:
743:Paul S. Wesson
718:
675:
668:
634:
630:
626:
620:
616:
612:
607:
603:
599:
592:
588:
585:
575:
538:
533:
521:
517:
511:
504:
492:
486:
482:
475:
470:
458:
452:
448:
441:
434:
423:
419:
408:
388:
384:
377:
373:
367:
363:
355:
348:
339:
332:
323:
313:
306:
302:
295:
291:
284:
280:
274:
270:
263:
260:
257:
248:
219:
216:
212:
211:
203:
198:
188:
178:
175:speed of light
137:Standard Model
64:
15:
9:
6:
4:
3:
2:
8430:
8419:
8416:
8414:
8411:
8409:
8408:Natural units
8406:
8405:
8403:
8393:
8388:
8383:
8381:
8376:
8371:
8370:
8367:
8353:
8345:
8342:
8341:
8337:
8329:
8326:
8324:
8321:
8319:
8316:
8315:
8314:
8311:
8310:
8308:
8304:
8294:
8291:
8289:
8286:
8284:
8281:
8279:
8276:
8274:
8271:
8269:
8268:Dual graviton
8266:
8264:
8261:
8259:
8256:
8255:
8253:
8249:
8243:
8240:
8239:
8237:
8235:
8231:
8225:
8222:
8220:
8217:
8216:
8214:
8212:
8208:
8202:
8199:
8197:
8194:
8192:
8189:
8187:
8184:
8183:
8181:
8179:
8178:String theory
8175:
8172:
8168:
8162:
8159:
8157:
8154:
8152:
8149:
8147:
8144:
8142:
8139:
8137:
8134:
8132:
8129:
8127:
8124:
8122:
8119:
8118:
8116:
8114:
8110:
8104:
8101:
8099:
8096:
8094:
8091:
8089:
8086:
8085:
8083:
8081:
8075:
8069:
8066:
8064:
8061:
8059:
8056:
8054:
8051:
8049:
8046:
8044:
8041:
8040:
8038:
8036:
8032:
8026:
8023:
8021:
8018:
8016:
8013:
8011:
8008:
8006:
8003:
8001:
7998:
7996:
7993:
7991:
7988:
7986:
7983:
7981:
7978:
7976:
7973:
7971:
7968:
7966:
7963:
7961:
7958:
7957:
7955:
7951:
7947:
7940:
7935:
7933:
7928:
7926:
7921:
7920:
7917:
7905:
7902:
7900:
7897:
7895:
7892:
7891:
7889:
7885:
7879:
7876:
7874:
7871:
7869:
7866:
7865:
7863:
7861:List articles
7859:
7853:
7850:
7848:
7845:
7843:
7840:
7838:
7835:
7833:
7830:
7828:
7825:
7823:
7820:
7818:
7815:
7813:
7810:
7809:
7807:
7803:
7793:
7790:
7788:
7785:
7783:
7780:
7778:
7775:
7773:
7770:
7768:
7765:
7763:
7760:
7758:
7755:
7753:
7750:
7749:
7747:
7745:South America
7743:
7737:
7734:
7732:
7729:
7727:
7724:
7722:
7719:
7717:
7714:
7712:
7709:
7707:
7704:
7703:
7701:
7699:North America
7697:
7691:
7688:
7686:
7683:
7681:
7680:South African
7678:
7676:
7673:
7671:
7668:
7666:
7663:
7661:
7658:
7656:
7653:
7651:
7648:
7646:
7643:
7641:
7638:
7636:
7633:
7631:
7628:
7626:
7623:
7622:
7620:
7616:
7610:
7607:
7605:
7602:
7600:
7597:
7595:
7592:
7590:
7587:
7585:
7582:
7580:
7577:
7575:
7572:
7570:
7567:
7565:
7562:
7560:
7557:
7555:
7552:
7550:
7547:
7545:
7542:
7540:
7537:
7535:
7532:
7530:
7527:
7525:
7522:
7520:
7517:
7515:
7512:
7511:
7509:
7505:
7499:
7496:
7494:
7491:
7489:
7486:
7484:
7481:
7479:
7476:
7474:
7471:
7469:
7466:
7464:
7461:
7459:
7456:
7454:
7451:
7449:
7446:
7444:
7441:
7439:
7436:
7434:
7431:
7429:
7428:Luxembourgian
7426:
7424:
7421:
7419:
7416:
7414:
7411:
7409:
7406:
7404:
7401:
7397:
7394:
7393:
7392:
7389:
7387:
7384:
7380:
7377:
7375:
7372:
7371:
7370:
7367:
7365:
7362:
7360:
7357:
7353:
7350:
7348:
7345:
7344:
7343:
7340:
7338:
7335:
7333:
7330:
7328:
7325:
7323:
7320:
7318:
7315:
7314:
7312:
7308:
7302:
7301:gravitational
7299:
7296:
7293:
7290:
7287:
7284:
7281:
7280:
7278:
7274:
7271:
7267:
7256:
7253:
7251:
7248:
7246:
7243:
7242:
7240:
7236:
7230:
7227:
7225:
7222:
7220:
7217:
7215:
7212:
7211:
7209:
7205:
7202:
7198:
7188:
7185:
7183:
7180:
7178:
7175:
7173:
7170:
7168:
7165:
7163:
7160:
7159:
7157:
7155:
7151:
7144:
7141:
7139:
7136:
7134:
7131:
7129:
7126:
7124:
7121:
7119:
7118:Apothecaries'
7116:
7115:
7113:
7109:
7101:
7098:
7097:
7096:
7093:
7091:
7088:
7086:
7083:
7081:
7078:
7077:
7075:
7071:
7068:
7064:
7060:
7053:
7048:
7046:
7041:
7039:
7034:
7033:
7030:
7023:
7020:
7017:
7013:
7010:
7009:
6996:
6992:
6988:
6982:
6978:
6977:
6969:
6961:
6957:
6953:
6947:
6943:
6942:
6937:
6931:
6915:
6911:
6909:9780792336709
6905:
6901:
6897:
6890:
6882:
6878:
6874:
6870:
6866:
6863:
6862:
6857:
6851:
6842:
6837:
6833:
6827:
6820:
6818:
6811:
6805:
6801:
6794:
6787:
6782:
6775:
6771:
6767:
6763:
6758:
6753:
6749:
6745:
6741:
6737:
6730:
6721:
6716:
6712:
6708:
6703:
6698:
6694:
6690:
6689:
6684:
6680:
6673:
6665:
6661:
6657:
6653:
6649:
6645:
6640:
6635:
6631:
6627:
6626:
6618:
6610:
6606:
6602:
6598:
6594:
6590:
6586:
6582:
6581:
6573:
6565:
6561:
6557:
6551:
6547:
6540:
6524:
6520:
6516:
6512:
6508:
6504:
6500:
6495:
6490:
6486:
6482:
6478:
6474:
6468:
6460:
6456:
6452:
6448:
6444:
6440:
6436:
6432:
6427:
6426:gr-qc/0205067
6422:
6419:(4): 045001.
6418:
6414:
6410:
6406:
6399:
6391:
6387:
6383:
6379:
6375:
6371:
6366:
6361:
6357:
6353:
6352:
6343:
6327:
6323:
6319:
6313:
6305:
6301:
6297:
6293:
6289:
6285:
6281:
6277:
6272:
6267:
6263:
6260:
6259:
6251:
6243:
6239:
6234:
6229:
6225:
6221:
6216:
6211:
6207:
6203:
6198:
6193:
6189:
6185:
6184:
6179:
6175:
6169:
6161:
6157:
6153:
6147:
6143:
6139:
6135:
6131:
6125:
6109:
6105:
6099:
6095:
6091:
6087:
6086:
6078:
6067:
6063:
6059:
6055:
6051:
6047:
6043:
6039:
6035:
6031:
6027:
6026:
6018:
6014:
6007:
5991:
5987:
5983:
5979:
5975:
5971:
5967:
5963:
5959:
5958:
5957:Physics Today
5953:
5949:
5943:
5935:
5931:
5927:
5923:
5921:0-8176-3479-7
5917:
5913:
5912:
5907:
5901:
5885:
5881:
5877:
5873:
5869:
5865:
5861:
5857:
5853:
5849:
5845:
5841:
5837:
5831:
5823:
5819:
5815:
5811:
5807:
5803:
5802:
5797:
5791:
5784:
5771:
5767:
5763:
5759:
5755:
5751:
5747:
5743:
5739:
5734:
5729:
5725:
5721:
5720:
5715:
5708:
5692:
5688:
5684:
5680:
5676:
5675:Siegel, Ethan
5670:
5662:
5658:
5654:
5650:
5646:
5642:
5637:
5632:
5628:
5624:
5623:
5615:
5613:
5596:
5592:
5588:
5584:
5578:
5574:
5570:
5566:
5562:
5556:
5554:
5545:
5541:
5537:
5533:
5529:
5525:
5520:
5515:
5511:
5507:
5506:
5498:
5490:
5486:
5482:
5476:
5472:
5468:
5467:
5462:
5458:
5452:
5444:
5442:0-375-42221-8
5438:
5434:
5430:
5424:
5422:
5405:
5401:
5395:
5391:
5390:
5382:
5376:
5372:
5358:
5354:
5350:
5346:
5339:
5337:
5327:
5322:
5318:
5314:
5310:
5306:
5305:
5300:
5296:
5290:
5274:
5270:
5266:
5262:
5258:
5254:
5250:
5246:
5242:
5241:
5236:
5232:
5226:
5210:
5206:
5205:
5200:
5193:
5184:
5179:
5175:
5169:
5161:
5157:
5153:
5149:
5145:
5141:
5140:
5132:
5123:
5118:
5114:
5110:
5106:
5102:
5101:
5100:Physics Today
5096:
5092:
5086:
5078:
5072:
5068:
5067:
5062:
5056:
5049:
5044:
5040:
5036:
5030:
5026:
5022:
5018:
5017:
5012:
5006:
5004:
4995:
4991:
4987:
4983:
4976:
4960:
4956:
4952:
4946:
4930:
4926:
4922:
4916:
4897:
4893:
4887:
4883:
4876:
4875:
4867:
4851:
4847:
4841:
4837:
4833:
4828:
4827:gr-qc/0610061
4823:
4819:
4818:
4810:
4794:
4790:
4786:
4779:
4763:
4759:
4753:
4749:
4745:
4744:
4739:
4732:
4724:
4720:
4716:
4710:
4706:
4702:
4696:
4694:
4692:
4690:
4688:
4671:
4667:
4663:
4657:
4655:
4638:
4634:
4630:
4624:
4608:
4604:
4600:
4594:
4578:
4574:
4570:
4564:
4562:
4542:
4538:
4531:
4530:
4522:
4520:
4518:
4510:
4509:
4504:
4503:
4498:
4494:
4490:
4486:
4472:
4468:
4465:(in German).
4464:
4460:
4456:
4450:
4448:
4446:
4429:
4425:
4421:
4417:
4413:
4409:
4405:
4401:
4397:
4396:Barrow, J. D.
4391:
4383:
4379:
4375:
4371:
4367:
4363:
4359:
4355:
4354:
4349:
4348:Wesson, P. S.
4343:
4335:
4329:
4321:
4317:
4313:
4311:0-7167-0334-3
4307:
4303:
4302:
4297:
4293:
4289:
4283:
4281:
4272:
4268:
4264:
4260:
4256:
4252:
4248:
4247:
4246:Physics Today
4242:
4236:
4234:
4229:
4215:
4211:
4207:
4202:
4195:
4194:
4189:
4185:
4184:Frank Wilczek
4179:
4175:
4165:
4162:
4160:
4157:
4155:
4152:
4150:
4147:
4145:
4142:
4141:
4128:
4118:
4111:
4096:
4093:are known as
4090:
4081:
4077:
4073:
4069:
4043:
4037:
4028:
4018:
4011:
4001:
3993:
3990:
3989:event horizon
3983:
3979:
3975:
3968:
3961:
3953:
3941:
3936:
3932:
3929:
3925:
3919:
3911:
3893:
3889:
3885:
3882:equation) of
3881:
3880:Lorentz force
3877:
3873:
3869:
3865:
3858:
3855:
3851:
3844:
3840:
3832:
3828:
3824:(rather than
3822:
3817:
3813:
3808:
3805:
3804:
3782:
3777:
3771:Normalizing 4
3770:
3769:
3768:
3758:
3754:
3751:
3747:
3736:
3733:
3726:
3719:
3715:
3711:
3707:
3701:
3699:
3692:
3685:
3677:
3616:
3609:
3602:
3597:
3595:
3587:
3583:
3579:
3575:
3568:
3564:
3560:
3559:gravitational
3556:
3552:
3548:
3544:
3540:
3536:
3529:
3525:
3521:
3512:
3503:
3499:
3495:
3490:
3485:
3477:
3473:
3468:
3463:
3458:
3453:
3449:
3445:
3438:
3434:
3430:
3426:
3421:
3419:
3408:
3406:
3402:
3398:
3391:
3387:
3368:
3358:
3355:
3351:
3329:
3325:
3321:
3318:
3314:
3309:
3304:
3300:
3294:
3283:
3278:
3269:
3262:
3253:
3244:
3234:
3232:
3227:
3221:
3211:
3208:
3201:
3194:Planck energy
3191:
3188:
3178:
3174:
3171:required for
3170:
3151:
3137:
3129:
3125:
3124:string theory
3120:
3118:
3113:
3112:event horizon
3109:
3104:
3102:
3098:
3094:
3086:
3066:
3048:
3044:
3039:
3036:
3029:
3018:
3009:
2993:Planck length
2985:
2983:
2979:
2975:
2971:
2962:
2958:
2948:
2944:
2942:
2939:
2938:
2934:
2924:
2920:
2918:
2915:
2914:
2911:
2905:
2900:
2896:
2894:
2891:
2890:
2886:
2881:
2874:
2870:
2868:
2865:
2864:
2860:
2856:
2851:
2847:
2842:
2838:
2836:
2833:
2832:
2829:
2825:
2820:
2816:
2814:
2811:
2810:
2806:
2801:
2797:
2795:
2792:
2791:
2787:
2782:
2780:
2774:
2773:
2767:
2764:
2759:
2755:
2751:
2747:
2743:
2739:
2738:unified force
2735:
2731:
2724:
2720:
2717:, before the
2716:
2712:
2708:
2704:
2698:
2692:
2682:
2680:
2676:
2672:
2668:
2664:
2663:string theory
2660:
2656:
2652:
2647:
2645:
2641:
2630:
2628:
2624:
2620:
2616:
2601:
2596:
2586:
2582:
2578:
2567:
2559:
2557:
2553:
2547:
2545:
2541:
2540:proton charge
2537:
2533:
2529:
2525:
2518:
2516:
2509:
2507:
2506:Frank Wilczek
2503:
2502:Planck length
2499:
2495:
2491:
2487:
2483:
2473:
2471:
2470:22 micrograms
2466:
2464:
2460:
2456:
2452:
2447:
2437:
2428:
2410:
2407:
2401:
2397:
2390:
2379:
2375:
2370:
2361:
2353:
2350:
2347:
2346:
2342:
2333:
2314:
2310:
2306:
2300:
2296:
2290:
2283:
2274:
2263:
2259:
2253:
2246:
2237:
2227:
2221:
2212:
2204:
2201:
2198:
2197:
2193:
2184:
2168:
2163:
2159:
2153:
2141:
2131:
2126:
2121:
2110:
2100:
2094:
2085:
2077:
2074:
2071:
2070:
2066:
2057:
2040:
2034:
2030:
2026:
2019:
2008:
2004:
1999:
1994:
1990:
1980:
1976:
1967:
1959:
1956:
1953:
1952:
1948:
1943:
1926:
1920:
1916:
1912:
1905:
1894:
1890:
1885:
1882:
1873:
1865:
1862:
1859:
1858:
1854:
1845:
1826:
1822:
1815:
1807:
1794:
1788:
1785:
1779:
1772:
1768:
1763:
1760:
1754:
1749:
1744:
1735:
1727:
1724:
1721:
1720:
1716:
1707:
1689:
1685:
1680:
1677:
1671:
1666:
1657:
1649:
1646:
1643:
1642:
1638:
1634:
1631:
1628:
1627:
1621:
1615:Derived units
1612:
1610:
1594:
1574:
1571:
1565:
1560:
1555:
1544:
1539:
1537:
1514:
1507:
1500:
1496:
1492:
1488:
1483:
1470:
1467:
1461:
1458:
1448:
1445:
1441:
1437:
1434:
1431:
1426:
1418:
1414:
1410:
1407:
1402:
1393:
1366:
1358:
1354:
1331:
1311:
1302:
1298:
1291:
1287:
1283:
1276:
1267:
1259:
1256:
1253:
1250:
1249:
1232:
1213:
1209:
1204:
1201:
1194:
1185:
1177:
1174:
1171:
1168:
1167:
1150:
1133:
1129:
1126:
1119:
1110:
1102:
1099:
1096:
1093:
1092:
1075:
1056:
1052:
1047:
1044:
1037:
1028:
1020:
1017:
1014:
1011:
1010:
1006:
1002:
999:
996:
993:
992:
986:
972:
952:
930:
927:
917:
913:
909:
905:
881:
860:
840:
820:
810:
805:
803:
797:
795:
791:
787:
783:
779:
776:
772:
768:
764:
760:
756:
755:natural units
746:
744:
738:
734:
729:
725:
717:
713:
706:
700:
693:
688:
681:
674:
667:
660:
656:
632:
628:
624:
618:
614:
610:
605:
601:
597:
590:
586:
583:
574:
571:
567:
563:
559:
536:
531:
519:
515:
509:
502:
490:
484:
480:
473:
468:
456:
450:
446:
439:
432:
421:
417:
407:
386:
382:
375:
371:
365:
361:
353:
346:
337:
330:
321:
311:
304:
300:
293:
289:
282:
278:
272:
268:
261:
258:
255:
247:
245:
239:
237:
233:
229:
225:
215:
209:
202:
199:
196:
192:
189:
186:
182:
179:
176:
172:
169:
168:
167:
165:
160:
158:
154:
150:
146:
142:
138:
130:
118:
106:
102:
97:
95:
91:
87:
83:
79:
75:
74:natural units
71:
67:
63:
58:
57:
52:
51:
46:
45:
40:
36:
32:
28:
24:
19:
8343:
8306:Applications
8196:Supergravity
8103:Unruh effect
8010:Quantum foam
8005:Planck units
8004:
8000:IR/UV mixing
7975:Causal patch
7842:Mesopotamian
7736:Puerto Rican
7176:
7133:Astronomical
6975:
6968:
6940:
6930:
6918:. Retrieved
6899:
6889:
6867:(2): 87–90.
6864:
6859:
6850:
6826:
6813:
6799:
6793:
6780:
6777:
6739:
6735:
6729:
6692:
6686:
6679:Visser, Matt
6672:
6629:
6623:
6617:
6584:
6578:
6572:
6545:
6539:
6527:. Retrieved
6484:
6480:
6467:
6416:
6412:
6405:Visser, Matt
6398:
6355:
6349:
6342:
6330:. Retrieved
6321:
6312:
6261:
6256:
6250:
6187:
6181:
6168:
6133:
6124:
6112:. Retrieved
6084:
6077:
6066:the original
6032:(5): 48–55.
6029:
6023:
6015:(May 2005).
6006:
5994:. Retrieved
5961:
5955:
5950:(May 2009).
5942:
5934:the original
5910:
5900:
5888:. Retrieved
5847:
5843:
5830:
5805:
5799:
5790:
5781:
5774:. Retrieved
5723:
5717:
5707:
5695:. Retrieved
5682:
5669:
5626:
5620:
5599:. Retrieved
5572:
5509:
5503:
5497:
5464:
5451:
5432:
5408:. Retrieved
5388:
5381:
5361:. Retrieved
5348:
5308:
5304:Scholarpedia
5302:
5289:
5277:. Retrieved
5244:
5238:
5225:
5213:. Retrieved
5204:CERN Courier
5202:
5192:
5168:
5143:
5137:
5131:
5107:(6): 12–13.
5104:
5098:
5085:
5065:
5055:
5046:
5025:172, 434–435
5015:
5011:Zee, Anthony
4985:
4981:
4975:
4963:. Retrieved
4954:
4945:
4933:. Retrieved
4924:
4915:
4903:. Retrieved
4873:
4866:
4854:. Retrieved
4816:
4809:
4797:. Retrieved
4788:
4778:
4766:. Retrieved
4742:
4731:
4704:
4674:. Retrieved
4665:
4641:. Retrieved
4632:
4623:
4611:. Retrieved
4602:
4593:
4581:. Retrieved
4572:
4548:. Retrieved
4541:the original
4528:
4507:
4501:
4496:
4492:
4488:
4475:. Retrieved
4466:
4462:
4432:. Retrieved
4407:
4403:
4390:
4357:
4351:
4342:
4304:. New York.
4300:
4250:
4244:
4201:
4191:
4178:
4126:
4116:
4109:
4094:
4088:
4067:
4041:
4035:
4016:
4009:
3999:
3991:
3981:
3973:
3969:Eliminates 4
3959:
3951:
3934:
3930:
3917:
3909:
3888:mass density
3863:
3859:Eliminates 4
3849:
3845:Eliminates 4
3838:
3830:
3826:
3820:
3815:
3811:
3780:
3775:
3756:
3752:
3742:
3734:
3724:
3717:
3713:
3709:
3705:
3702:
3697:
3691:rationalized
3690:
3683:
3614:
3607:
3600:
3598:
3581:
3573:
3566:
3555:flux density
3534:
3527:
3514:The factor 4
3513:
3509:
3497:
3493:
3488:
3483:
3475:
3471:
3466:
3461:
3451:
3447:
3443:
3436:
3432:
3428:
3422:
3414:
3404:
3389:
3366:
3364:
3356:
3352:
3240:
3228:
3219:
3199:
3197:
3157:
3121:
3105:
3067:
2996:
2981:
2973:
2967:
2946:
2922:
2898:
2879:
2872:
2850:solar masses
2840:
2818:
2799:
2788:Equivalents
2777:present-day
2765:
2757:
2722:
2710:
2707:Planck epoch
2706:
2700:
2685:In cosmology
2648:
2636:
2585:energy scale
2574:
2571:Planck scale
2561:
2548:
2520:
2511:
2479:
2476:Significance
2467:
2455:Planck epoch
2443:
2349:acceleration
1945:6.5249
1635:Approximate
1618:
1543:Stoney units
1541:Compared to
1540:
1535:
1512:
1505:
1498:
1494:
1490:
1486:
1484:
1350:
1169:Planck time
1094:Planck mass
965:rather than
812:
808:
799:
777:
770:
766:
763:Stoney units
752:
736:
732:
727:
723:
715:
711:
704:
698:
691:
686:
679:
672:
665:
658:
654:
652:
565:
561:
555:
405:
240:
221:
218:Introduction
213:
200:
190:
180:
170:
161:
101:Planck scale
100:
98:
61:
54:
48:
42:
31:Planck units
30:
20:
18:
8263:Causal sets
8113:Black holes
7706:Costa Rican
7670:Seychellois
7579:Singaporean
7374:Traditional
7229:Metrication
7167:Geometrised
7123:Avoirdupois
6695:(11): 403.
6332:21 December
5311:(5): 7117.
5023:. pp.
4856:31 December
4799:22 February
4768:9 September
4738:Deza, Elena
4550:31 December
4469:: 440–480.
4455:Planck, Max
4301:Gravitation
4193:Gravitation
4144:cGh physics
3894:, and with
3547:Gauss's law
3154:Planck time
3010:defined as:
2893:Temperature
2839:approx. 10
2828:light-years
2775:Property of
2719:time passed
2613:) at which
2544:proton mass
2515:quintillion
1639:equivalent
1632:Expression
1255:temperature
1000:Expression
916:temperature
109:10 GeV
8413:Max Planck
8402:Categories
8328:Multiverse
8170:Approaches
8048:CGHS model
8035:Toy models
7970:CA-duality
7792:Venezuelan
7777:Paraguayan
7731:Nicaraguan
7609:Vietnamese
7584:Sri Lankan
7569:Philippine
7529:Indonesian
7453:Portuguese
7347:Winchester
7250:Comparison
7200:Background
7138:Electrical
6995:1202475208
6920:16 January
6841:1807.04363
6702:2102.01831
6271:2005.11572
5964:(5): 8–9.
5733:1705.09749
5561:Baez, John
5371:Planck era
5215:4 November
5174:Witten, Ed
4961:. May 2024
4931:. May 2024
4672:. May 2024
4639:. May 2024
4609:. May 2024
4579:. May 2024
4360:(2): 117.
4224:References
4078:, and the
3916:replacing
3890:replacing
3551:divergence
3549:, and the
3526:of radius
3455:, and the
2855:10 protons
2798:8.08 × 10
2711:Planck era
2611:10 kg
2552:Heisenberg
2526:, because
2494:base units
1160:10 kg
997:Dimension
782:Max Planck
773:, and the
224:base units
177:in vacuum,
131:of around
107:of around
90:Max Planck
82:free space
8344:See also:
8283:Spin foam
8063:RST model
7787:Uruguayan
7772:Colombian
7762:Brazilian
7752:Argentine
7690:Tanzanian
7660:Mauritian
7630:Ethiopian
7594:Taiwanese
7564:Pakistani
7544:Mongolian
7519:Cambodian
7438:Norwegian
7408:Icelandic
7403:Hungarian
7396:Byzantine
7352:Exchequer
7100:Hong Kong
6960:317496332
6774:810438317
6757:1203.3827
6664:118154613
6609:120924238
6564:785233016
6494:1403.2080
6459:119094398
6451:0556-2821
6390:119092012
6304:218870394
6224:2367-3613
6197:1203.6191
6160:245575026
5986:0031-9228
5880:123012079
5872:1827-6121
5836:Regge, T.
5766:119022491
5758:0002-9505
5636:1001.1205
5591:924701824
5519:1105.3105
5043:659549695
4424:0035-8738
4382:120784299
4328:cite book
4257:: 12–13.
4070:from the
3852:from the
3382:10 K
3322:×
3315:≈
3078:10 m
3037:ℏ
3019:ℓ
2897:1.9 × 10
2871:1.8 × 10
2817:5.4 × 10
2408:ℏ
2307:ℏ
2260:ℏ
2213:ρ
2127:ℏ
2027:ℏ
2005:ℏ
1913:ℏ
1891:ℏ
1805:ℏ
1761:ℏ
1678:ℏ
1595:α
1572:≈
1566:α
1459:≈
1446:−
1438:×
1432:≈
1424:ℏ
1415:ϵ
1411:π
1341:10 K
1284:ℏ
1242:10 s
1202:ℏ
1127:ℏ
1085:10 m
1045:ℏ
953:ℏ
931:π
242:example,
162:The four
133:10 m
125:10 s
113:10 J
99:The term
8191:M-theory
8151:ER = EPR
7990:Graviton
7873:Obsolete
7868:Humorous
7822:Egyptian
7782:Peruvian
7757:Bolivian
7721:Honduran
7685:Tunisian
7665:Moroccan
7655:Malagasy
7640:Eritrean
7635:Egyptian
7625:Algerian
7554:Nepalese
7534:Japanese
7468:Scottish
7458:Romanian
7359:Estonian
7269:Historic
7245:Overview
7214:Overview
7111:Specific
6938:(2009).
6914:Archived
6688:Universe
6529:16 April
6523:Archived
6519:14373748
6326:Archived
6296:32639827
6242:28179841
6190:(1): 2.
6114:22 March
6108:Archived
6062:10872062
6054:15882021
5996:22 March
5990:Archived
5930:24011430
5890:22 March
5884:Archived
5770:Archived
5691:Archived
5661:55181581
5601:20 March
5595:Archived
5563:(2001).
5544:55125081
5489:87028148
5463:(1986).
5431:(2002).
5410:10 April
5404:Archived
5375:Universe
5363:20 March
5357:Archived
5297:(2008).
5279:20 March
5273:Archived
5269:55539246
5233:(1975).
5209:Archived
5093:(2001).
5063:(2005).
5013:(2010).
4896:Archived
4882:Springer
4850:Archived
4793:Archived
4762:Archived
4748:Springer
4740:(2016).
4723:58568857
4703:(2004).
4471:Archived
4457:(1899).
4434:16 April
4428:Archived
4298:(1973).
4138:See also
4029:Setting
3809:becomes
3405:decrease
3218:10
3185:10
2813:Diameter
2715:Big Bang
2667:M-theory
2593:10
2451:Big Bang
2434:10
2339:10
2190:10
2063:10
1861:momentum
1851:10
1713:10
1587:, where
1435:1.875546
629:′
619:′
606:′
587:′
157:Big Bang
105:energies
8392:Science
8380:Physics
8366:Portals
7904:Modulor
7878:Unusual
7847:Persian
7805:Ancient
7767:Chilean
7726:Mexican
7716:Haitian
7645:Guinean
7549:Myanmar
7488:Swedish
7483:Spanish
7473:Serbian
7463:Russian
7443:Ottoman
7433:Maltese
7423:Latvian
7418:Italian
7364:Finnish
7342:English
7322:Cypriot
7317:Cornish
7224:History
7219:Outline
7154:Natural
7095:Chinese
7073:General
7066:Current
7018:(NIST).
6869:Bibcode
6762:Bibcode
6746:, NASA
6707:Bibcode
6644:Bibcode
6589:Bibcode
6499:Bibcode
6431:Bibcode
6370:Bibcode
6358:: 144.
6276:Bibcode
6233:5255898
6202:Bibcode
6034:Bibcode
5966:Bibcode
5852:Bibcode
5810:Bibcode
5776:9 April
5738:Bibcode
5697:26 June
5641:Bibcode
5524:Bibcode
5313:Bibcode
5249:Bibcode
5148:Bibcode
5109:Bibcode
4990:Bibcode
4988:: 152.
4412:Bibcode
4362:Bibcode
4259:Bibcode
4122:)/2 = 2
4099:√
4061:
4046:
3964:
3944:
3914:
3896:
3886:, with
3866:in the
3800:
3785:
3672:
3657:
3653:
3638:
3634:
3619:
3167:is the
3081:
2935:≈ 10 m
2906:2.725 K
2867:Density
2740:of the
2651:gravity
2619:gravity
2587:around
2496:in the
2200:density
1607:is the
1524:
1455: C
1344:
1245:
1163:
1088:
1007:units)
1003:Value (
678:′, and
129:lengths
8251:Others
7899:N-body
7837:Indian
7812:Arabic
7675:Somali
7650:Libyan
7618:Africa
7589:Syrian
7539:Korean
7524:Indian
7514:Afghan
7478:Slovak
7448:Polish
7386:German
7369:French
7332:Danish
7310:Europe
7276:Metric
7207:Metric
7187:Stoney
7177:Planck
7162:Atomic
6993:
6983:
6958:
6948:
6906:
6806:
6772:
6662:
6607:
6562:
6552:
6517:
6487:: 74.
6457:
6449:
6388:
6302:
6294:
6240:
6230:
6222:
6158:
6148:
6100:
6060:
6052:
5984:
5928:
5918:
5878:
5870:
5764:
5756:
5687:Forbes
5659:
5589:
5579:
5542:
5487:
5477:
5439:
5396:
5267:
5073:
5041:
5031:
4965:18 May
4935:18 May
4905:31 May
4888:
4842:
4754:
4721:
4711:
4676:18 May
4643:18 May
4613:18 May
4583:18 May
4477:23 May
4422:
4410:: 24.
4380:
4320:585119
4318:
4308:
4253:(10).
3524:sphere
3335:
3319:1.2103
3177:vacuum
3093:proton
3008:length
2705:, the
2677:, and
2490:second
2430:5.5608
2335:5.1550
2186:1.2103
2075:(LMT)
2059:1.9561
1957:(LMT)
1955:energy
1947:kg⋅m/s
1863:(LMT)
1847:4.2217
1723:volume
1709:2.6121
1465:
1016:length
914:, and
904:length
873:, and
786:action
206:, the
193:, the
183:, the
173:, the
78:nature
59:, and
33:are a
7887:Other
7852:Roman
7832:Hindu
7827:Greek
7711:Cuban
7599:Tatar
7559:Omani
7498:Welsh
7493:Swiss
7413:Irish
7391:Greek
7337:Dutch
7327:Czech
7297:(CGS)
7291:(MTS)
7285:(MKS)
7257:(FPS)
7238:UK/US
6836:arXiv
6752:arXiv
6750:: 5.
6742:(1).
6697:arXiv
6660:S2CID
6634:arXiv
6605:S2CID
6515:S2CID
6489:arXiv
6455:S2CID
6421:arXiv
6386:S2CID
6360:arXiv
6300:S2CID
6266:arXiv
6192:arXiv
6156:S2CID
6069:(PDF)
6058:S2CID
6020:(PDF)
5876:S2CID
5762:S2CID
5728:arXiv
5657:S2CID
5631:arXiv
5567:. In
5540:S2CID
5514:arXiv
5265:S2CID
5178:arXiv
4899:(PDF)
4878:(PDF)
4822:arXiv
4544:(PDF)
4533:(PDF)
4378:S2CID
3610:(or 8
3603:but 4
3435:) + (
3374:1.416
3243:force
3173:light
3070:1.616
2945:≈ 10
2921:≈ 10
2604:2.176
2486:metre
2351:(LT)
2202:(LM)
2073:force
1333:1.416
1234:5.391
1152:2.176
1077:1.616
994:Name
236:metre
197:, and
7604:Thai
7574:Pegu
7507:Asia
7145:(US)
7128:Troy
6991:OCLC
6981:ISBN
6956:OCLC
6946:ISBN
6922:2018
6904:ISBN
6804:ISBN
6770:OCLC
6560:OCLC
6550:ISBN
6531:2022
6447:ISSN
6334:2016
6292:PMID
6238:PMID
6220:ISSN
6146:ISBN
6116:2022
6098:ISBN
6050:PMID
5998:2022
5982:ISSN
5926:OCLC
5916:ISBN
5892:2022
5868:ISSN
5778:2022
5754:ISSN
5699:2019
5603:2022
5587:OCLC
5577:ISBN
5485:LCCN
5475:ISBN
5437:ISBN
5412:2010
5394:ISBN
5365:2022
5281:2022
5217:2021
5071:ISBN
5048:out.
5039:OCLC
5029:ISBN
4967:2024
4959:NIST
4937:2024
4929:NIST
4907:2020
4886:ISBN
4858:2019
4840:ISBN
4801:2021
4770:2020
4752:ISBN
4719:OCLC
4709:ISBN
4678:2024
4670:NIST
4645:2024
4637:NIST
4615:2024
4607:NIST
4585:2024
4577:NIST
4552:2019
4505:and
4495:and
4479:2020
4436:2022
4420:ISSN
4334:link
4316:OCLC
4306:ISBN
4186:and
3834:= −4
3636:(or
3561:and
3539:flux
3530:is 4
3378:(16)
3181:5.39
3169:time
3074:(18)
2835:Mass
2730:time
2665:and
2589:1.22
2579:and
2534:are
2530:and
2528:mass
2488:and
2341:kg/m
1725:(L)
1647:(L)
1645:area
1575:11.7
1526:and
1511:and
1497:and
1462:11.7
1337:(16)
1257:(Θ)
1238:(60)
1175:(T)
1173:time
1156:(24)
1100:(M)
1098:mass
1081:(18)
1018:(L)
912:mass
908:time
792:for
143:and
127:and
117:time
25:and
6877:doi
6715:doi
6652:doi
6597:doi
6507:doi
6439:doi
6378:doi
6356:441
6284:doi
6262:124
6228:PMC
6210:doi
6138:doi
6090:doi
6042:doi
6030:292
5974:doi
5860:doi
5818:doi
5746:doi
5649:doi
5532:doi
5321:doi
5257:doi
5156:doi
5144:135
5117:doi
4832:doi
4370:doi
4267:doi
4115:= (
4091:= 1
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3803:):
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2794:Age
2763:).
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