8386:
8359:
8398:
249:. 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.
3130:". 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.
813:... 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.
1366:, 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
562:
6825:
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.
2560:
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,
1630:
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
820:... 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".
6358:
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
5793:
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
2532:
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
2459:
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
2648:
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
252:
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
3521:
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
3364:
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
583:
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
6789:
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
2341:
412:
5058:
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
3125:
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
2192:
3360:
1853:
1492:
3426:
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
2523:
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
2065:
422:
3666:) 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
3201:. 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.
1951:
3754:
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
2436:
6985:
1339:
2218:
262:
3259:
3075:
659:
1399:
5701:
1240:
1083:
7260:
3431:. 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.
2483:: 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.
1715:
1158:
2668:
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
2091:
1741:
795:
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
1596:
557:{\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}}}}
3368:
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.
2653:, 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
1973:
954:
3244:
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.
741:, 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 "
1394:
911:
225:
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.
7060:
2979:
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
1616:
114:
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
7305:
5922:
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
974:
1879:
7026:
5689:
4969:
4939:
4680:
4647:
4617:
4587:
4344:
3114:, though whether those concepts are in fact simultaneously applicable is open to debate. (The same heuristic argument simultaneously motivates the Planck mass.)
7255:
3159:
2573:
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
994:
882:
862:
842:
3216:
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
2739:, 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
2367:
6027:
5920:
7299:
3024:
146:(approximately the energy-equivalent of the Planck mass, the Planck time and the Planck length, respectively). At the Planck scale, the predictions of the
4906:
7827:
7574:
4860:
7947:
7852:
2336:{\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}}}}
756:
wrote that, "Mathematically it is an acceptable trick which saves labour. Physically it represents a loss of information and can lead to confusion."
8088:
254:
5219:
1273:
7847:
7564:
7384:
2707:
8053:
5250:
407:{\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}}}}
8385:
4481:
7883:
7878:
7832:
7483:
7473:
7374:
6995:
6960:
6818:
6564:
6160:
6112:
5591:
5489:
5408:
5085:
5043:
4854:
4723:
2920:
2903:
7888:
7458:
7396:
7053:
5476:
2515:
and Planck time are conceptually linked at a fundamental physical level. Consequently, natural units help physicists to reframe questions.
6924:
5414:
4772:
3774:
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
7822:
7609:
7862:
7837:
7690:
7438:
6118:
6000:
5515:
5025:
4900:
4766:
3685:
in the nondimensionalized form of
Coulomb's law as well, so that the nondimensionalized Maxwell's equations for electromagnetism and
8362:
7746:
7265:
6918:
5930:
5451:
5367:
4320:
7975:
4438:
1191:
1034:
6336:
5894:
4892:
4758:
4538:
3410:
at the Planck temperature might contain Planck-scale black holes, constantly being formed from thermal radiation and decaying via
2468:, i.e., when the universe was less than one Planck time old. Describing the universe during the Planck epoch requires a theory of
8078:
7716:
7680:
7589:
7327:
2577:, where 𝐺 is the gravitational constant in natural units. A similar limitation applies to the precise synchronization of clocks.
7802:
7787:
7741:
7619:
7594:
7579:
7539:
7463:
7046:
2845:
2187:{\displaystyle F_{\text{P}}={\frac {E_{\text{P}}}{l_{\text{P}}}}={\frac {\hbar }{l_{\text{P}}t_{\text{P}}}}={\frac {c^{4}}{G}}}
1663:
1116:
5605:
3355:{\displaystyle F_{\text{P}}={\frac {m_{\text{P}}c}{t_{\text{P}}}}={\frac {c^{4}}{G}}\approx \mathrm {1.2103\times 10^{44}~N} }
8268:
8136:
7797:
7782:
7772:
7762:
7700:
7670:
7640:
7604:
7529:
7448:
7423:
7418:
7413:
7406:
7110:
6590:
5780:
45:
8063:
8025:
5283:
1848:{\displaystyle l_{\text{P}}^{3}=\left({\frac {\hbar G}{c^{3}}}\right)^{\frac {3}{2}}={\sqrt {\frac {(\hbar G)^{3}}{c^{9}}}}}
6266:
Wendel, Garrett; Martínez, Luis; Bojowald, Martin (19 June 2020). "Physical
Implications of a Fundamental Period of Time".
1545:. Hence, the uncertainties in the SI values of the Planck units derive almost entirely from uncertainty in the SI value of
79:(described further below). Expressing one of these physical constants in terms of Planck units yields a numerical value of
8151:
7940:
7792:
7767:
7731:
7695:
7675:
7665:
7650:
7645:
7635:
7544:
7468:
7369:
7100:
7090:
6754:
6533:
4265:
1487:{\displaystyle q_{\text{P}}={\sqrt {4\pi \epsilon _{0}\hbar c}}\approx 1.875546\times 10^{-18}{\text{ C}}\approx 11.7\ e.}
1363:
1015:
238:
8376:
3435:
8156:
7857:
7777:
7736:
7726:
7655:
7559:
7498:
7493:
7453:
7443:
7433:
7428:
7332:
7234:
7229:
7148:
7143:
7105:
6193:
3217:
2660:
While physicists have a fairly good understanding of the other fundamental interactions of forces on the quantum level,
8234:
8035:
7182:
4086:
3588:
2610:
589:
7685:
7660:
7599:
7549:
7534:
7524:
7488:
7379:
7342:
7311:
5729:
5632:
3095:
2632:
become significant. At this scale, present descriptions and theories of sub-atomic particle interactions in terms of
1559:
4803:
3406:; a quantum theory of gravity would be required to model the extreme energies attained. Hypothetically, a system in
8131:
7721:
7569:
7503:
7401:
7347:
7337:
5031:
3817:
3522:
fundamental equations of physics, is not evident, and the values of the Planck units are sensitive to this choice.
3107:
2701:
2465:
163:
8244:
8221:
8171:
8141:
7614:
7153:
6361:
6076:
5944:
4224:
4164:
3988:
3241:
3118:
8252:
8428:
8298:
7933:
6758:
4883:
4082:
3938:
2650:
2446:
2351:
580:
8098:
7990:
3580:
around a point charge will be distributed uniformly over that sphere. From this, it follows that a factor of 4
2060:{\displaystyle E_{\text{P}}=m_{\text{P}}c^{2}={\frac {\hbar }{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{G}}}}
8288:
8030:
7177:
7128:
6871:
6268:
5481:
4826:
4220:
4169:
3950:
3878:
3686:
2760:
2685:
1619:
769:
234:
205:
66:
6845:(17 July 2018). "Scrambling Time and Causal Structure of the Photon Sphere of a Schwarzschild Black Hole".
5919:(1992). "First Steps of Quantum Gravity and the Planck Values". In Eisenstaedt, Jean; Kox, Anne J. (eds.).
3595:
would change if space were higher-dimensional; the correct expressions can be deduced from the geometry of
8418:
7970:
6635:
5806:
5444:
The
Constants of Nature: From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe
4306:
3886:
3760:
2561:
or, similarly, times that measured of the order of Planck time, did carry special implications related to
2525:
2495:
arbitrariness, but do still involve some arbitrary choices in terms of the defining constants. Unlike the
195:
60:
3610:
Hence a substantial body of physical theory developed since Planck's 1899 paper suggests normalizing not
3399:
reaches the Planck length. There are no known physical models able to describe temperatures greater than
8196:
8108:
8005:
7995:
7069:
6946:
5209:
5147:
Mead, Chester Alden (10 August 1964). "Possible
Connection Between Gravitation and Fundamental Length".
4363:
3428:
3138:
2988:
2927:
2566:
96:
3604:
3576:
produced by point objects have spherical symmetry, and so the electric flux through a sphere of radius
2763:, where gravitation is separated from the unified force of the Standard Model, in turn followed by the
233:
Any system of measurement may be assigned a mutually independent set of base quantities and associated
8229:
6879:
6772:
6717:
6654:
6599:
6509:
6441:
6380:
6286:
6212:
6044:
5976:
5862:
5820:
5748:
5651:
5534:
5363:
5323:
5259:
5158:
5119:
5000:
4517:
4422:
4372:
4269:
4159:
3111:
2756:
2681:
2633:
2546:
932:
804:
568:
151:
6559:. International series of monographs on physics. Vol. 155. Oxford University Press. p. 5.
3256:
in the Planck system if the Planck units of time, length, and mass are considered to be base units.
8423:
7842:
7362:
7293:
6483:
6184:
6035:
5075:
4469:
4090:
3882:
3573:
3569:
3530:
3407:
2980:
2877:
2865:
2823:
2804:
2789:
2456:
7032:
1372:
8283:
8211:
8166:
7357:
7133:
6846:
6762:
6745:
Afshordi, Niayesh (1 March 2012). "Where will
Einstein fail? Leasing for Gravity and cosmology".
6707:
6698:
6670:
6644:
6615:
6525:
6499:
6465:
6431:
6396:
6370:
6310:
6276:
6202:
6166:
6140:
6068:
5886:
5772:
5738:
5667:
5641:
5550:
5524:
5275:
5188:
4832:
4511:
4388:
4338:
4310:
4216:
4203:
3756:
3553:
3220:
2764:
2713:
2591:
2534:
2473:
887:
800:
218:
174:
155:
71:
37:
4831:. Fundamental Theories of Physics. Vol. 119. Dordrecht: Kluwer Academic. pp. 347–352.
3689:
both take the same form as those for electromagnetism in SI, which do not have any factors of 4
1396:
is added to the list of constants used by Planck. Thus, this charge unit is typically given by
772:, noting that electric charge is quantized, derived units of length, time, and mass, now named
8328:
8323:
8103:
8068:
7001:
6991:
6966:
6956:
6914:
6906:
6814:
6780:
6570:
6560:
6457:
6302:
6248:
6230:
6156:
6108:
6060:
5992:
5936:
5926:
5878:
5764:
5597:
5587:
5495:
5485:
5447:
5404:
5398:
5081:
5049:
5039:
4896:
4850:
4762:
4752:
4729:
4719:
4430:
4326:
4316:
4298:
4174:
3411:
3396:
2748:
2665:
2562:
2550:
785:
49:
3117:
The Planck length is a distance scale of interest in speculations about quantum gravity. The
1946:{\displaystyle m_{\text{P}}c={\frac {\hbar }{l_{\text{P}}}}={\sqrt {\frac {\hbar c^{3}}{G}}}}
1601:
8146:
7389:
6887:
6725:
6662:
6607:
6517:
6449:
6388:
6294:
6238:
6220:
6148:
6100:
6052:
5984:
5870:
5828:
5756:
5659:
5542:
5331:
5267:
5166:
5127:
4842:
4380:
4277:
3894:
3864:
2869:
2654:
2587:
2557:
is far less than the unit mass in a system that treats both forces as having the same form.
1367:
959:
796:
242:
33:
2479:
Several quantities are not "extreme" in magnitude, such as the Planck mass, which is about
584:
symbols of physical quantity, without being scaled explicitly by their corresponding unit:
8333:
7956:
7554:
6827:
6096:
Quantum Field Theory II: Introductions to
Quantum Gravity, Supersymmetry and String Theory
6023:
5962:
5958:
5916:
5811:
5579:
5471:
5241:
5149:
5035:
4711:
4495:
4198:
2951:
2744:
2669:
2637:
2625:
2542:
2492:
2469:
159:
104:
5355:
733:, but not as a direct equality of quantities. This may seem to be "setting the constants
6883:
6776:
6721:
6658:
6603:
6513:
6445:
6384:
6290:
6216:
6048:
5980:
5866:
5824:
5752:
5655:
5538:
5327:
5263:
5162:
5123:
5004:
4426:
4410:
4376:
4273:
791:
would be numerically equal to 1. In 1899, one year before the advent of quantum theory,
8402:
8390:
8303:
7904:
7478:
7095:
6328:
6243:
6188:
5467:
5439:
4551:
4406:
4358:
3902:
3557:
3467:
3144:
2752:
2549:
quantities. Rather, the disparity of magnitude of force is a manifestation of that the
2431:{\displaystyle a_{\text{P}}={\frac {c}{t_{\text{P}}}}={\sqrt {\frac {c^{7}}{\hbar G}}}}
979:
867:
847:
827:
753:
185:
147:
54:
6419:
6056:
8412:
8278:
8188:
7352:
7224:
7192:
7164:
6866:
6674:
6619:
6588:
de
Sabbata, Venzo; Sivaram, C. (1993). "On limiting field strengths in gravitation".
6469:
6400:
6314:
6170:
6094:
5967:
5890:
5776:
5305:
5110:
5101:
5071:
4392:
4256:
4251:
4194:
4118:. Also, the Bekenstein–Hawking formula for the entropy of a black hole simplifies to
3999:
3890:
3134:
3122:
2777:
Table 3 lists properties of the observable universe today expressed in Planck units.
2673:
2516:
2512:
2202:
1957:
765:
84:
6529:
6072:
5671:
5554:
5513:
Barrow, John D.; Shaw, Douglas J. (2011). "The value of the cosmological constant".
5279:
4672:
3418:
its temperature by creating larger black holes, whose
Hawking temperature is lower.
2743:
is meaningful for values smaller than the Planck time. It is generally assumed that
8397:
8206:
8113:
8020:
8010:
7985:
7909:
7197:
7172:
7022:
6298:
6099:. Graduate Texts in Physics. Cham: Springer International Publishing. p. 187.
5846:
5685:
5575:
5314:
5214:
3898:
3565:
3127:
2605:
2595:
2504:
2359:
1725:
1553:
773:
5724:
4639:
4609:
4579:
4494:
pp. 478–80 contain the first appearance of the Planck base units, and of the
3778:
appearing in the equations of physics are to be eliminated via the normalization.
1631:
for empirical or practical use and there are large uncertainties in their values.
5245:
4931:
2759:. Immeasurably hot and dense, the state of the Planck epoch was succeeded by the
8273:
7508:
7239:
7138:
6689:
6415:
6329:"HiRes – The High Resolution Fly's Eye Ultra High Energy Cosmic Ray Observatory"
5021:
4961:
4474:
Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin
4154:
2838:
2729:
2689:
2554:
1863:
1265:
926:
6453:
6152:
177:
that, by definition, have a numeric value 1 when expressed in these units are:
8338:
8123:
8058:
7980:
7005:
6891:
6842:
6666:
6633:
Gibbons, G. W. (2002). "The Maximum Tension Principle in General Relativity".
6487:
6104:
5546:
5381:
5336:
5309:
4748:
4465:
4302:
3707:. When applied additionally to gravitation and Planck units, these are called
3561:
2860:
792:
162:
are expected to dominate. One example is represented by the conditions in the
100:
80:
6970:
6784:
6574:
6461:
6234:
5996:
5882:
5768:
5601:
5170:
5053:
4434:
1334:{\displaystyle T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{Gk_{\text{B}}^{2}}}}}
8293:
8161:
8073:
8045:
6730:
6693:
5940:
5571:
5184:
4733:
4547:
3106:. It can be motivated in various ways, such as considering a particle whose
17:
6521:
6306:
6252:
6064:
5832:
4540:
Natural Systems of Units. To the Centenary Anniversary of the Planck System
6950:
4846:
4330:
8201:
8000:
7038:
6375:
5385:
4795:
3607:
when relating the gravitational potential to the distribution of matter.
3596:
2725:
2677:
2480:
2461:
1871:
167:
6359:
beyond the expected spectral cutoff due to cosmic microwave radiation".
6225:
4548:
Proceedings Of The XXII Workshop On High Energy Physics And Field Theory
7914:
6987:
Curvature of Space and Time, with an Introduction to Geometric Analysis
6649:
6611:
5874:
5271:
5193:
4384:
2661:
2649:
which the effects of quantum gravity can no longer be ignored in other
2629:
2210:
6436:
5988:
5760:
5663:
5132:
5105:
4837:
4281:
3137:
are modeled to be on the order of the Planck length. In theories with
2747:
dominate physical interactions at this time scale. At this scale, the
807:. Planck underlined the universality of the new unit system, writing:
7584:
5850:
5697:
3889:. These equations have the same form as Maxwell's equations (and the
3534:
3197:
3187:
3103:
3070:{\displaystyle \ell _{\mathrm {P} }={\sqrt {\frac {\hbar G}{c^{3}}}}}
3018:
2971:
2500:
1965:
1733:
1026:
914:
139:
115:
92:
88:
3599:.) Likewise for Newton's law of universal gravitation: a factor of 4
2636:
break down and become inadequate, due to the impact of the apparent
6851:
6712:
6392:
6281:
5743:
237:, from which all other quantities and units may be derived. In the
7925:
7033:
The Planck scale: relativity meets quantum mechanics meets gravity
6907:"A Model of Topological Quantization of the Electromagnetic Field"
6767:
6504:
6207:
5646:
5529:
3253:
3252:
The Planck unit of force may be thought of as the derived unit of
3183:
2496:
2083:
2075:
1000:
Table 1: Modern values for Planck's original choice of quantities
824:
Planck considered only the units based on the universal constants
799:, now usually known as the Planck constant, which appeared in the
246:
5499:
4361:(1980). "The application of dimensional analysis to cosmology".
3591:. (Both the numerical factor and the power of the dependence on
3549:
3179:
2740:
2538:
1655:
1183:
1108:
922:
918:
127:
7929:
7042:
5630:
Adler, Ronald J. (2010). "Six easy roads to the Planck scale".
2987:) squared. Barrow and Shaw proposed a modified theory in which
6492:
Symmetry, Integrability and Geometry: Methods and Applications
3223:
had a measured energy of about 50 J, equivalent to about
2991:
is a field evolving in such a way that its value remains Λ ~
1235:{\displaystyle t_{\text{P}}={\sqrt {\frac {\hbar G}{c^{5}}}}}
1078:{\displaystyle l_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}}
929:. His definitions differ from the modern ones by a factor of
6830:
would be necessary, but such a theory has yet to be written.
103:, they are relevant in research on unified theories such as
6990:. Providence, Rhode Island: American Mathematical Society.
2508:
1647:
3533:
because in three-dimensional space, the surface area of a
3395:. At this temperature, the wavelength of light emitted by
3141:, the Planck length calculated from the observed value of
3094:(the two digits enclosed by parentheses are the estimated
5690:"What Is The Smallest Possible Distance In The Universe?"
4885:
Quantum Field Theory I: Basics in Mathematics and Physics
3161:
can be smaller than the true, fundamental Planck length.
1710:{\displaystyle l_{\text{P}}^{2}={\frac {\hbar G}{c^{3}}}}
1153:{\displaystyle m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}}
2455:
Some Planck units, such as of time and length, are many
1526:
in SI units respectively have relative uncertainties of
6143:(December 2021). "From the S Matrix to String Theory".
4966:
The NIST Reference on Constants, Units, and Uncertainty
4936:
The NIST Reference on Constants, Units, and Uncertainty
4677:
The NIST Reference on Constants, Units, and Uncertainty
4644:
The NIST Reference on Constants, Units, and Uncertainty
4614:
The NIST Reference on Constants, Units, and Uncertainty
4584:
The NIST Reference on Constants, Units, and Uncertainty
3746:
There are several possible alternative normalizations.
3098:
associated with the reported numerical value) or about
2767:, which ended after about 10 seconds (or about 10
4962:"2022 CODATA Value: Newtonian constant of gravitation"
1562:
8374:
5576:"Higher-Dimensional Algebra and Planck-Scale Physics"
3693:. When this is applied to electromagnetic constants,
3262:
3147:
3027:
2370:
2221:
2094:
1976:
1882:
1744:
1666:
1604:
1402:
1375:
1276:
1194:
1119:
1037:
982:
962:
935:
890:
870:
850:
830:
592:
524:
488:
454:
425:
265:
6189:"Minimal Length Scale Scenarios for Quantum Gravity"
575:
system of quantities, but the second equation, with
8316:
8261:
8243:
8220:
8187:
8180:
8122:
8087:
8044:
7963:
7897:
7871:
7815:
7755:
7709:
7628:
7517:
7320:
7286:
7279:
7248:
7217:
7210:
7163:
7121:
7083:
7076:
4828:
The Landscape of Theoretical Physics: A Global View
4415:
Quarterly Journal of the Royal Astronomical Society
3587:will appear in the denominator of Coulomb's law in
2657:at which a black hole could be formed by collapse.
6420:"Conservation laws in "doubly special relativity""
3766:in Newton's law. In theories emerging after 1899,
3714:The rationalized Planck units are defined so that
3354:
3153:
3069:
2430:
2335:
2186:
2059:
1945:
1847:
1709:
1610:
1590:
1486:
1388:
1333:
1234:
1152:
1077:
988:
968:
948:
905:
876:
856:
836:
653:
556:
406:
99:. Originally proposed in 1899 by German physicist
7025:, including the Planck units, as reported by the
6694:"Counterexamples to the maximum force conjecture"
5349:
5347:
4991:Ray, T. P. (1981). "Stoney's Fundamental Units".
4932:"2022 CODATA Value: vacuum electric permittivity"
4093:, for gravitation. Planck units modified so that
3770:nearly always appears in formulae multiplied by 4
2672:is necessary. Approaches to this problem include
654:{\displaystyle F'={\frac {m_{1}'m_{2}'}{r'^{2}}}}
5586:. Cambridge University Press. pp. 172–195.
5106:"Scaling Mount Planck I: A View from the Bottom"
3186:to travel a distance of 1 Planck length in
1635:Table 2: Coherent derived units of Planck units
1591:{\textstyle {\sqrt {{1}/{\alpha }}}\approx 11.7}
5794:one gains intuition into a new physical regime.
4667:
4665:
2571:
2521:
1556:, Planck base units are all larger by a factor
818:
809:
245:include length with the associated unit of the
48:defined exclusively in terms of four universal
7027:National Institute of Standards and Technology
5434:
5432:
4574:
4572:
2664:is problematic, and cannot be integrated with
7941:
7054:
6952:General Relativity and the Einstein Equations
6909:. In M. Ferrero; Alwyn van der Merwe (eds.).
6747:Bulletin of the Astronomical Society of India
5210:"Can experiment access Planck-scale physics?"
4460:
4458:
4456:
4293:
4291:
3987:from the Bekenstein–Hawking formula (for the
3628:) to 1. Doing so would introduce a factor of
776:in his honor. Stoney chose his units so that
8:
5851:"Gravitational fields and quantum mechanics"
5625:
5623:
5584:Physics Meets Philosophy at the Planck Scale
5397:Kolb, Edward W.; Turner, Michael S. (1994).
5157:(3B). American Physical Society: B849–B862.
3190:, which is a time interval of approximately
2472:that would incorporate quantum effects into
811:
5566:
5564:
3759:in 1915). Hence Planck normalized to 1 the
8184:
7948:
7934:
7926:
7283:
7214:
7080:
7061:
7047:
7039:
4343:: CS1 maint: location missing publisher (
3953:in free space to 1 (normally expressed as
3119:Bekenstein–Hawking entropy of a black hole
2781:Table 3: Today's universe in Planck units
87:, defined using fundamental properties of
6850:
6766:
6729:
6711:
6648:
6503:
6435:
6374:
6280:
6242:
6224:
6206:
6145:Geoffrey Chew: Architect of the Bootstrap
5742:
5645:
5528:
5335:
5192:
5131:
4836:
4254:(2005). "On Absolute Units, I: Choices".
3339:
3328:
3314:
3308:
3297:
3283:
3276:
3267:
3261:
3146:
3058:
3043:
3033:
3032:
3026:
2609:(the Planck energy, corresponding to the
2553:is approximately the unit charge but the
2411:
2404:
2393:
2384:
2375:
2369:
2324:
2310:
2304:
2293:
2288:
2277:
2267:
2256:
2251:
2241:
2235:
2226:
2220:
2173:
2167:
2155:
2145:
2135:
2124:
2114:
2108:
2099:
2093:
2044:
2033:
2022:
2013:
2004:
1994:
1981:
1975:
1930:
1919:
1908:
1899:
1887:
1881:
1836:
1825:
1808:
1794:
1782:
1768:
1754:
1749:
1743:
1699:
1685:
1676:
1671:
1665:
1603:
1575:
1570:
1565:
1563:
1561:
1464:
1455:
1428:
1416:
1407:
1401:
1380:
1374:
1321:
1316:
1301:
1290:
1281:
1275:
1223:
1208:
1199:
1193:
1133:
1124:
1118:
1066:
1051:
1042:
1036:
981:
961:
936:
934:
896:
895:
889:
869:
849:
829:
696:being the dimensionless ratio quantities
642:
624:
611:
604:
591:
546:
533:
523:
504:
494:
487:
470:
460:
453:
446:
435:
426:
424:
396:
385:
375:
368:
356:
351:
340:
335:
325:
318:
303:
292:
282:
275:
264:
5925:. Boston: Birkhäuser. pp. 364–379.
4108:, because the Planck mass is divided by
2995:throughout the history of the universe.
2779:
1633:
998:
170:, approximately 13.8 billion years ago.
27:Units defined only by physical constants
8381:
6911:Fundamental Problems in Quantum Physics
4673:"2022 CODATA Value: Planck temperature"
4240:
4186:
3414:. Adding energy to such a system might
3047:
2708:Time-variation of fundamental constants
2418:
2317:
2270:
2137:
2037:
2015:
1923:
1901:
1815:
1771:
1688:
1434:
1294:
1212:
1137:
1055:
963:
639:
6905:Rañada, Antonio F. (31 October 1995).
6813:. W. H. Freeman Company. p. 117.
5704:from the original on 18 September 2021
5251:Communications in Mathematical Physics
4470:"Über irreversible Strahlungsvorgänge"
4336:
700:the standard quantities, written e.g.
6927:from the original on 1 September 2020
5783:from the original on 30 December 2017
5725:"Three new roads to the Planck scale"
5370:from the original on 28 November 2018
5222:from the original on 30 November 2020
5016:
5014:
4863:from the original on 5 September 2021
4706:
4704:
4702:
4700:
4698:
4532:
4530:
4528:
4484:from the original on 17 November 2020
3711:and are seen in high-energy physics.
2921:cosmic microwave background radiation
956:, because the modern definitions use
255:Newton's law of universal gravitation
95:) rather than properties of a chosen
7:
6147:. World Scientific. pp. 72–83.
5477:The Anthropic Cosmological Principle
4441:from the original on 20 January 2022
4246:
4244:
3881:(GEM) equations, which hold in weak
3517:Alternative choices of normalization
2640:of gravity within current theories.
2476:. Such a theory does not yet exist.
6955:. Oxford: Oxford University Press.
6339:from the original on 15 August 2009
3365:Newtonian attraction between them.
2511:system for historical reasons, the
7091:International System of Units (SI)
7023:Value of the fundamental constants
6536:from the original on 19 March 2022
6121:from the original on 24 March 2022
6003:from the original on 22 March 2022
5897:from the original on 24 March 2022
5723:Faraoni, Valerio (November 2017).
5516:General Relativity and Gravitation
5246:"Particle Creation by Black Holes"
5208:Bingham, Robert (4 October 2006).
5027:Quantum Field Theory in a Nutshell
4580:"2022 CODATA Value: Planck length"
4550:. pp. 287–296. Archived from
3548:. This, along with the concept of
3348:
3034:
897:
25:
8363:Template:Quantum mechanics topics
6869:(1983). "Kaluza-Klein Monopole".
6057:10.1038/scientificamerican0505-48
5417:from the original on 6 March 2021
5187:(2002). "Quest For Unification".
4912:from the original on 19 June 2020
4806:from the original on 3 March 2021
4775:from the original on 6 March 2021
8396:
8384:
8358:
8357:
8079:Topological quantum field theory
5608:from the original on 8 July 2021
5286:from the original on 5 July 2014
4640:"2022 CODATA Value: Planck time"
4610:"2022 CODATA Value: Planck mass"
4510:in this paper correspond to the
4223:propagates at the same speed as
8064:Jackiw–Teitelboim gravity
5384:" at the very beginning of the
4716:A First Course in String Theory
2818:4.35 × 10 s or 1.38 × 10 years
949:{\displaystyle {\sqrt {2\pi }}}
913:to arrive at natural units for
158:are not expected to apply, and
8269:Causal dynamical triangulation
8137:Black hole information paradox
6591:Foundations of Physics Letters
6299:10.1103/PhysRevLett.124.241301
6093:Manoukian, Edouard B. (2016).
4718:. Cambridge University Press.
3700:, this unit system is called "
3121:is one-fourth the area of its
2732:was equal to the Planck time,
1822:
1812:
46:system of units of measurement
1:
8235:Wheeler–DeWitt equation
7101:US customary units (USCS/USC)
7035:from 'Einstein Light' at UNSW
6755:Astronomical Society of India
5963:"What's bad about this habit"
5380:Discusses "Planck time" and "
5080:. New York: Alfred A. Knopf.
4411:"Natural Units Before Planck"
4266:American Institute of Physics
2724:is the earliest stage of the
2535:comparing apples with oranges
1364:International System of Units
768:was introduced in 1874, when
239:International System of Units
91:(specifically, properties of
8157:Cosmic censorship hypothesis
7976:Batalin–Vilkovisky formalism
6194:Living Reviews in Relativity
5403:. Basic Books. p. 447.
4201:do so, as does the textbook
3789:to 1 (and therefore setting
3422:Nondimensionalized equations
3218:ultra-high-energy cosmic ray
1515:are exact and the values of
1389:{\displaystyle k_{\text{e}}}
664:This last equation (without
6826:To go even farther back, a
5730:American Journal of Physics
5633:American Journal of Physics
3008:The Planck length, denoted
1496:In SI units, the values of
906:{\displaystyle k_{\rm {B}}}
8445:
8253:Hartle–Hawking state
8152:Bousso's holographic bound
8132:Black hole complementarity
6454:10.1103/PhysRevD.68.045001
6153:10.1142/9789811219832_0013
5032:Princeton University Press
4993:Irish Astronomical Journal
4882:Zeidler, Eberhard (2006).
4498:, which Planck denoted by
3597:higher-dimensional spheres
3240:Proposals for theories of
3108:reduced Compton wavelength
2745:quantum effects of gravity
2705:
2702:Chronology of the universe
2699:
166:of our universe after the
160:quantum effects of gravity
8349:
8245:Euclidean quantum gravity
8222:Canonical quantum gravity
8172:Gravitational singularity
8142:Black-hole thermodynamics
8026:Ryu–Takayanagi conjecture
7154:English Engineering Units
6913:. Springer. p. 271.
6892:10.1103/PhysRevLett.51.87
6828:quantum theory of gravity
6488:"The Soccer-Ball Problem"
6362:The Astrophysical Journal
6105:10.1007/978-3-319-33852-1
5809:(January 1955). "Geons".
5547:10.1007/s10714-011-1199-1
5337:10.4249/scholarpedia.7117
4800:The Physics Hypertextbook
4754:Encyclopedia of Distances
4225:electromagnetic radiation
4165:Doubly special relativity
4074:). This would eliminate 8
4008:) which is simplified to
3709:rationalized Planck units
3242:doubly special relativity
2594:, the Planck scale is an
2575:𝛥𝑥 ≳ √𝐺 = 1.6 × 10 cm
2491:Planck units have little
1362:Unlike the case with the
579:absent, is relating only
8299:Superfluid vacuum theory
8054:2+1D topological gravity
6759:Astrophysics Data System
5582:; Huggett, Nick (eds.).
5171:10.1103/PhysRev.135.B849
4083:Einstein field equations
3939:characteristic impedance
3552:, are the basis for the
3436:energy–momentum relation
3128:foam at the Planck scale
3102:times the diameter of a
2868:(sometimes known as the
2757:unified with gravitation
2651:fundamental interactions
581:dimensionless quantities
569:dimensionally consistent
8289:Noncommutative geometry
8036:Weinberg–Witten theorem
8031:Trans-Planckian problem
7193:Quantum chromodynamical
6811:The Hour of Our Delight
6809:Reeves, Hubert (1991).
6731:10.3390/universe7110403
6667:10.1023/A:1022370717626
5482:Oxford University Press
5356:"Birth of the Universe"
4537:Tomilin, K. A. (1999).
4221:gravitational radiation
4170:Trans-Planckian problem
4087:Einstein–Hilbert action
4051:(and therefore setting
3989:entropy of a black hole
3951:gravitational radiation
3818:Gauss's law for gravity
3687:gravitoelectromagnetism
3376:The Planck temperature
2837:8.7 × 10 m or 9.2 × 10
2761:grand unification epoch
2686:noncommutative geometry
2655:approximate lower limit
2644:Relationship to gravity
2464:does not extend to the
1620:fine-structure constant
1611:{\displaystyle \alpha }
770:George Johnstone Stoney
206:reduced Planck constant
83:. They are a system of
7971:AdS/CFT correspondence
7306:centimetre–gram–second
7070:Systems of measurement
6947:Choquet-Bruhat, Yvonne
6636:Foundations of Physics
6555:Kiefer, Claus (2012).
6522:10.3842/SIGMA.2014.074
5833:10.1103/PhysRev.97.511
4825:Pavšic, Matej (2001).
3887:locally flat spacetime
3879:gravitoelectromagnetic
3761:gravitational constant
3750:Gravitational constant
3356:
3155:
3139:large extra dimensions
3071:
2859:3 × 10 kg or 1.5 × 10
2579:
2530:
2432:
2337:
2188:
2061:
1947:
1849:
1711:
1612:
1592:
1488:
1390:
1335:
1236:
1154:
1079:
990:
970:
969:{\displaystyle \hbar }
950:
907:
878:
858:
838:
822:
817:
812:
760:History and definition
655:
558:
408:
196:gravitational constant
8197:Bosonic string theory
8109:Semiclassical gravity
8006:Holographic principle
7996:Gravitational anomaly
7828:Biblical and Talmudic
7294:metre–kilogram–second
6984:Stavrov, Iva (2020).
6028:"Quantum Black Holes"
4847:10.1007/0-306-47136-1
4796:"Blackbody Radiation"
4364:Space Science Reviews
3991:in terms of its mass
3603:naturally appears in
3434:Examples include the
3429:nondimensionalization
3357:
3156:
3110:is comparable to its
3072:
2999:Analysis of the units
2928:Cosmological constant
2863:(only counting stars)
2706:Further information:
2638:non-renormalizability
2567:uncertainty principle
2433:
2338:
2189:
2062:
1948:
1850:
1712:
1613:
1593:
1489:
1391:
1336:
1237:
1155:
1080:
991:
971:
951:
908:
879:
859:
839:
698:corresponding to
656:
571:and equally valid in
559:
417:can be expressed as:
409:
8230:Loop quantum gravity
8089:Quantum field theory
6484:Hossenfelder, Sabine
6185:Hossenfelder, Sabine
6082:on 14 February 2019.
5364:University of Oregon
4747:Deza, Michel Marie;
4160:Dimensional analysis
4106:reduced Planck units
3998:and the area of its
3883:gravitational fields
3574:electrostatic fields
3564:operator applied to
3260:
3248:Planck unit of force
3145:
3112:Schwarzschild radius
3025:
2682:loop quantum gravity
2634:quantum field theory
2613:of the Planck mass,
2519:puts it succinctly:
2368:
2219:
2092:
1974:
1880:
1742:
1664:
1602:
1560:
1400:
1373:
1274:
1192:
1117:
1035:
980:
960:
933:
888:
868:
848:
828:
805:black-body radiation
590:
423:
263:
152:quantum field theory
130:intervals of around
8099:Bunch–Davies vacuum
8091:in curved spacetime
7991:Faddeev–Popov ghost
6884:1983PhRvL..51...87S
6777:2012BASI...40....1A
6722:2021Univ....7..403J
6692:(3 February 2021).
6659:2002FoPh...32.1891G
6604:1993FoPhL...6..561D
6514:2014SIGMA..10..074H
6446:2003PhRvD..68d5001J
6385:1995ApJ...441..144B
6291:2020PhRvL.124x1301W
6226:10.12942/lrr-2013-2
6217:2013LRR....16....2H
6049:2005SciAm.292e..48C
6036:Scientific American
6024:Giddings, Steven B.
5981:2009PhT....62e...8M
5867:1958NCim....7..215R
5825:1955PhRv...97..511W
5753:2017AmJPh..85..865F
5656:2010AmJPh..78..925A
5539:2011GReGr..43.2555B
5328:2008SchpJ...3.7117R
5264:1975CMaPh..43..199H
5163:1964PhRv..135..849M
5124:2001PhT....54f..12W
5077:The Road to Reality
5030:(second ed.).
5005:1981IrAJ...15..152R
4557:on 12 December 2020
4427:1983QJRAS..24...24B
4377:1980SSRv...27..109W
4274:2005PhT....58j..12W
4091:Friedmann equations
3531:theoretical physics
3412:Hawking evaporation
3408:thermal equilibrium
2981:age of the universe
2970:≈ 10 s ≈ 10 (km/s)/
2919:temperature of the
2790:observable universe
2782:
2457:orders of magnitude
2298:
2261:
1759:
1681:
1636:
1326:
1262:Planck temperature
1001:
632:
619:
567:Both equations are
361:
345:
241:, for example, the
175:universal constants
8284:Group field theory
8212:Superstring theory
8167:Firewall (physics)
7300:metre–tonne–second
7096:UK imperial system
6612:10.1007/BF00662806
6333:www.cosmic-ray.org
6022:Carr, Bernard J.;
5875:10.1007/BF02744199
5849:(1 January 1958).
5694:Starts with a Bang
5446:. Pantheon Books.
5400:The Early Universe
5354:Schombert, James.
5272:10.1007/BF02345020
4385:10.1007/bf00212237
4299:Misner, Charles W.
4217:General relativity
4193:For example, both
3757:general relativity
3605:Poisson's equation
3554:inverse-square law
3372:Planck temperature
3352:
3209:The Planck energy
3151:
3067:
2794:Approximate number
2780:
2765:inflationary epoch
2714:Big Bang cosmology
2592:physical cosmology
2474:general relativity
2428:
2333:
2284:
2247:
2184:
2057:
1943:
1845:
1745:
1707:
1667:
1634:
1608:
1588:
1484:
1386:
1331:
1312:
1232:
1150:
1075:
999:
986:
966:
946:
903:
874:
854:
834:
801:Wien approximation
651:
620:
607:
554:
540:
511:
477:
404:
347:
331:
243:SI base quantities
219:Boltzmann constant
156:general relativity
50:physical constants
38:physical cosmology
8372:
8371:
8329:Eternal inflation
8324:Quantum cosmology
8312:
8311:
8104:Hawking radiation
8069:Liouville gravity
7923:
7922:
7811:
7810:
7275:
7274:
7266:Foot–pound–second
7206:
7205:
7183:Heaviside–Lorentz
6997:978-1-4704-6313-7
6962:978-0-19-155226-7
6820:978-0-7167-2220-5
6643:(12): 1891–1901.
6566:978-0-191-62885-6
6424:Physical Review D
6418:(4 August 2003).
6187:(December 2013).
6162:978-981-12-1982-5
6114:978-3-319-33851-4
5989:10.1063/1.3141952
5947:on 25 April 2019.
5761:10.1119/1.4994804
5664:10.1119/1.3439650
5593:978-0-521-66280-2
5523:(10): 2555–2560.
5491:978-0-19-282147-8
5410:978-0-201-62674-2
5360:HC 441: Cosmology
5310:"Quantum gravity"
5133:10.1063/1.1387576
5087:978-0-679-45443-4
5045:978-0-691-14034-6
4856:978-0-7923-7006-2
4725:978-0-521-83143-7
4282:10.1063/1.2138392
4181:Explanatory notes
4175:Zero-point energy
3853:in Planck units).
3589:rationalized form
3529:is ubiquitous in
3397:thermal radiation
3347:
3323:
3303:
3300:
3286:
3270:
3154:{\displaystyle G}
3065:
3064:
2977:
2976:
2755:is assumed to be
2690:causal set theory
2666:quantum mechanics
2611:energy equivalent
2503:, which exist as
2453:
2452:
2426:
2425:
2399:
2396:
2378:
2331:
2299:
2291:
2280:
2262:
2254:
2244:
2229:
2182:
2162:
2158:
2148:
2130:
2127:
2117:
2102:
2055:
2054:
2028:
2025:
1997:
1984:
1941:
1940:
1914:
1911:
1890:
1843:
1842:
1802:
1788:
1752:
1705:
1674:
1580:
1477:
1467:
1440:
1410:
1383:
1360:
1359:
1329:
1328:
1319:
1284:
1230:
1229:
1202:
1148:
1147:
1127:
1073:
1072:
1045:
989:{\displaystyle h}
944:
877:{\displaystyle c}
857:{\displaystyle h}
837:{\displaystyle G}
649:
552:
539:
536:
510:
507:
476:
473:
441:
438:
402:
362:
354:
338:
328:
309:
16:(Redirected from
8436:
8401:
8400:
8389:
8388:
8380:
8361:
8360:
8185:
8147:Bekenstein bound
7964:Central concepts
7950:
7943:
7936:
7927:
7390:Mesures usuelles
7284:
7215:
7081:
7063:
7056:
7049:
7040:
7010:
7009:
6981:
6975:
6974:
6943:
6937:
6936:
6934:
6932:
6902:
6896:
6895:
6872:Phys. Rev. Lett.
6863:
6857:
6856:
6854:
6839:
6833:
6832:
6806:
6800:
6799:
6770:
6742:
6736:
6735:
6733:
6715:
6685:
6679:
6678:
6652:
6630:
6624:
6623:
6585:
6579:
6578:
6552:
6546:
6545:
6543:
6541:
6507:
6480:
6474:
6473:
6439:
6411:
6405:
6404:
6378:
6376:astro-ph/9410067
6355:
6349:
6348:
6346:
6344:
6325:
6319:
6318:
6284:
6269:Phys. Rev. Lett.
6263:
6257:
6256:
6246:
6228:
6210:
6181:
6175:
6174:
6141:Schwarz, John H.
6137:
6131:
6130:
6128:
6126:
6090:
6084:
6083:
6081:
6075:. Archived from
6032:
6019:
6013:
6012:
6010:
6008:
5959:Mermin, N. David
5955:
5949:
5948:
5943:. Archived from
5917:Gorelik, Gennady
5913:
5907:
5906:
5904:
5902:
5855:Il Nuovo Cimento
5843:
5837:
5836:
5803:
5797:
5796:
5790:
5788:
5746:
5720:
5714:
5713:
5711:
5709:
5688:(26 June 2019).
5682:
5676:
5675:
5649:
5627:
5618:
5617:
5615:
5613:
5580:Callender, Craig
5568:
5559:
5558:
5532:
5510:
5504:
5503:
5480:(1st ed.).
5472:Tipler, Frank J.
5464:
5458:
5457:
5436:
5427:
5426:
5424:
5422:
5394:
5388:
5379:
5377:
5375:
5351:
5342:
5341:
5339:
5302:
5296:
5295:
5293:
5291:
5238:
5232:
5231:
5229:
5227:
5205:
5199:
5198:
5196:
5181:
5175:
5174:
5144:
5138:
5137:
5135:
5098:
5092:
5091:
5068:
5062:
5061:
5018:
5009:
5008:
4988:
4982:
4981:
4979:
4977:
4958:
4952:
4951:
4949:
4947:
4928:
4922:
4921:
4919:
4917:
4911:
4890:
4879:
4873:
4872:
4870:
4868:
4840:
4822:
4816:
4815:
4813:
4811:
4791:
4785:
4784:
4782:
4780:
4744:
4738:
4737:
4712:Zwiebach, Barton
4708:
4693:
4692:
4690:
4688:
4669:
4660:
4659:
4657:
4655:
4636:
4630:
4629:
4627:
4625:
4606:
4600:
4599:
4597:
4595:
4576:
4567:
4566:
4564:
4562:
4556:
4545:
4534:
4523:
4522:in this article.
4493:
4491:
4489:
4462:
4451:
4450:
4448:
4446:
4409:(1 March 1983).
4403:
4397:
4396:
4355:
4349:
4348:
4342:
4334:
4307:Wheeler, John A.
4295:
4286:
4285:
4248:
4228:
4214:
4208:
4191:
4143:
4136:
4117:
4116:
4115:
4103:
4098:
4077:
4073:
4071:
4070:
4069:
4064:
4061:
4050:
4045:
4034:
4019:
3983:
3976:
3974:
3973:
3968:
3965:
3961:
3926:
3924:
3923:
3919:
3914:
3911:
3895:electromagnetism
3873:
3865:Poisson equation
3859:
3852:
3848:
3834:
3813:
3812:
3810:
3809:
3808:
3803:
3800:
3785:
3777:
3773:
3742:
3706:
3692:
3684:
3682:
3681:
3680:
3675:
3672:
3665:
3663:
3662:
3661:
3656:
3653:
3646:
3644:
3643:
3642:
3637:
3634:
3624:
3617:
3602:
3583:
3544:
3528:
3512:
3491:, which becomes
3490:
3465:
3452:, which becomes
3451:
3394:
3392:
3388:
3361:
3359:
3358:
3353:
3351:
3345:
3344:
3343:
3324:
3319:
3318:
3309:
3304:
3302:
3301:
3298:
3292:
3288:
3287:
3284:
3277:
3272:
3271:
3268:
3236:
3228:
3221:observed in 1991
3200:
3195:
3177:
3169:The Planck time
3160:
3158:
3157:
3152:
3101:
3093:
3090:
3088:
3084:
3076:
3074:
3073:
3068:
3066:
3063:
3062:
3053:
3045:
3044:
3039:
3038:
3037:
3016:
2967:
2966:
2943:
2942:
2870:Eddington number
2796:of Planck units
2783:
2623:
2621:
2618:
2608:
2603:
2588:particle physics
2576:
2449:
2444:
2437:
2435:
2434:
2429:
2427:
2424:
2416:
2415:
2406:
2405:
2400:
2398:
2397:
2394:
2385:
2380:
2379:
2376:
2354:
2349:
2342:
2340:
2339:
2334:
2332:
2330:
2329:
2328:
2315:
2314:
2305:
2300:
2297:
2292:
2289:
2283:
2282:
2281:
2278:
2268:
2263:
2260:
2255:
2252:
2246:
2245:
2242:
2236:
2231:
2230:
2227:
2205:
2200:
2193:
2191:
2190:
2185:
2183:
2178:
2177:
2168:
2163:
2161:
2160:
2159:
2156:
2150:
2149:
2146:
2136:
2131:
2129:
2128:
2125:
2119:
2118:
2115:
2109:
2104:
2103:
2100:
2078:
2073:
2066:
2064:
2063:
2058:
2056:
2050:
2049:
2048:
2035:
2034:
2029:
2027:
2026:
2023:
2014:
2009:
2008:
1999:
1998:
1995:
1986:
1985:
1982:
1960:
1952:
1950:
1949:
1944:
1942:
1936:
1935:
1934:
1921:
1920:
1915:
1913:
1912:
1909:
1900:
1892:
1891:
1888:
1866:
1861:
1854:
1852:
1851:
1846:
1844:
1841:
1840:
1831:
1830:
1829:
1810:
1809:
1804:
1803:
1795:
1793:
1789:
1787:
1786:
1777:
1769:
1758:
1753:
1750:
1728:
1723:
1716:
1714:
1713:
1708:
1706:
1704:
1703:
1694:
1686:
1680:
1675:
1672:
1640:Derived unit of
1637:
1617:
1615:
1614:
1609:
1597:
1595:
1594:
1589:
1581:
1579:
1574:
1569:
1564:
1544:
1542:
1536:
1533:
1531:
1493:
1491:
1490:
1485:
1475:
1468:
1465:
1463:
1462:
1441:
1433:
1432:
1417:
1412:
1411:
1408:
1395:
1393:
1392:
1387:
1385:
1384:
1381:
1368:Coulomb constant
1356:
1353:
1351:
1347:
1340:
1338:
1337:
1332:
1330:
1327:
1325:
1320:
1317:
1307:
1306:
1305:
1292:
1291:
1286:
1285:
1282:
1257:
1254:
1252:
1248:
1241:
1239:
1238:
1233:
1231:
1228:
1227:
1218:
1210:
1209:
1204:
1203:
1200:
1175:
1172:
1170:
1166:
1159:
1157:
1156:
1151:
1149:
1143:
1135:
1134:
1129:
1128:
1125:
1100:
1097:
1095:
1091:
1084:
1082:
1081:
1076:
1074:
1071:
1070:
1061:
1053:
1052:
1047:
1046:
1043:
1002:
995:
993:
992:
987:
975:
973:
972:
967:
955:
953:
952:
947:
945:
937:
912:
910:
909:
904:
902:
901:
900:
883:
881:
880:
875:
863:
861:
860:
855:
843:
841:
840:
835:
815:
751:
732:
720:
712:
707:
695:
674:
668:) is valid with
660:
658:
657:
652:
650:
648:
647:
646:
633:
628:
615:
605:
600:
563:
561:
560:
555:
553:
551:
550:
545:
541:
538:
537:
534:
525:
517:
516:
512:
509:
508:
505:
499:
498:
489:
482:
478:
475:
474:
471:
465:
464:
455:
447:
442:
440:
439:
436:
427:
413:
411:
410:
405:
403:
401:
400:
391:
390:
389:
380:
379:
369:
367:
363:
360:
355:
352:
346:
344:
339:
336:
330:
329:
326:
319:
310:
308:
307:
298:
297:
296:
287:
286:
276:
164:first 10 seconds
145:
137:
135:
125:
121:
97:prototype object
34:particle physics
21:
8444:
8443:
8439:
8438:
8437:
8435:
8434:
8433:
8429:Quantum gravity
8409:
8408:
8407:
8395:
8383:
8375:
8373:
8368:
8367:
8345:
8334:FRW/CFT duality
8308:
8257:
8239:
8216:
8176:
8118:
8090:
8083:
8040:
7959:
7957:Quantum gravity
7954:
7924:
7919:
7893:
7867:
7807:
7751:
7705:
7624:
7513:
7316:
7271:
7244:
7202:
7159:
7117:
7072:
7067:
7019:
7014:
7013:
6998:
6983:
6982:
6978:
6963:
6945:
6944:
6940:
6930:
6928:
6921:
6904:
6903:
6899:
6865:
6864:
6860:
6841:
6840:
6836:
6821:
6808:
6807:
6803:
6796:
6744:
6743:
6739:
6687:
6686:
6682:
6632:
6631:
6627:
6587:
6586:
6582:
6567:
6557:Quantum Gravity
6554:
6553:
6549:
6539:
6537:
6486:(9 July 2014).
6482:
6481:
6477:
6413:
6412:
6408:
6357:
6356:
6352:
6342:
6340:
6327:
6326:
6322:
6265:
6264:
6260:
6183:
6182:
6178:
6163:
6139:
6138:
6134:
6124:
6122:
6115:
6092:
6091:
6087:
6079:
6030:
6021:
6020:
6016:
6006:
6004:
5957:
5956:
5952:
5933:
5915:
5914:
5910:
5900:
5898:
5845:
5844:
5840:
5812:Physical Review
5805:
5804:
5800:
5786:
5784:
5737:(11): 865–869.
5722:
5721:
5717:
5707:
5705:
5684:
5683:
5679:
5629:
5628:
5621:
5611:
5609:
5594:
5570:
5569:
5562:
5512:
5511:
5507:
5492:
5468:Barrow, John D.
5466:
5465:
5461:
5454:
5440:Barrow, John D.
5438:
5437:
5430:
5420:
5418:
5411:
5396:
5395:
5391:
5373:
5371:
5353:
5352:
5345:
5304:
5303:
5299:
5289:
5287:
5240:
5239:
5235:
5225:
5223:
5207:
5206:
5202:
5183:
5182:
5178:
5150:Physical Review
5146:
5145:
5141:
5100:
5099:
5095:
5088:
5070:
5069:
5065:
5046:
5020:
5019:
5012:
4990:
4989:
4985:
4975:
4973:
4960:
4959:
4955:
4945:
4943:
4930:
4929:
4925:
4915:
4913:
4909:
4903:
4895:. p. 953.
4888:
4881:
4880:
4876:
4866:
4864:
4857:
4824:
4823:
4819:
4809:
4807:
4793:
4792:
4788:
4778:
4776:
4769:
4761:. p. 602.
4746:
4745:
4741:
4726:
4710:
4709:
4696:
4686:
4684:
4671:
4670:
4663:
4653:
4651:
4638:
4637:
4633:
4623:
4621:
4608:
4607:
4603:
4593:
4591:
4578:
4577:
4570:
4560:
4558:
4554:
4543:
4536:
4535:
4526:
4496:Planck constant
4487:
4485:
4464:
4463:
4454:
4444:
4442:
4405:
4404:
4400:
4357:
4356:
4352:
4335:
4323:
4297:
4296:
4289:
4250:
4249:
4242:
4237:
4232:
4231:
4215:
4211:
4199:Barton Zwiebach
4192:
4188:
4183:
4151:
4142:
4134:
4132:
4125:
4119:
4113:
4111:
4109:
4096:
4094:
4075:
4067:
4065:
4062:
4059:
4058:
4056:
4043:
4041:
4032:
4025:
4017:
4015:
4009:
4007:
3997:
3981:
3969:
3966:
3959:
3957:
3956:
3954:
3948:
3937:Normalizes the
3933:
3917:
3915:
3912:
3909:
3908:
3906:
3871:
3857:
3846:
3844:
3836:
3829:
3821:
3806:
3804:
3801:
3798:
3797:
3795:
3790:
3783:
3775:
3771:
3752:
3740:
3733:
3715:
3704:
3699:
3690:
3678:
3676:
3673:
3670:
3669:
3667:
3659:
3657:
3654:
3651:
3650:
3648:
3640:
3638:
3635:
3632:
3631:
3629:
3622:
3615:
3600:
3581:
3568:. For example,
3542:
3526:
3519:
3502:
3492:
3480:
3470:
3453:
3438:
3424:
3405:
3390:
3386:
3384:
3382:
3374:
3335:
3310:
3293:
3279:
3278:
3263:
3258:
3257:
3250:
3235:
3226:
3224:
3215:
3207:
3193:
3191:
3176:
3173:
3170:
3167:
3143:
3142:
3133:The strings of
3099:
3091:
3086:
3082:
3080:
3079:It is equal to
3054:
3046:
3028:
3023:
3022:
3017:, is a unit of
3015:
3012:
3009:
3006:
3001:
2965:
2962:
2961:
2960:
2952:Hubble constant
2941:
2938:
2937:
2936:
2918:
2914:
2895:
2888:
2864:
2856:
2834:
2815:
2795:
2787:
2773:
2738:
2710:
2704:
2698:
2670:quantum gravity
2646:
2626:quantum effects
2619:
2616:
2614:
2601:
2599:
2584:
2574:
2547:incommensurable
2543:electric charge
2493:anthropocentric
2489:
2470:quantum gravity
2442:
2440:
2417:
2407:
2389:
2371:
2366:
2365:
2347:
2345:
2320:
2316:
2306:
2273:
2269:
2237:
2222:
2217:
2216:
2198:
2196:
2169:
2151:
2141:
2140:
2120:
2110:
2095:
2090:
2089:
2071:
2069:
2040:
2036:
2018:
2000:
1990:
1977:
1972:
1971:
1955:
1926:
1922:
1904:
1883:
1878:
1877:
1859:
1857:
1832:
1821:
1811:
1778:
1770:
1764:
1763:
1740:
1739:
1721:
1719:
1695:
1687:
1662:
1661:
1628:
1600:
1599:
1558:
1557:
1540:
1538:
1534:
1529:
1527:
1521:
1514:
1451:
1424:
1403:
1398:
1397:
1376:
1371:
1370:
1354:
1349:
1345:
1343:
1308:
1297:
1293:
1277:
1272:
1271:
1255:
1250:
1246:
1244:
1219:
1211:
1195:
1190:
1189:
1173:
1168:
1164:
1162:
1136:
1120:
1115:
1114:
1098:
1093:
1089:
1087:
1062:
1054:
1038:
1033:
1032:
978:
977:
958:
957:
931:
930:
891:
886:
885:
866:
865:
846:
845:
826:
825:
786:electron charge
764:The concept of
762:
742:
731:
718:
714:
705:
701:
693:
688:
681:
672:
662:
638:
634:
606:
593:
588:
587:
565:
529:
519:
518:
500:
490:
483:
466:
456:
449:
448:
431:
421:
420:
415:
392:
381:
371:
370:
321:
320:
314:
299:
288:
278:
277:
261:
260:
231:
216:
143:
133:
131:
123:
119:
105:quantum gravity
77:
28:
23:
22:
15:
12:
11:
5:
8442:
8440:
8432:
8431:
8426:
8421:
8411:
8410:
8406:
8405:
8393:
8370:
8369:
8366:
8365:
8351:
8350:
8347:
8346:
8344:
8343:
8342:
8341:
8336:
8331:
8320:
8318:
8314:
8313:
8310:
8309:
8307:
8306:
8304:Twistor theory
8301:
8296:
8291:
8286:
8281:
8276:
8271:
8265:
8263:
8259:
8258:
8256:
8255:
8249:
8247:
8241:
8240:
8238:
8237:
8232:
8226:
8224:
8218:
8217:
8215:
8214:
8209:
8204:
8199:
8193:
8191:
8182:
8178:
8177:
8175:
8174:
8169:
8164:
8159:
8154:
8149:
8144:
8139:
8134:
8128:
8126:
8120:
8119:
8117:
8116:
8111:
8106:
8101:
8095:
8093:
8085:
8084:
8082:
8081:
8076:
8071:
8066:
8061:
8056:
8050:
8048:
8042:
8041:
8039:
8038:
8033:
8028:
8023:
8018:
8013:
8008:
8003:
7998:
7993:
7988:
7983:
7978:
7973:
7967:
7965:
7961:
7960:
7955:
7953:
7952:
7945:
7938:
7930:
7921:
7920:
7918:
7917:
7912:
7907:
7905:Absolute scale
7901:
7899:
7895:
7894:
7892:
7891:
7886:
7881:
7875:
7873:
7869:
7868:
7866:
7865:
7860:
7855:
7850:
7845:
7840:
7835:
7830:
7825:
7819:
7817:
7813:
7812:
7809:
7808:
7806:
7805:
7800:
7795:
7790:
7785:
7780:
7775:
7770:
7765:
7759:
7757:
7753:
7752:
7750:
7749:
7744:
7739:
7734:
7729:
7724:
7719:
7713:
7711:
7707:
7706:
7704:
7703:
7698:
7693:
7688:
7683:
7678:
7673:
7668:
7663:
7658:
7653:
7648:
7643:
7638:
7632:
7630:
7626:
7625:
7623:
7622:
7617:
7612:
7607:
7602:
7597:
7592:
7587:
7582:
7577:
7572:
7567:
7562:
7557:
7552:
7547:
7542:
7537:
7532:
7527:
7521:
7519:
7515:
7514:
7512:
7511:
7506:
7501:
7496:
7491:
7486:
7481:
7476:
7471:
7466:
7461:
7456:
7451:
7446:
7441:
7436:
7431:
7426:
7421:
7416:
7411:
7410:
7409:
7399:
7394:
7393:
7392:
7387:
7377:
7372:
7367:
7366:
7365:
7360:
7350:
7345:
7340:
7335:
7330:
7324:
7322:
7318:
7317:
7315:
7314:
7309:
7303:
7297:
7290:
7288:
7281:
7277:
7276:
7273:
7272:
7270:
7269:
7263:
7258:
7252:
7250:
7246:
7245:
7243:
7242:
7237:
7232:
7227:
7221:
7219:
7212:
7208:
7207:
7204:
7203:
7201:
7200:
7195:
7190:
7185:
7180:
7175:
7169:
7167:
7161:
7160:
7158:
7157:
7151:
7146:
7141:
7136:
7131:
7125:
7123:
7119:
7118:
7116:
7115:
7114:
7113:
7103:
7098:
7093:
7087:
7085:
7078:
7074:
7073:
7068:
7066:
7065:
7058:
7051:
7043:
7037:
7036:
7030:
7018:
7017:External links
7015:
7012:
7011:
6996:
6976:
6961:
6938:
6919:
6897:
6867:Sorkin, Rafael
6858:
6843:Shor, Peter W.
6834:
6819:
6801:
6794:
6737:
6688:Jowsey, Aden;
6680:
6650:hep-th/0210109
6625:
6598:(6): 561–570.
6580:
6565:
6547:
6475:
6414:Judes, Simon;
6406:
6393:10.1086/175344
6350:
6320:
6275:(24): 241301.
6258:
6176:
6161:
6132:
6113:
6085:
6014:
5950:
5931:
5908:
5861:(2): 215–221.
5838:
5819:(2): 511–536.
5807:Wheeler, J. A.
5798:
5715:
5677:
5640:(9): 925–932.
5619:
5592:
5560:
5505:
5490:
5459:
5452:
5428:
5409:
5389:
5343:
5306:Rovelli, Carlo
5297:
5258:(3): 199–220.
5242:Hawking, S. W.
5233:
5200:
5194:hep-ph/0207124
5176:
5139:
5102:Wilczek, Frank
5093:
5086:
5072:Penrose, Roger
5063:
5044:
5010:
4983:
4953:
4923:
4902:978-3540347620
4901:
4874:
4855:
4817:
4794:Elert, Glenn.
4786:
4768:978-3662528433
4767:
4739:
4724:
4694:
4661:
4631:
4601:
4568:
4524:
4452:
4398:
4350:
4321:
4303:Thorne, Kip S.
4287:
4252:Wilczek, Frank
4239:
4238:
4236:
4233:
4230:
4229:
4219:predicts that
4209:
4185:
4184:
4182:
4179:
4178:
4177:
4172:
4167:
4162:
4157:
4150:
4147:
4146:
4145:
4140:
4130:
4123:
4038:
4037:
4036:
4030:
4023:
4013:
4005:
3995:
3978:
3944:
3935:
3931:
3903:charge density
3868:
3854:
3840:
3825:
3751:
3748:
3738:
3731:
3697:
3518:
3515:
3498:
3476:
3468:Dirac equation
3423:
3420:
3403:
3380:
3373:
3370:
3350:
3342:
3338:
3334:
3331:
3327:
3322:
3317:
3313:
3307:
3296:
3291:
3282:
3275:
3266:
3249:
3246:
3233:
3213:
3206:
3203:
3174:
3171:
3166:
3163:
3150:
3096:standard error
3061:
3057:
3052:
3049:
3042:
3036:
3031:
3013:
3010:
3005:
3002:
3000:
2997:
2975:
2974:
2968:
2963:
2954:
2948:
2947:
2944:
2939:
2930:
2924:
2923:
2915:
2912:
2906:
2900:
2899:
2898:9.9 × 10 kg⋅m
2896:
2893:
2886:
2880:
2874:
2873:
2857:
2854:
2848:
2842:
2841:
2835:
2832:
2826:
2820:
2819:
2816:
2813:
2807:
2801:
2800:
2797:
2792:
2771:
2753:Standard Model
2736:
2700:Main article:
2697:
2694:
2645:
2642:
2583:
2580:
2488:
2485:
2451:
2450:
2438:
2423:
2420:
2414:
2410:
2403:
2392:
2388:
2383:
2374:
2363:
2356:
2355:
2343:
2327:
2323:
2319:
2313:
2309:
2303:
2296:
2287:
2276:
2272:
2266:
2259:
2250:
2240:
2234:
2225:
2214:
2207:
2206:
2194:
2181:
2176:
2172:
2166:
2154:
2144:
2139:
2134:
2123:
2113:
2107:
2098:
2087:
2080:
2079:
2067:
2053:
2047:
2043:
2039:
2032:
2021:
2017:
2012:
2007:
2003:
1993:
1989:
1980:
1969:
1962:
1961:
1953:
1939:
1933:
1929:
1925:
1918:
1907:
1903:
1898:
1895:
1886:
1875:
1868:
1867:
1855:
1839:
1835:
1828:
1824:
1820:
1817:
1814:
1807:
1801:
1798:
1792:
1785:
1781:
1776:
1773:
1767:
1762:
1757:
1748:
1737:
1730:
1729:
1717:
1702:
1698:
1693:
1690:
1684:
1679:
1670:
1659:
1652:
1651:
1644:
1641:
1627:
1624:
1607:
1587:
1584:
1578:
1573:
1568:
1519:
1512:
1483:
1480:
1474:
1471:
1461:
1458:
1454:
1450:
1447:
1444:
1439:
1436:
1431:
1427:
1423:
1420:
1415:
1406:
1379:
1358:
1357:
1341:
1324:
1315:
1311:
1304:
1300:
1296:
1289:
1280:
1269:
1263:
1259:
1258:
1242:
1226:
1222:
1217:
1214:
1207:
1198:
1187:
1181:
1177:
1176:
1160:
1146:
1142:
1139:
1132:
1123:
1112:
1106:
1102:
1101:
1085:
1069:
1065:
1060:
1057:
1050:
1041:
1030:
1024:
1023:Planck length
1020:
1019:
1012:
1009:
1006:
985:
965:
943:
940:
899:
894:
873:
853:
833:
761:
758:
754:Paul S. Wesson
729:
686:
679:
645:
641:
637:
631:
627:
623:
618:
614:
610:
603:
599:
596:
586:
549:
544:
532:
528:
522:
515:
503:
497:
493:
486:
481:
469:
463:
459:
452:
445:
434:
430:
419:
399:
395:
388:
384:
378:
374:
366:
359:
350:
343:
334:
324:
317:
313:
306:
302:
295:
291:
285:
281:
274:
271:
268:
259:
230:
227:
223:
222:
214:
209:
199:
189:
186:speed of light
148:Standard Model
75:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8441:
8430:
8427:
8425:
8422:
8420:
8419:Natural units
8417:
8416:
8414:
8404:
8399:
8394:
8392:
8387:
8382:
8378:
8364:
8356:
8353:
8352:
8348:
8340:
8337:
8335:
8332:
8330:
8327:
8326:
8325:
8322:
8321:
8319:
8315:
8305:
8302:
8300:
8297:
8295:
8292:
8290:
8287:
8285:
8282:
8280:
8279:Dual graviton
8277:
8275:
8272:
8270:
8267:
8266:
8264:
8260:
8254:
8251:
8250:
8248:
8246:
8242:
8236:
8233:
8231:
8228:
8227:
8225:
8223:
8219:
8213:
8210:
8208:
8205:
8203:
8200:
8198:
8195:
8194:
8192:
8190:
8189:String theory
8186:
8183:
8179:
8173:
8170:
8168:
8165:
8163:
8160:
8158:
8155:
8153:
8150:
8148:
8145:
8143:
8140:
8138:
8135:
8133:
8130:
8129:
8127:
8125:
8121:
8115:
8112:
8110:
8107:
8105:
8102:
8100:
8097:
8096:
8094:
8092:
8086:
8080:
8077:
8075:
8072:
8070:
8067:
8065:
8062:
8060:
8057:
8055:
8052:
8051:
8049:
8047:
8043:
8037:
8034:
8032:
8029:
8027:
8024:
8022:
8019:
8017:
8014:
8012:
8009:
8007:
8004:
8002:
7999:
7997:
7994:
7992:
7989:
7987:
7984:
7982:
7979:
7977:
7974:
7972:
7969:
7968:
7966:
7962:
7958:
7951:
7946:
7944:
7939:
7937:
7932:
7931:
7928:
7916:
7913:
7911:
7908:
7906:
7903:
7902:
7900:
7896:
7890:
7887:
7885:
7882:
7880:
7877:
7876:
7874:
7872:List articles
7870:
7864:
7861:
7859:
7856:
7854:
7851:
7849:
7846:
7844:
7841:
7839:
7836:
7834:
7831:
7829:
7826:
7824:
7821:
7820:
7818:
7814:
7804:
7801:
7799:
7796:
7794:
7791:
7789:
7786:
7784:
7781:
7779:
7776:
7774:
7771:
7769:
7766:
7764:
7761:
7760:
7758:
7756:South America
7754:
7748:
7745:
7743:
7740:
7738:
7735:
7733:
7730:
7728:
7725:
7723:
7720:
7718:
7715:
7714:
7712:
7710:North America
7708:
7702:
7699:
7697:
7694:
7692:
7691:South African
7689:
7687:
7684:
7682:
7679:
7677:
7674:
7672:
7669:
7667:
7664:
7662:
7659:
7657:
7654:
7652:
7649:
7647:
7644:
7642:
7639:
7637:
7634:
7633:
7631:
7627:
7621:
7618:
7616:
7613:
7611:
7608:
7606:
7603:
7601:
7598:
7596:
7593:
7591:
7588:
7586:
7583:
7581:
7578:
7576:
7573:
7571:
7568:
7566:
7563:
7561:
7558:
7556:
7553:
7551:
7548:
7546:
7543:
7541:
7538:
7536:
7533:
7531:
7528:
7526:
7523:
7522:
7520:
7516:
7510:
7507:
7505:
7502:
7500:
7497:
7495:
7492:
7490:
7487:
7485:
7482:
7480:
7477:
7475:
7472:
7470:
7467:
7465:
7462:
7460:
7457:
7455:
7452:
7450:
7447:
7445:
7442:
7440:
7439:Luxembourgian
7437:
7435:
7432:
7430:
7427:
7425:
7422:
7420:
7417:
7415:
7412:
7408:
7405:
7404:
7403:
7400:
7398:
7395:
7391:
7388:
7386:
7383:
7382:
7381:
7378:
7376:
7373:
7371:
7368:
7364:
7361:
7359:
7356:
7355:
7354:
7351:
7349:
7346:
7344:
7341:
7339:
7336:
7334:
7331:
7329:
7326:
7325:
7323:
7319:
7313:
7312:gravitational
7310:
7307:
7304:
7301:
7298:
7295:
7292:
7291:
7289:
7285:
7282:
7278:
7267:
7264:
7262:
7259:
7257:
7254:
7253:
7251:
7247:
7241:
7238:
7236:
7233:
7231:
7228:
7226:
7223:
7222:
7220:
7216:
7213:
7209:
7199:
7196:
7194:
7191:
7189:
7186:
7184:
7181:
7179:
7176:
7174:
7171:
7170:
7168:
7166:
7162:
7155:
7152:
7150:
7147:
7145:
7142:
7140:
7137:
7135:
7132:
7130:
7129:Apothecaries'
7127:
7126:
7124:
7120:
7112:
7109:
7108:
7107:
7104:
7102:
7099:
7097:
7094:
7092:
7089:
7088:
7086:
7082:
7079:
7075:
7071:
7064:
7059:
7057:
7052:
7050:
7045:
7044:
7041:
7034:
7031:
7028:
7024:
7021:
7020:
7016:
7007:
7003:
6999:
6993:
6989:
6988:
6980:
6977:
6972:
6968:
6964:
6958:
6954:
6953:
6948:
6942:
6939:
6926:
6922:
6920:9780792336709
6916:
6912:
6908:
6901:
6898:
6893:
6889:
6885:
6881:
6877:
6874:
6873:
6868:
6862:
6859:
6853:
6848:
6844:
6838:
6835:
6831:
6829:
6822:
6816:
6812:
6805:
6802:
6798:
6793:
6786:
6782:
6778:
6774:
6769:
6764:
6760:
6756:
6752:
6748:
6741:
6738:
6732:
6727:
6723:
6719:
6714:
6709:
6705:
6701:
6700:
6695:
6691:
6684:
6681:
6676:
6672:
6668:
6664:
6660:
6656:
6651:
6646:
6642:
6638:
6637:
6629:
6626:
6621:
6617:
6613:
6609:
6605:
6601:
6597:
6593:
6592:
6584:
6581:
6576:
6572:
6568:
6562:
6558:
6551:
6548:
6535:
6531:
6527:
6523:
6519:
6515:
6511:
6506:
6501:
6497:
6493:
6489:
6485:
6479:
6476:
6471:
6467:
6463:
6459:
6455:
6451:
6447:
6443:
6438:
6437:gr-qc/0205067
6433:
6430:(4): 045001.
6429:
6425:
6421:
6417:
6410:
6407:
6402:
6398:
6394:
6390:
6386:
6382:
6377:
6372:
6368:
6364:
6363:
6354:
6351:
6338:
6334:
6330:
6324:
6321:
6316:
6312:
6308:
6304:
6300:
6296:
6292:
6288:
6283:
6278:
6274:
6271:
6270:
6262:
6259:
6254:
6250:
6245:
6240:
6236:
6232:
6227:
6222:
6218:
6214:
6209:
6204:
6200:
6196:
6195:
6190:
6186:
6180:
6177:
6172:
6168:
6164:
6158:
6154:
6150:
6146:
6142:
6136:
6133:
6120:
6116:
6110:
6106:
6102:
6098:
6097:
6089:
6086:
6078:
6074:
6070:
6066:
6062:
6058:
6054:
6050:
6046:
6042:
6038:
6037:
6029:
6025:
6018:
6015:
6002:
5998:
5994:
5990:
5986:
5982:
5978:
5974:
5970:
5969:
5968:Physics Today
5964:
5960:
5954:
5951:
5946:
5942:
5938:
5934:
5932:0-8176-3479-7
5928:
5924:
5923:
5918:
5912:
5909:
5896:
5892:
5888:
5884:
5880:
5876:
5872:
5868:
5864:
5860:
5856:
5852:
5848:
5842:
5839:
5834:
5830:
5826:
5822:
5818:
5814:
5813:
5808:
5802:
5799:
5795:
5782:
5778:
5774:
5770:
5766:
5762:
5758:
5754:
5750:
5745:
5740:
5736:
5732:
5731:
5726:
5719:
5716:
5703:
5699:
5695:
5691:
5687:
5686:Siegel, Ethan
5681:
5678:
5673:
5669:
5665:
5661:
5657:
5653:
5648:
5643:
5639:
5635:
5634:
5626:
5624:
5620:
5607:
5603:
5599:
5595:
5589:
5585:
5581:
5577:
5573:
5567:
5565:
5561:
5556:
5552:
5548:
5544:
5540:
5536:
5531:
5526:
5522:
5518:
5517:
5509:
5506:
5501:
5497:
5493:
5487:
5483:
5479:
5478:
5473:
5469:
5463:
5460:
5455:
5453:0-375-42221-8
5449:
5445:
5441:
5435:
5433:
5429:
5416:
5412:
5406:
5402:
5401:
5393:
5390:
5387:
5383:
5369:
5365:
5361:
5357:
5350:
5348:
5344:
5338:
5333:
5329:
5325:
5321:
5317:
5316:
5311:
5307:
5301:
5298:
5285:
5281:
5277:
5273:
5269:
5265:
5261:
5257:
5253:
5252:
5247:
5243:
5237:
5234:
5221:
5217:
5216:
5211:
5204:
5201:
5195:
5190:
5186:
5180:
5177:
5172:
5168:
5164:
5160:
5156:
5152:
5151:
5143:
5140:
5134:
5129:
5125:
5121:
5117:
5113:
5112:
5111:Physics Today
5107:
5103:
5097:
5094:
5089:
5083:
5079:
5078:
5073:
5067:
5064:
5060:
5055:
5051:
5047:
5041:
5037:
5033:
5029:
5028:
5023:
5017:
5015:
5011:
5006:
5002:
4998:
4994:
4987:
4984:
4971:
4967:
4963:
4957:
4954:
4941:
4937:
4933:
4927:
4924:
4908:
4904:
4898:
4894:
4887:
4886:
4878:
4875:
4862:
4858:
4852:
4848:
4844:
4839:
4838:gr-qc/0610061
4834:
4830:
4829:
4821:
4818:
4805:
4801:
4797:
4790:
4787:
4774:
4770:
4764:
4760:
4756:
4755:
4750:
4743:
4740:
4735:
4731:
4727:
4721:
4717:
4713:
4707:
4705:
4703:
4701:
4699:
4695:
4682:
4678:
4674:
4668:
4666:
4662:
4649:
4645:
4641:
4635:
4632:
4619:
4615:
4611:
4605:
4602:
4589:
4585:
4581:
4575:
4573:
4569:
4553:
4549:
4542:
4541:
4533:
4531:
4529:
4525:
4521:
4520:
4515:
4514:
4509:
4505:
4501:
4497:
4483:
4479:
4476:(in German).
4475:
4471:
4467:
4461:
4459:
4457:
4453:
4440:
4436:
4432:
4428:
4424:
4420:
4416:
4412:
4408:
4407:Barrow, J. D.
4402:
4399:
4394:
4390:
4386:
4382:
4378:
4374:
4370:
4366:
4365:
4360:
4359:Wesson, P. S.
4354:
4351:
4346:
4340:
4332:
4328:
4324:
4322:0-7167-0334-3
4318:
4314:
4313:
4308:
4304:
4300:
4294:
4292:
4288:
4283:
4279:
4275:
4271:
4267:
4263:
4259:
4258:
4257:Physics Today
4253:
4247:
4245:
4241:
4234:
4226:
4222:
4218:
4213:
4210:
4206:
4205:
4200:
4196:
4195:Frank Wilczek
4190:
4187:
4180:
4176:
4173:
4171:
4168:
4166:
4163:
4161:
4158:
4156:
4153:
4152:
4148:
4139:
4129:
4122:
4107:
4104:are known as
4101:
4092:
4088:
4084:
4080:
4054:
4048:
4039:
4029:
4022:
4012:
4004:
4001:
4000:event horizon
3994:
3990:
3986:
3979:
3972:
3964:
3952:
3947:
3943:
3940:
3936:
3930:
3922:
3904:
3900:
3896:
3893:equation) of
3892:
3891:Lorentz force
3888:
3884:
3880:
3876:
3869:
3866:
3862:
3855:
3851:
3843:
3839:
3835:(rather than
3833:
3828:
3824:
3819:
3816:
3815:
3793:
3788:
3782:Normalizing 4
3781:
3780:
3779:
3769:
3765:
3762:
3758:
3749:
3747:
3744:
3737:
3730:
3726:
3722:
3718:
3712:
3710:
3703:
3696:
3688:
3627:
3620:
3613:
3608:
3606:
3598:
3594:
3590:
3586:
3579:
3575:
3571:
3570:gravitational
3567:
3563:
3559:
3555:
3551:
3547:
3540:
3536:
3532:
3523:
3516:
3514:
3510:
3506:
3501:
3496:
3488:
3484:
3479:
3474:
3469:
3464:
3460:
3456:
3449:
3445:
3441:
3437:
3432:
3430:
3421:
3419:
3417:
3413:
3409:
3402:
3398:
3379:
3371:
3369:
3366:
3362:
3340:
3336:
3332:
3329:
3325:
3320:
3315:
3311:
3305:
3294:
3289:
3280:
3273:
3264:
3255:
3247:
3245:
3243:
3238:
3232:
3222:
3219:
3212:
3205:Planck energy
3204:
3202:
3199:
3189:
3185:
3182:required for
3181:
3164:
3162:
3148:
3140:
3136:
3135:string theory
3131:
3129:
3124:
3123:event horizon
3120:
3115:
3113:
3109:
3105:
3097:
3077:
3059:
3055:
3050:
3040:
3029:
3020:
3004:Planck length
3003:
2998:
2996:
2994:
2990:
2986:
2982:
2973:
2969:
2959:
2955:
2953:
2950:
2949:
2945:
2935:
2931:
2929:
2926:
2925:
2922:
2916:
2911:
2907:
2905:
2902:
2901:
2897:
2892:
2885:
2881:
2879:
2876:
2875:
2871:
2867:
2862:
2858:
2853:
2849:
2847:
2844:
2843:
2840:
2836:
2831:
2827:
2825:
2822:
2821:
2817:
2812:
2808:
2806:
2803:
2802:
2798:
2793:
2791:
2785:
2784:
2778:
2775:
2770:
2766:
2762:
2758:
2754:
2750:
2749:unified force
2746:
2742:
2735:
2731:
2728:, before the
2727:
2723:
2719:
2715:
2709:
2703:
2695:
2693:
2691:
2687:
2683:
2679:
2675:
2674:string theory
2671:
2667:
2663:
2658:
2656:
2652:
2643:
2641:
2639:
2635:
2631:
2627:
2612:
2607:
2597:
2593:
2589:
2581:
2578:
2570:
2568:
2564:
2558:
2556:
2552:
2551:proton charge
2548:
2544:
2540:
2536:
2529:
2527:
2520:
2518:
2517:Frank Wilczek
2514:
2513:Planck length
2510:
2506:
2502:
2498:
2494:
2486:
2484:
2482:
2481:22 micrograms
2477:
2475:
2471:
2467:
2463:
2458:
2448:
2439:
2421:
2412:
2408:
2401:
2390:
2386:
2381:
2372:
2364:
2361:
2358:
2357:
2353:
2344:
2325:
2321:
2311:
2307:
2301:
2294:
2285:
2274:
2264:
2257:
2248:
2238:
2232:
2223:
2215:
2212:
2209:
2208:
2204:
2195:
2179:
2174:
2170:
2164:
2152:
2142:
2132:
2121:
2111:
2105:
2096:
2088:
2085:
2082:
2081:
2077:
2068:
2051:
2045:
2041:
2030:
2019:
2010:
2005:
2001:
1991:
1987:
1978:
1970:
1967:
1964:
1963:
1959:
1954:
1937:
1931:
1927:
1916:
1905:
1896:
1893:
1884:
1876:
1873:
1870:
1869:
1865:
1856:
1837:
1833:
1826:
1818:
1805:
1799:
1796:
1790:
1783:
1779:
1774:
1765:
1760:
1755:
1746:
1738:
1735:
1732:
1731:
1727:
1718:
1700:
1696:
1691:
1682:
1677:
1668:
1660:
1657:
1654:
1653:
1649:
1645:
1642:
1639:
1638:
1632:
1626:Derived units
1625:
1623:
1621:
1605:
1585:
1582:
1576:
1571:
1566:
1555:
1550:
1548:
1525:
1518:
1511:
1507:
1503:
1499:
1494:
1481:
1478:
1472:
1469:
1459:
1456:
1452:
1448:
1445:
1442:
1437:
1429:
1425:
1421:
1418:
1413:
1404:
1377:
1369:
1365:
1342:
1322:
1313:
1309:
1302:
1298:
1287:
1278:
1270:
1267:
1264:
1261:
1260:
1243:
1224:
1220:
1215:
1205:
1196:
1188:
1185:
1182:
1179:
1178:
1161:
1144:
1140:
1130:
1121:
1113:
1110:
1107:
1104:
1103:
1086:
1067:
1063:
1058:
1048:
1039:
1031:
1028:
1025:
1022:
1021:
1017:
1013:
1010:
1007:
1004:
1003:
997:
983:
941:
938:
928:
924:
920:
916:
892:
871:
851:
831:
821:
816:
814:
808:
806:
802:
798:
794:
790:
787:
783:
779:
775:
771:
767:
766:natural units
759:
757:
755:
749:
745:
740:
736:
728:
724:
717:
711:
704:
699:
692:
685:
678:
671:
667:
643:
635:
629:
625:
621:
616:
612:
608:
601:
597:
594:
585:
582:
578:
574:
570:
547:
542:
530:
526:
520:
513:
501:
495:
491:
484:
479:
467:
461:
457:
450:
443:
432:
428:
418:
397:
393:
386:
382:
376:
372:
364:
357:
348:
341:
332:
322:
315:
311:
304:
300:
293:
289:
283:
279:
272:
269:
266:
258:
256:
250:
248:
244:
240:
236:
228:
226:
220:
213:
210:
207:
203:
200:
197:
193:
190:
187:
183:
180:
179:
178:
176:
171:
169:
165:
161:
157:
153:
149:
141:
129:
117:
113:
108:
106:
102:
98:
94:
90:
86:
85:natural units
82:
78:
74:
69:
68:
63:
62:
57:
56:
51:
47:
43:
39:
35:
30:
19:
8354:
8317:Applications
8207:Supergravity
8114:Unruh effect
8021:Quantum foam
8016:Planck units
8015:
8011:IR/UV mixing
7986:Causal patch
7853:Mesopotamian
7747:Puerto Rican
7187:
7144:Astronomical
6986:
6979:
6951:
6941:
6929:. Retrieved
6910:
6900:
6878:(2): 87–90.
6875:
6870:
6861:
6837:
6824:
6810:
6804:
6791:
6788:
6750:
6746:
6740:
6703:
6697:
6690:Visser, Matt
6683:
6640:
6634:
6628:
6595:
6589:
6583:
6556:
6550:
6538:. Retrieved
6495:
6491:
6478:
6427:
6423:
6416:Visser, Matt
6409:
6366:
6360:
6353:
6341:. Retrieved
6332:
6323:
6272:
6267:
6261:
6198:
6192:
6179:
6144:
6135:
6123:. Retrieved
6095:
6088:
6077:the original
6043:(5): 48–55.
6040:
6034:
6026:(May 2005).
6017:
6005:. Retrieved
5972:
5966:
5961:(May 2009).
5953:
5945:the original
5921:
5911:
5899:. Retrieved
5858:
5854:
5841:
5816:
5810:
5801:
5792:
5785:. Retrieved
5734:
5728:
5718:
5706:. Retrieved
5693:
5680:
5637:
5631:
5610:. Retrieved
5583:
5520:
5514:
5508:
5475:
5462:
5443:
5419:. Retrieved
5399:
5392:
5372:. Retrieved
5359:
5319:
5315:Scholarpedia
5313:
5300:
5288:. Retrieved
5255:
5249:
5236:
5224:. Retrieved
5215:CERN Courier
5213:
5203:
5179:
5154:
5148:
5142:
5118:(6): 12–13.
5115:
5109:
5096:
5076:
5066:
5057:
5036:172, 434–435
5026:
5022:Zee, Anthony
4996:
4992:
4986:
4974:. Retrieved
4965:
4956:
4944:. Retrieved
4935:
4926:
4914:. Retrieved
4884:
4877:
4865:. Retrieved
4827:
4820:
4808:. Retrieved
4799:
4789:
4777:. Retrieved
4753:
4742:
4715:
4685:. Retrieved
4676:
4652:. Retrieved
4643:
4634:
4622:. Retrieved
4613:
4604:
4592:. Retrieved
4583:
4559:. Retrieved
4552:the original
4539:
4518:
4512:
4507:
4503:
4499:
4486:. Retrieved
4477:
4473:
4443:. Retrieved
4418:
4414:
4401:
4368:
4362:
4353:
4315:. New York.
4311:
4261:
4255:
4212:
4202:
4189:
4137:
4127:
4120:
4105:
4099:
4078:
4052:
4046:
4027:
4020:
4010:
4002:
3992:
3984:
3980:Eliminates 4
3970:
3962:
3945:
3941:
3928:
3920:
3899:mass density
3874:
3870:Eliminates 4
3860:
3856:Eliminates 4
3849:
3841:
3837:
3831:
3826:
3822:
3791:
3786:
3767:
3763:
3753:
3745:
3735:
3728:
3724:
3720:
3716:
3713:
3708:
3702:rationalized
3701:
3694:
3625:
3618:
3611:
3609:
3592:
3584:
3577:
3566:flux density
3545:
3538:
3525:The factor 4
3524:
3520:
3508:
3504:
3499:
3494:
3486:
3482:
3477:
3472:
3462:
3458:
3454:
3447:
3443:
3439:
3433:
3425:
3415:
3400:
3377:
3375:
3367:
3363:
3251:
3239:
3230:
3210:
3208:
3168:
3132:
3116:
3078:
3007:
2992:
2984:
2978:
2957:
2933:
2909:
2890:
2883:
2861:solar masses
2851:
2829:
2810:
2799:Equivalents
2788:present-day
2776:
2768:
2733:
2721:
2718:Planck epoch
2717:
2711:
2696:In cosmology
2659:
2647:
2596:energy scale
2585:
2582:Planck scale
2572:
2559:
2531:
2522:
2490:
2487:Significance
2478:
2466:Planck epoch
2454:
2360:acceleration
1956:6.5249
1646:Approximate
1629:
1554:Stoney units
1552:Compared to
1551:
1546:
1523:
1516:
1509:
1505:
1501:
1497:
1495:
1361:
1180:Planck time
1105:Planck mass
976:rather than
823:
819:
810:
788:
781:
777:
774:Stoney units
763:
747:
743:
738:
734:
726:
722:
715:
709:
702:
697:
690:
683:
676:
669:
665:
663:
576:
572:
566:
416:
251:
232:
229:Introduction
224:
211:
201:
191:
181:
172:
112:Planck scale
111:
109:
72:
65:
59:
53:
42:Planck units
41:
31:
29:
18:Planck scale
8274:Causal sets
8124:Black holes
7717:Costa Rican
7681:Seychellois
7590:Singaporean
7385:Traditional
7240:Metrication
7178:Geometrised
7134:Avoirdupois
6706:(11): 403.
6343:21 December
5322:(5): 7117.
5034:. pp.
4867:31 December
4810:22 February
4779:9 September
4749:Deza, Elena
4561:31 December
4480:: 440–480.
4466:Planck, Max
4312:Gravitation
4204:Gravitation
4155:cGh physics
3905:, and with
3558:Gauss's law
3165:Planck time
3021:defined as:
2904:Temperature
2850:approx. 10
2839:light-years
2786:Property of
2730:time passed
2624:) at which
2555:proton mass
2526:quintillion
1650:equivalent
1643:Expression
1266:temperature
1011:Expression
927:temperature
120:10 GeV
8424:Max Planck
8413:Categories
8339:Multiverse
8181:Approaches
8059:CGHS model
8046:Toy models
7981:CA-duality
7803:Venezuelan
7788:Paraguayan
7742:Nicaraguan
7620:Vietnamese
7595:Sri Lankan
7580:Philippine
7540:Indonesian
7464:Portuguese
7358:Winchester
7261:Comparison
7211:Background
7149:Electrical
7006:1202475208
6931:16 January
6852:1807.04363
6713:2102.01831
6282:2005.11572
5975:(5): 8–9.
5744:1705.09749
5572:Baez, John
5382:Planck era
5226:4 November
5185:Witten, Ed
4972:. May 2024
4942:. May 2024
4683:. May 2024
4650:. May 2024
4620:. May 2024
4590:. May 2024
4371:(2): 117.
4235:References
4089:, and the
3927:replacing
3901:replacing
3562:divergence
3560:, and the
3537:of radius
3466:, and the
2866:10 protons
2809:8.08 × 10
2722:Planck era
2622:10 kg
2563:Heisenberg
2537:, because
2505:base units
1171:10 kg
1008:Dimension
793:Max Planck
784:, and the
235:base units
188:in vacuum,
142:of around
118:of around
101:Max Planck
93:free space
8355:See also:
8294:Spin foam
8074:RST model
7798:Uruguayan
7783:Colombian
7773:Brazilian
7763:Argentine
7701:Tanzanian
7671:Mauritian
7641:Ethiopian
7605:Taiwanese
7575:Pakistani
7555:Mongolian
7530:Cambodian
7449:Norwegian
7419:Icelandic
7414:Hungarian
7407:Byzantine
7363:Exchequer
7111:Hong Kong
6971:317496332
6785:810438317
6768:1203.3827
6675:118154613
6620:120924238
6575:785233016
6505:1403.2080
6470:119094398
6462:0556-2821
6401:119092012
6315:218870394
6235:2367-3613
6208:1203.6191
6171:245575026
5997:0031-9228
5891:123012079
5883:1827-6121
5847:Regge, T.
5777:119022491
5769:0002-9505
5647:1001.1205
5602:924701824
5530:1105.3105
5054:659549695
4435:0035-8738
4393:120784299
4339:cite book
4268:: 12–13.
4081:from the
3863:from the
3393:10 K
3333:×
3326:≈
3089:10 m
3048:ℏ
3030:ℓ
2908:1.9 × 10
2882:1.8 × 10
2828:5.4 × 10
2419:ℏ
2318:ℏ
2271:ℏ
2224:ρ
2138:ℏ
2038:ℏ
2016:ℏ
1924:ℏ
1902:ℏ
1816:ℏ
1772:ℏ
1689:ℏ
1606:α
1583:≈
1577:α
1470:≈
1457:−
1449:×
1443:≈
1435:ℏ
1426:ϵ
1422:π
1352:10 K
1295:ℏ
1253:10 s
1213:ℏ
1138:ℏ
1096:10 m
1056:ℏ
964:ℏ
942:π
253:example,
173:The four
144:10 m
136:10 s
124:10 J
110:The term
8202:M-theory
8162:ER = EPR
8001:Graviton
7884:Obsolete
7879:Humorous
7833:Egyptian
7793:Peruvian
7768:Bolivian
7732:Honduran
7696:Tunisian
7676:Moroccan
7666:Malagasy
7651:Eritrean
7646:Egyptian
7636:Algerian
7565:Nepalese
7545:Japanese
7479:Scottish
7469:Romanian
7370:Estonian
7280:Historic
7256:Overview
7225:Overview
7122:Specific
6949:(2009).
6925:Archived
6699:Universe
6540:16 April
6534:Archived
6530:14373748
6337:Archived
6307:32639827
6253:28179841
6201:(1): 2.
6125:22 March
6119:Archived
6073:10872062
6065:15882021
6007:22 March
6001:Archived
5941:24011430
5901:22 March
5895:Archived
5781:Archived
5702:Archived
5672:55181581
5612:20 March
5606:Archived
5574:(2001).
5555:55125081
5500:87028148
5474:(1986).
5442:(2002).
5421:10 April
5415:Archived
5386:Universe
5374:20 March
5368:Archived
5308:(2008).
5290:20 March
5284:Archived
5280:55539246
5244:(1975).
5220:Archived
5104:(2001).
5074:(2005).
5024:(2010).
4907:Archived
4893:Springer
4861:Archived
4804:Archived
4773:Archived
4759:Springer
4751:(2016).
4734:58568857
4714:(2004).
4482:Archived
4468:(1899).
4445:16 April
4439:Archived
4309:(1973).
4149:See also
4040:Setting
3820:becomes
3416:decrease
3229:10
3196:10
2824:Diameter
2726:Big Bang
2678:M-theory
2604:10
2462:Big Bang
2445:10
2350:10
2201:10
2074:10
1872:momentum
1862:10
1724:10
1598:, where
1446:1.875546
640:′
630:′
617:′
598:′
168:Big Bang
116:energies
8403:Science
8391:Physics
8377:Portals
7915:Modulor
7889:Unusual
7858:Persian
7816:Ancient
7778:Chilean
7737:Mexican
7727:Haitian
7656:Guinean
7560:Myanmar
7499:Swedish
7494:Spanish
7484:Serbian
7474:Russian
7454:Ottoman
7444:Maltese
7434:Latvian
7429:Italian
7375:Finnish
7353:English
7333:Cypriot
7328:Cornish
7235:History
7230:Outline
7165:Natural
7106:Chinese
7084:General
7077:Current
7029:(NIST).
6880:Bibcode
6773:Bibcode
6757:, NASA
6718:Bibcode
6655:Bibcode
6600:Bibcode
6510:Bibcode
6442:Bibcode
6381:Bibcode
6369:: 144.
6287:Bibcode
6244:5255898
6213:Bibcode
6045:Bibcode
5977:Bibcode
5863:Bibcode
5821:Bibcode
5787:9 April
5749:Bibcode
5708:26 June
5652:Bibcode
5535:Bibcode
5324:Bibcode
5260:Bibcode
5159:Bibcode
5120:Bibcode
5001:Bibcode
4999:: 152.
4423:Bibcode
4373:Bibcode
4270:Bibcode
4133:)/2 = 2
4110:√
4072:
4057:
3975:
3955:
3925:
3907:
3897:, with
3877:in the
3811:
3796:
3683:
3668:
3664:
3649:
3645:
3630:
3178:is the
3092:
2946:≈ 10 m
2917:2.725 K
2878:Density
2751:of the
2662:gravity
2630:gravity
2598:around
2507:in the
2211:density
1618:is the
1535:
1466: C
1355:
1256:
1174:
1099:
1018:units)
1014:Value (
689:′, and
140:lengths
8262:Others
7910:N-body
7848:Indian
7823:Arabic
7686:Somali
7661:Libyan
7629:Africa
7600:Syrian
7550:Korean
7535:Indian
7525:Afghan
7489:Slovak
7459:Polish
7397:German
7380:French
7343:Danish
7321:Europe
7287:Metric
7218:Metric
7198:Stoney
7188:Planck
7173:Atomic
7004:
6994:
6969:
6959:
6917:
6817:
6783:
6673:
6618:
6573:
6563:
6528:
6498:: 74.
6468:
6460:
6399:
6313:
6305:
6251:
6241:
6233:
6169:
6159:
6111:
6071:
6063:
5995:
5939:
5929:
5889:
5881:
5775:
5767:
5698:Forbes
5670:
5600:
5590:
5553:
5498:
5488:
5450:
5407:
5278:
5084:
5052:
5042:
4976:18 May
4946:18 May
4916:31 May
4899:
4853:
4765:
4732:
4722:
4687:18 May
4654:18 May
4624:18 May
4594:18 May
4488:23 May
4433:
4421:: 24.
4391:
4331:585119
4329:
4319:
4264:(10).
3535:sphere
3346:
3330:1.2103
3188:vacuum
3104:proton
3019:length
2716:, the
2688:, and
2501:second
2441:5.5608
2346:5.1550
2197:1.2103
2086:(LMT)
2070:1.9561
1968:(LMT)
1966:energy
1958:kg⋅m/s
1874:(LMT)
1858:4.2217
1734:volume
1720:2.6121
1476:
1027:length
925:, and
915:length
884:, and
797:action
217:, the
204:, the
194:, the
184:, the
89:nature
70:, and
44:are a
7898:Other
7863:Roman
7843:Hindu
7838:Greek
7722:Cuban
7610:Tatar
7570:Omani
7509:Welsh
7504:Swiss
7424:Irish
7402:Greek
7348:Dutch
7338:Czech
7308:(CGS)
7302:(MTS)
7296:(MKS)
7268:(FPS)
7249:UK/US
6847:arXiv
6763:arXiv
6761:: 5.
6753:(1).
6708:arXiv
6671:S2CID
6645:arXiv
6616:S2CID
6526:S2CID
6500:arXiv
6466:S2CID
6432:arXiv
6397:S2CID
6371:arXiv
6311:S2CID
6277:arXiv
6203:arXiv
6167:S2CID
6080:(PDF)
6069:S2CID
6031:(PDF)
5887:S2CID
5773:S2CID
5739:arXiv
5668:S2CID
5642:arXiv
5578:. In
5551:S2CID
5525:arXiv
5276:S2CID
5189:arXiv
4910:(PDF)
4889:(PDF)
4833:arXiv
4555:(PDF)
4544:(PDF)
4389:S2CID
3621:(or 8
3614:but 4
3446:) + (
3385:1.416
3254:force
3184:light
3081:1.616
2956:≈ 10
2932:≈ 10
2615:2.176
2497:metre
2362:(LT)
2213:(LM)
2084:force
1344:1.416
1245:5.391
1163:2.176
1088:1.616
1005:Name
247:metre
208:, and
7615:Thai
7585:Pegu
7518:Asia
7156:(US)
7139:Troy
7002:OCLC
6992:ISBN
6967:OCLC
6957:ISBN
6933:2018
6915:ISBN
6815:ISBN
6781:OCLC
6571:OCLC
6561:ISBN
6542:2022
6458:ISSN
6345:2016
6303:PMID
6249:PMID
6231:ISSN
6157:ISBN
6127:2022
6109:ISBN
6061:PMID
6009:2022
5993:ISSN
5937:OCLC
5927:ISBN
5903:2022
5879:ISSN
5789:2022
5765:ISSN
5710:2019
5614:2022
5598:OCLC
5588:ISBN
5496:LCCN
5486:ISBN
5448:ISBN
5423:2010
5405:ISBN
5376:2022
5292:2022
5228:2021
5082:ISBN
5059:out.
5050:OCLC
5040:ISBN
4978:2024
4970:NIST
4948:2024
4940:NIST
4918:2020
4897:ISBN
4869:2019
4851:ISBN
4812:2021
4781:2020
4763:ISBN
4730:OCLC
4720:ISBN
4689:2024
4681:NIST
4656:2024
4648:NIST
4626:2024
4618:NIST
4596:2024
4588:NIST
4563:2019
4516:and
4506:and
4490:2020
4447:2022
4431:ISSN
4345:link
4327:OCLC
4317:ISBN
4197:and
3845:= −4
3647:(or
3572:and
3550:flux
3541:is 4
3389:(16)
3192:5.39
3180:time
3085:(18)
2846:Mass
2741:time
2676:and
2600:1.22
2590:and
2545:are
2541:and
2539:mass
2499:and
2352:kg/m
1736:(L)
1658:(L)
1656:area
1586:11.7
1537:and
1522:and
1508:and
1473:11.7
1348:(16)
1268:(Θ)
1249:(60)
1186:(T)
1184:time
1167:(24)
1111:(M)
1109:mass
1092:(18)
1029:(L)
923:mass
919:time
803:for
154:and
138:and
128:time
36:and
6888:doi
6726:doi
6663:doi
6608:doi
6518:doi
6450:doi
6389:doi
6367:441
6295:doi
6273:124
6239:PMC
6221:doi
6149:doi
6101:doi
6053:doi
6041:292
5985:doi
5871:doi
5829:doi
5757:doi
5660:doi
5543:doi
5332:doi
5268:doi
5167:doi
5155:135
5128:doi
4843:doi
4381:doi
4278:doi
4126:= (
4102:= 1
4049:= 1
4026:= (
3949:of
3885:or
3830:= −
3814:):
3741:= 1
3719:= 4
3511:= 0
3489:= 0
3473:iħγ
3442:= (
3387:784
3383:is
3225:2.5
3083:255
2972:Mpc
2805:Age
2774:).
2720:or
2712:In
2628:of
2586:In
2565:'s
2447:m/s
1539:2.2
1528:1.6
1346:784
1247:247
1165:434
1090:255
752:",
750:= 1
713:or
682:′,
573:any
122:or
32:In
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2606:eV
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