8408:
8381:
8420:
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.
3152:". 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.
835:... 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.
1388:, 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
567:
6847:
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.
2582:
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,
1652:
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
842:... 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".
6380:
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
5815:
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
2554:
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
2481:
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
2670:
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
3543:
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
3386:
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
587:
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
6811:
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
2363:
415:
5080:
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
3147:
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
2214:
424:
3382:
1875:
1514:
3448:
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
2545:
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
2087:
3688:) 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
3223:. 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.
1973:
3776:
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
2458:
262:
7007:
1361:
2240:
666:
3281:
3097:
1421:
5723:
1262:
1105:
7282:
3453:. 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.
2505:: 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.
1737:
1180:
2690:
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
2113:
1763:
817:
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
1618:
562:{\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}}}.}
3390:
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.
2675:, 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
1995:
976:
3266:
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.
761:, 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 "
1416:
933:
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.
7082:
3001:
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
1638:
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
7327:
5944:
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
996:
1901:
7048:
5711:
4991:
4961:
4702:
4669:
4639:
4609:
4366:
3136:, though whether those concepts are in fact simultaneously applicable is open to debate. (The same heuristic argument simultaneously motivates the Planck mass.)
7277:
3181:
2595:
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
1016:
904:
884:
864:
3238:
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
2761:, 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
2389:
6049:
5942:
7321:
3046:
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
4928:
7849:
7596:
4882:
7969:
7874:
2358:{\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}}}}
778:
wrote that, "Mathematically it is an acceptable trick which saves labour. Physically it represents a loss of information and can lead to confusion."
8110:
254:
5241:
1295:
7869:
7586:
7406:
2729:
8075:
410:{\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}}},}
5272:
8407:
4503:
7905:
7900:
7854:
7505:
7495:
7396:
7017:
6982:
6840:
6586:
6182:
6134:
5613:
5511:
5430:
5107:
5065:
4876:
4745:
2942:
2925:
7910:
7480:
7418:
7075:
5498:
2537:
and Planck time are conceptually linked at a fundamental physical level. Consequently, natural units help physicists to reframe questions.
6946:
5436:
4794:
3796:
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
7844:
7631:
593:
7884:
7859:
7712:
7460:
6140:
6022:
5537:
5047:
4922:
4788:
3707:
in the nondimensionalized form of
Coulomb's law as well, so that the nondimensionalized Maxwell's equations for electromagnetism and
8384:
7768:
7287:
6940:
5952:
5473:
5389:
4342:
7997:
4460:
1213:
1056:
6358:
5916:
4914:
4780:
4560:
3432:
at the Planck temperature might contain Planck-scale black holes, constantly being formed from thermal radiation and decaying via
2490:, i.e., when the universe was less than one Planck time old. Describing the universe during the Planck epoch requires a theory of
8100:
7738:
7702:
7611:
7349:
2599:, where 𝐺 is the gravitational constant in natural units. A similar limitation applies to the precise synchronization of clocks.
7824:
7809:
7763:
7641:
7616:
7601:
7561:
7485:
7068:
2867:
2209:{\displaystyle F_{\text{P}}={\frac {E_{\text{P}}}{l_{\text{P}}}}={\frac {\hbar }{l_{\text{P}}t_{\text{P}}}}={\frac {c^{4}}{G}}}
1685:
1138:
5627:
3377:{\displaystyle F_{\text{P}}={\frac {m_{\text{P}}c}{t_{\text{P}}}}={\frac {c^{4}}{G}}\approx \mathrm {1.2103\times 10^{44}~N} }
8290:
8158:
7819:
7804:
7794:
7784:
7722:
7692:
7662:
7626:
7551:
7470:
7445:
7440:
7435:
7428:
7132:
6612:
5802:
45:
8085:
8047:
5305:
1870:{\displaystyle l_{\text{P}}^{3}=\left({\frac {\hbar G}{c^{3}}}\right)^{\frac {3}{2}}={\sqrt {\frac {(\hbar G)^{3}}{c^{9}}}}}
6288:
Wendel, Garrett; Martínez, Luis; Bojowald, Martin (19 June 2020). "Physical
Implications of a Fundamental Period of Time".
1567:. 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
8173:
7962:
7814:
7789:
7753:
7717:
7697:
7687:
7672:
7667:
7657:
7566:
7490:
7391:
7122:
7112:
6776:
6555:
4287:
1509:{\displaystyle q_{\text{P}}={\sqrt {4\pi \epsilon _{0}\hbar c}}\approx 1.875546\times 10^{-18}{\text{ C}}\approx 11.7\ e.}
1385:
1037:
238:
3457:
8178:
7879:
7799:
7758:
7748:
7677:
7581:
7520:
7515:
7475:
7465:
7455:
7450:
7354:
7256:
7251:
7170:
7165:
7127:
6215:
3239:
2682:
While physicists have a fairly good understanding of the other fundamental interactions of forces on the quantum level,
8256:
8057:
7204:
4108:
3610:
2632:
7707:
7682:
7621:
7571:
7556:
7546:
7510:
7401:
7364:
7333:
5751:
5654:
3117:
2654:
become significant. At this scale, present descriptions and theories of sub-atomic particle interactions in terms of
1581:
4825:
3428:; a quantum theory of gravity would be required to model the extreme energies attained. Hypothetically, a system in
8398:
8153:
7743:
7591:
7525:
7423:
7369:
7359:
5053:
3839:
3544:
fundamental equations of physics, is not evident, and the values of the Planck units are sensitive to this choice.
3129:
2723:
2487:
163:
8266:
8243:
8193:
8163:
7636:
7175:
6383:
6098:
5966:
4246:
4186:
4010:
3263:
3140:
8274:
8450:
8320:
7955:
6780:
4905:
4104:
3960:
2672:
2468:
2373:
584:
8120:
8012:
3602:
around a point charge will be distributed uniformly over that sphere. From this, it follows that a factor of 4
2082:{\displaystyle E_{\text{P}}=m_{\text{P}}c^{2}={\frac {\hbar }{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{G}}}}
8310:
8052:
7199:
7150:
6893:
6290:
5503:
4848:
4242:
4191:
3972:
3900:
3708:
2782:
2707:
1641:
791:
234:
205:
66:
6867:(17 July 2018). "Scrambling Time and Causal Structure of the Photon Sphere of a Schwarzschild Black Hole".
5941:(1992). "First Steps of Quantum Gravity and the Planck Values". In Eisenstaedt, Jean; Kox, Anne J. (eds.).
3617:
would change if space were higher-dimensional; the correct expressions can be deduced from the geometry of
8440:
7992:
6657:
5828:
5466:
The
Constants of Nature: From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe
4328:
3908:
3782:
2583:
or, similarly, times that measured of the order of Planck time, did carry special implications related to
2547:
2517:
arbitrariness, but do still involve some arbitrary choices in terms of the defining constants. Unlike the
195:
60:
3632:
Hence a substantial body of physical theory developed since Planck's 1899 paper suggests normalizing not
3421:
reaches the Planck length. There are no known physical models able to describe temperatures greater than
8218:
8130:
8027:
8017:
7091:
6968:
5231:
5169:
Mead, Chester Alden (10 August 1964). "Possible
Connection Between Gravitation and Fundamental Length".
4385:
3450:
3160:
3010:
2949:
2588:
96:
3626:
3598:
produced by point objects have spherical symmetry, and so the electric flux through a sphere of radius
2785:, 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
8251:
6901:
6794:
6739:
6676:
6621:
6531:
6463:
6402:
6308:
6234:
6066:
5998:
5884:
5842:
5770:
5673:
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5385:
5345:
5281:
5180:
5141:
5022:
4539:
4444:
4394:
4291:
4181:
3133:
2778:
2703:
2655:
2568:
954:
826:
572:
151:
6581:. International series of monographs on physics. Vol. 155. Oxford University Press. p. 5.
3278:
in the Planck system if the Planck units of time, length, and mass are considered to be base units.
8445:
7864:
7384:
7315:
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6206:
6057:
5097:
4491:
4112:
3904:
3595:
3591:
3552:
3429:
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2887:
2845:
2826:
2811:
2478:
7054:
1394:
8305:
8233:
8188:
7379:
7155:
6868:
6784:
6767:
Afshordi, Niayesh (1 March 2012). "Where will
Einstein fail? Leasing for Gravity and cosmology".
6729:
6720:
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6666:
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6453:
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5908:
5794:
5760:
5689:
5663:
5572:
5546:
5297:
5210:
4854:
4533:
4410:
4360:
4332:
4238:
4225:
3778:
3575:
3242:
2786:
2735:
2613:
2556:
2495:
909:
822:
218:
174:
155:
71:
37:
4853:. Fundamental Theories of Physics. Vol. 119. Dordrecht: Kluwer Academic. pp. 347–352.
3711:
both take the same form as those for electromagnetism in SI, which do not have any factors of 4
1418:
is added to the list of constants used by Planck. Thus, this charge unit is typically given by
794:, noting that electric charge is quantized, derived units of length, time, and mass, now named
8350:
8345:
8125:
8090:
7023:
7013:
6988:
6978:
6936:
6928:
6836:
6802:
6592:
6582:
6479:
6324:
6270:
6252:
6178:
6130:
6082:
6014:
5958:
5948:
5900:
5786:
5619:
5609:
5517:
5507:
5469:
5426:
5420:
5103:
5071:
5061:
4918:
4872:
4784:
4774:
4751:
4741:
4452:
4348:
4338:
4320:
4196:
3433:
3418:
2770:
2687:
2584:
2572:
807:
49:
3139:
The Planck length is a distance scale of interest in speculations about quantum gravity. The
1968:{\displaystyle m_{\text{P}}c={\frac {\hbar }{l_{\text{P}}}}={\sqrt {\frac {\hbar c^{3}}{G}}}}
1623:
8168:
7411:
6909:
6747:
6684:
6629:
6539:
6471:
6410:
6316:
6260:
6242:
6170:
6122:
6074:
6006:
5892:
5850:
5778:
5681:
5564:
5353:
5289:
5188:
5149:
4864:
4402:
4299:
3916:
3886:
2891:
2676:
2609:
2579:
is far less than the unit mass in a system that treats both forces as having the same form.
1389:
981:
818:
242:
33:
2501:
Several quantities are not "extreme" in magnitude, such as the Planck mass, which is about
588:
symbols of physical quantity, without being scaled explicitly by their corresponding unit:
8355:
7978:
7576:
6849:
6118:
Quantum Field Theory II: Introductions to
Quantum Gravity, Supersymmetry and String Theory
6045:
5984:
5980:
5938:
5833:
5601:
5493:
5263:
5171:
5057:
4733:
4517:
4220:
2973:
2766:
2691:
2659:
2647:
2564:
2514:
2491:
159:
104:
5377:
753:, but not as a direct equality of quantities. This may seem to be "setting the constants
6905:
6798:
6743:
6680:
6625:
6535:
6467:
6406:
6312:
6238:
6070:
6002:
5888:
5846:
5774:
5677:
5560:
5349:
5285:
5184:
5145:
5026:
4448:
4432:
4398:
4295:
813:
would be numerically equal to 1. In 1899, one year before the advent of quantum theory,
8424:
8412:
8325:
7926:
7500:
7117:
6350:
6265:
6210:
5489:
5461:
4573:
4428:
4380:
3924:
3579:
3489:
3166:
2774:
2571:
quantities. Rather, the disparity of magnitude of force is a manifestation of that the
2453:{\displaystyle a_{\text{P}}={\frac {c}{t_{\text{P}}}}={\sqrt {\frac {c^{7}}{\hbar G}}}}
1001:
889:
869:
849:
775:
185:
147:
54:
6441:
6078:
8434:
8300:
8210:
7374:
7246:
7214:
7186:
6888:
6696:
6641:
6610:
de
Sabbata, Venzo; Sivaram, C. (1993). "On limiting field strengths in gravitation".
6491:
6422:
6336:
6192:
6116:
5989:
5912:
5798:
5327:
5132:
5123:
5093:
4414:
4278:
4273:
4216:
4140:. Also, the Bekenstein–Hawking formula for the entropy of a black hole simplifies to
4021:
3912:
3156:
3144:
2799:
Table 3 lists properties of the observable universe today expressed in Planck units.
2695:
2538:
2534:
2224:
1979:
787:
84:
6551:
6094:
5693:
5576:
5535:
Barrow, John D.; Shaw, Douglas J. (2011). "The value of the cosmological constant".
5301:
4694:
3440:
its temperature by creating larger black holes, whose
Hawking temperature is lower.
2765:
is meaningful for values smaller than the Planck time. It is generally assumed that
8419:
8228:
8135:
8042:
8032:
8007:
7931:
7219:
7194:
7044:
6320:
6121:. Graduate Texts in Physics. Cham: Springer International Publishing. p. 187.
5868:
5707:
5597:
5336:
5236:
3920:
3587:
3149:
2627:
2617:
2526:
2381:
1747:
1575:
795:
5746:
4661:
4631:
4601:
4516:
pp. 478–80 contain the first appearance of the Planck base units, and of the
3800:
appearing in the equations of physics are to be eliminated via the normalization.
1653:
for empirical or practical use and there are large uncertainties in their values.
17:
5267:
4953:
2781:. Immeasurably hot and dense, the state of the Planck epoch was succeeded by the
8295:
7530:
7261:
7160:
6711:
6437:
6351:"HiRes – The High Resolution Fly's Eye Ultra High Energy Cosmic Ray Observatory"
5043:
4983:
4496:
Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin
4176:
2860:
2751:
2711:
2576:
1885:
1287:
948:
6475:
6174:
177:
that, by definition, have a numeric value 1 when expressed in these units are:
8360:
8145:
8080:
8002:
7027:
6913:
6864:
6688:
6655:
Gibbons, G. W. (2002). "The Maximum Tension Principle in General Relativity".
6509:
6126:
5568:
5403:
5358:
5331:
4770:
4487:
4324:
3729:. When applied additionally to gravitation and Planck units, these are called
3583:
2882:
814:
162:
are expected to dominate. One example is represented by the conditions in the
100:
80:
6992:
6806:
6596:
6483:
6256:
6018:
5904:
5790:
5623:
5192:
5075:
4456:
1356:{\displaystyle T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{Gk_{\text{B}}^{2}}}}}
8315:
8183:
8095:
8067:
6752:
6715:
5962:
5593:
5206:
4755:
4569:
3128:. It can be motivated in various ways, such as considering a particle whose
6543:
6328:
6274:
6086:
5854:
4562:
Natural Systems of Units. To the Centenary Anniversary of the Planck System
6972:
4868:
4352:
8223:
8022:
7060:
6397:
5407:
4817:
3629:
when relating the gravitational potential to the distribution of matter.
3618:
2747:
2699:
2502:
2483:
1893:
167:
6381:
beyond the expected spectral cutoff due to cosmic microwave radiation".
6247:
4570:
Proceedings Of The XXII Workshop On High Energy Physics And Field Theory
7936:
7009:
Curvature of Space and Time, with an Introduction to Geometric Analysis
6671:
6633:
5896:
5293:
5215:
4406:
2683:
2671:
which the effects of quantum gravity can no longer be ignored in other
2651:
2232:
6458:
6010:
5782:
5685:
5154:
5127:
4859:
4303:
3159:
are modeled to be on the order of the Planck length. In theories with
2769:
dominate physical interactions at this time scale. At this scale, the
829:. Planck underlined the universality of the new unit system, writing:
7606:
5872:
5719:
3911:. These equations have the same form as Maxwell's equations (and the
3556:
3219:
3209:
3125:
3092:{\displaystyle \ell _{\mathrm {P} }={\sqrt {\frac {\hbar G}{c^{3}}}}}
3040:
2993:
2522:
1987:
1755:
1048:
936:
139:
115:
92:
88:
3621:.) Likewise for Newton's law of universal gravitation: a factor of 4
2658:
break down and become inadequate, due to the impact of the apparent
6873:
6734:
6414:
6303:
5765:
237:, from which all other quantities and units may be derived. In the
7947:
7055:
The Planck scale: relativity meets quantum mechanics meets gravity
6929:"A Model of Topological Quantization of the Electromagnetic Field"
6789:
6526:
6229:
5668:
5551:
3275:
3274:
The Planck unit of force may be thought of as the derived unit of
3205:
2518:
2105:
2097:
1022:
Table 1: Modern values for Planck's original choice of quantities
846:
Planck considered only the units based on the universal constants
821:, now usually known as the Planck constant, which appeared in the
246:
5521:
4383:(1980). "The application of dimensional analysis to cosmology".
3613:. (Both the numerical factor and the power of the dependence on
3571:
3201:
2762:
2560:
1677:
1205:
1130:
944:
940:
127:
7951:
7064:
5652:
Adler, Ronald J. (2010). "Six easy roads to the Planck scale".
3009:) squared. Barrow and Shaw proposed a modified theory in which
6514:
Symmetry, Integrability and Geometry: Methods and Applications
3245:
had a measured energy of about 50 J, equivalent to about
3013:
is a field evolving in such a way that its value remains Λ ~
1257:{\displaystyle t_{\text{P}}={\sqrt {\frac {\hbar G}{c^{5}}}}}
1100:{\displaystyle l_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}}
951:. His definitions differ from the modern ones by a factor of
6852:
would be necessary, but such a theory has yet to be written.
103:, they are relevant in research on unified theories such as
7012:. Providence, Rhode Island: American Mathematical Society.
2530:
1669:
3555:
because in three-dimensional space, the surface area of a
3417:. At this temperature, the wavelength of light emitted by
3163:, the Planck length calculated from the observed value of
3116:(the two digits enclosed by parentheses are the estimated
5712:"What Is The Smallest Possible Distance In The Universe?"
4907:
Quantum Field Theory I: Basics in Mathematics and Physics
3183:
can be smaller than the true, fundamental Planck length.
1732:{\displaystyle l_{\text{P}}^{2}={\frac {\hbar G}{c^{3}}}}
1175:{\displaystyle m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}}
2477:
Some Planck units, such as of time and length, are many
1548:
in SI units respectively have relative uncertainties of
6165:(December 2021). "From the S Matrix to String Theory".
4988:
The NIST Reference on Constants, Units, and Uncertainty
4958:
The NIST Reference on Constants, Units, and Uncertainty
4699:
The NIST Reference on Constants, Units, and Uncertainty
4666:
The NIST Reference on Constants, Units, and Uncertainty
4636:
The NIST Reference on Constants, Units, and Uncertainty
4606:
The NIST Reference on Constants, Units, and Uncertainty
3768:
There are several possible alternative normalizations.
3120:
associated with the reported numerical value) or about
2789:, which ended after about 10 seconds (or about 10
4984:"2022 CODATA Value: Newtonian constant of gravitation"
1584:
8396:
5598:"Higher-Dimensional Algebra and Planck-Scale Physics"
3715:. When this is applied to electromagnetic constants,
3284:
3169:
3049:
2392:
2243:
2116:
1998:
1904:
1766:
1688:
1626:
1424:
1397:
1298:
1216:
1141:
1059:
1004:
984:
957:
912:
892:
872:
852:
596:
526:
490:
456:
427:
265:
6211:"Minimal Length Scale Scenarios for Quantum Gravity"
579:
system of quantities, but the second equation, with
8338:
8283:
8265:
8242:
8209:
8202:
8144:
8109:
8066:
7985:
7919:
7893:
7837:
7777:
7731:
7650:
7539:
7342:
7308:
7301:
7270:
7239:
7232:
7185:
7143:
7105:
7098:
4850:
The Landscape of Theoretical Physics: A Global View
4437:
Quarterly Journal of the Royal Astronomical Society
3609:will appear in the denominator of Coulomb's law in
2679:at which a black hole could be formed by collapse.
6442:"Conservation laws in "doubly special relativity""
3788:in Newton's law. In theories emerging after 1899,
3736:The rationalized Planck units are defined so that
3376:
3175:
3091:
2452:
2357:
2208:
2081:
1967:
1869:
1731:
1632:
1612:
1508:
1410:
1355:
1256:
1174:
1099:
1010:
990:
970:
927:
898:
878:
858:
661:{\displaystyle F'={\frac {m_{1}'m_{2}'}{r'^{2}}}.}
660:
561:
409:
99:. Originally proposed in 1899 by German physicist
7047:, including the Planck units, as reported by the
6716:"Counterexamples to the maximum force conjecture"
5371:
5369:
5013:Ray, T. P. (1981). "Stoney's Fundamental Units".
4954:"2022 CODATA Value: vacuum electric permittivity"
4115:, for gravitation. Planck units modified so that
3792:nearly always appears in formulae multiplied by 4
2694:is necessary. Approaches to this problem include
5608:. Cambridge University Press. pp. 172–195.
5128:"Scaling Mount Planck I: A View from the Bottom"
3208:to travel a distance of 1 Planck length in
1657:Table 2: Coherent derived units of Planck units
1613:{\textstyle {\sqrt {{1}/{\alpha }}}\approx 11.7}
5816:one gains intuition into a new physical regime.
4689:
4687:
2593:
2543:
1578:, Planck base units are all larger by a factor
840:
831:
245:include length with the associated unit of the
48:defined exclusively in terms of four universal
7049:National Institute of Standards and Technology
5456:
5454:
4596:
4594:
2686:is problematic, and cannot be integrated with
7963:
7076:
6974:General Relativity and the Einstein Equations
6931:. In M. Ferrero; Alwyn van der Merwe (eds.).
6769:Bulletin of the Astronomical Society of India
5232:"Can experiment access Planck-scale physics?"
4482:
4480:
4478:
4315:
4313:
4009:from the Bekenstein–Hawking formula (for the
3650:) to 1. Doing so would introduce a factor of
798:in his honor. Stoney chose his units so that
8:
5873:"Gravitational fields and quantum mechanics"
5647:
5645:
5606:Physics Meets Philosophy at the Planck Scale
5419:Kolb, Edward W.; Turner, Michael S. (1994).
5179:(3B). American Physical Society: B849–B862.
3212:, which is a time interval of approximately
2494:that would incorporate quantum effects into
833:
5588:
5586:
3781:in 1915). Hence Planck normalized to 1 the
8206:
7970:
7956:
7948:
7305:
7236:
7102:
7083:
7069:
7061:
4365:: CS1 maint: location missing publisher (
3975:in free space to 1 (normally expressed as
3141:Bekenstein–Hawking entropy of a black hole
2803:Table 3: Today's universe in Planck units
87:, defined using fundamental properties of
6872:
6788:
6751:
6733:
6670:
6525:
6457:
6396:
6302:
6264:
6246:
6228:
6167:Geoffrey Chew: Architect of the Bootstrap
5764:
5667:
5550:
5357:
5214:
5153:
4858:
4276:(2005). "On Absolute Units, I: Choices".
3361:
3350:
3336:
3330:
3319:
3305:
3298:
3289:
3283:
3168:
3080:
3065:
3055:
3054:
3048:
2631:(the Planck energy, corresponding to the
2575:is approximately the unit charge but the
2433:
2426:
2415:
2406:
2397:
2391:
2346:
2332:
2326:
2315:
2310:
2299:
2289:
2278:
2273:
2263:
2257:
2248:
2242:
2195:
2189:
2177:
2167:
2157:
2146:
2136:
2130:
2121:
2115:
2066:
2055:
2044:
2035:
2026:
2016:
2003:
1997:
1952:
1941:
1930:
1921:
1909:
1903:
1858:
1847:
1830:
1816:
1804:
1790:
1776:
1771:
1765:
1721:
1707:
1698:
1693:
1687:
1625:
1597:
1592:
1587:
1585:
1583:
1486:
1477:
1450:
1438:
1429:
1423:
1402:
1396:
1343:
1338:
1323:
1312:
1303:
1297:
1245:
1230:
1221:
1215:
1155:
1146:
1140:
1088:
1073:
1064:
1058:
1003:
983:
958:
956:
918:
917:
911:
891:
871:
851:
712:being the dimensionless ratio quantities
646:
628:
615:
608:
595:
548:
535:
525:
506:
496:
489:
472:
462:
455:
448:
437:
428:
426:
396:
385:
375:
368:
356:
351:
340:
335:
325:
318:
303:
292:
282:
275:
264:
5947:. Boston: Birkhäuser. pp. 364–379.
4130:, because the Planck mass is divided by
3017:throughout the history of the universe.
2801:
1655:
1020:
170:, approximately 13.8 billion years ago.
27:Units defined only by physical constants
8403:
6933:Fundamental Problems in Quantum Physics
4695:"2022 CODATA Value: Planck temperature"
4262:
4208:
3436:. Adding energy to such a system might
3069:
2730:Time-variation of fundamental constants
2440:
2339:
2292:
2159:
2059:
2037:
1945:
1923:
1837:
1793:
1710:
1456:
1316:
1234:
1159:
1077:
985:
643:
6927:Rañada, Antonio F. (31 October 1995).
6835:. W. H. Freeman Company. p. 117.
5726:from the original on 18 September 2021
5273:Communications in Mathematical Physics
4492:"Über irreversible Strahlungsvorgänge"
4358:
716:the standard quantities, written e.g.
6949:from the original on 1 September 2020
5805:from the original on 30 December 2017
5747:"Three new roads to the Planck scale"
5392:from the original on 28 November 2018
5244:from the original on 30 November 2020
5038:
5036:
4885:from the original on 5 September 2021
4728:
4726:
4724:
4722:
4720:
4554:
4552:
4550:
4506:from the original on 17 November 2020
3733:and are seen in high-energy physics.
2943:cosmic microwave background radiation
978:, because the modern definitions use
255:Newton's law of universal gravitation
95:) rather than properties of a chosen
7:
6169:. World Scientific. pp. 72–83.
5499:The Anthropic Cosmological Principle
4463:from the original on 20 January 2022
4268:
4266:
3903:(GEM) equations, which hold in weak
3539:Alternative choices of normalization
2662:of gravity within current theories.
2498:. Such a theory does not yet exist.
6977:. Oxford: Oxford University Press.
6361:from the original on 15 August 2009
3387:Newtonian attraction between them.
2533:system for historical reasons, the
7113:International System of Units (SI)
7045:Value of the fundamental constants
6558:from the original on 19 March 2022
6143:from the original on 24 March 2022
6025:from the original on 22 March 2022
5919:from the original on 24 March 2022
5745:Faraoni, Valerio (November 2017).
5538:General Relativity and Gravitation
5268:"Particle Creation by Black Holes"
5230:Bingham, Robert (4 October 2006).
5049:Quantum Field Theory in a Nutshell
4602:"2022 CODATA Value: Planck length"
4572:. pp. 287–296. Archived from
3570:. This, along with the concept of
3370:
3056:
919:
25:
8385:Template:Quantum mechanics topics
6891:(1983). "Kaluza-Klein Monopole".
6079:10.1038/scientificamerican0505-48
5439:from the original on 6 March 2021
5209:(2002). "Quest For Unification".
4934:from the original on 19 June 2020
4828:from the original on 3 March 2021
4797:from the original on 6 March 2021
8418:
8406:
8380:
8379:
8101:Topological quantum field theory
5630:from the original on 8 July 2021
5308:from the original on 5 July 2014
4662:"2022 CODATA Value: Planck time"
4632:"2022 CODATA Value: Planck mass"
4532:in this paper correspond to the
4245:propagates at the same speed as
8086:Jackiw–Teitelboim gravity
5406:" at the very beginning of the
4738:A First Course in String Theory
2840:4.35 × 10 s or 1.38 × 10 years
971:{\displaystyle {\sqrt {2\pi }}}
935:to arrive at natural units for
158:are not expected to apply, and
8291:Causal dynamical triangulation
8159:Black hole information paradox
6613:Foundations of Physics Letters
6321:10.1103/PhysRevLett.124.241301
6115:Manoukian, Edouard B. (2016).
4740:. Cambridge University Press.
3722:, this unit system is called "
3143:is one-fourth the area of its
2754:was equal to the Planck time,
1844:
1834:
46:system of units of measurement
1:
8257:Wheeler–DeWitt equation
7123:US customary units (USCS/USC)
7057:from 'Einstein Light' at UNSW
6777:Astronomical Society of India
5985:"What's bad about this habit"
5402:Discusses "Planck time" and "
5102:. New York: Alfred A. Knopf.
4433:"Natural Units Before Planck"
4288:American Institute of Physics
2746:is the earliest stage of the
2557:comparing apples with oranges
1386:International System of Units
790:was introduced in 1874, when
239:International System of Units
91:(specifically, properties of
8179:Cosmic censorship hypothesis
7998:Batalin–Vilkovisky formalism
6216:Living Reviews in Relativity
5425:. Basic Books. p. 447.
4223:do so, as does the textbook
3811:to 1 (and therefore setting
3444:Nondimensionalized equations
3240:ultra-high-energy cosmic ray
1537:are exact and the values of
1411:{\displaystyle k_{\text{e}}}
670:This last equation (without
6848:To go even farther back, a
5752:American Journal of Physics
5655:American Journal of Physics
3030:The Planck length, denoted
1518:In SI units, the values of
928:{\displaystyle k_{\rm {B}}}
8467:
8275:Hartle–Hawking state
8174:Bousso's holographic bound
8154:Black hole complementarity
6476:10.1103/PhysRevD.68.045001
6175:10.1142/9789811219832_0013
5054:Princeton University Press
5015:Irish Astronomical Journal
4904:Zeidler, Eberhard (2006).
4520:, which Planck denoted by
3619:higher-dimensional spheres
3262:Proposals for theories of
3130:reduced Compton wavelength
2767:quantum effects of gravity
2727:
2724:Chronology of the universe
2721:
166:of our universe after the
160:quantum effects of gravity
8371:
8267:Euclidean quantum gravity
8244:Canonical quantum gravity
8194:Gravitational singularity
8164:Black-hole thermodynamics
8048:Ryu–Takayanagi conjecture
7176:English Engineering Units
6935:. Springer. p. 271.
6914:10.1103/PhysRevLett.51.87
6850:quantum theory of gravity
6510:"The Soccer-Ball Problem"
6384:The Astrophysical Journal
6127:10.1007/978-3-319-33852-1
5831:(January 1955). "Geons".
5569:10.1007/s10714-011-1199-1
5359:10.4249/scholarpedia.7117
4822:The Physics Hypertextbook
4776:Encyclopedia of Distances
4247:electromagnetic radiation
4187:Doubly special relativity
4096:). This would eliminate 8
4030:) which is simplified to
3731:rationalized Planck units
3264:doubly special relativity
2616:, the Planck scale is an
2597:𝛥𝑥 ≳ √𝐺 = 1.6 × 10 cm
2513:Planck units have little
1384:Unlike the case with the
583:absent, is relating only
8321:Superfluid vacuum theory
8076:2+1D topological gravity
6781:Astrophysics Data System
5604:; Huggett, Nick (eds.).
5193:10.1103/PhysRev.135.B849
4105:Einstein field equations
3961:characteristic impedance
3574:, are the basis for the
3458:energy–momentum relation
3150:foam at the Planck scale
3124:times the diameter of a
2890:(sometimes known as the
2779:unified with gravitation
2673:fundamental interactions
585:dimensionless quantities
573:dimensionally consistent
8311:Noncommutative geometry
8058:Weinberg–Witten theorem
8053:Trans-Planckian problem
7215:Quantum chromodynamical
6833:The Hour of Our Delight
6831:Reeves, Hubert (1991).
6753:10.3390/universe7110403
6689:10.1023/A:1022370717626
5504:Oxford University Press
5378:"Birth of the Universe"
4559:Tomilin, K. A. (1999).
4243:gravitational radiation
4192:Trans-Planckian problem
4109:Einstein–Hilbert action
4073:(and therefore setting
4011:entropy of a black hole
3973:gravitational radiation
3840:Gauss's law for gravity
3709:gravitoelectromagnetism
3398:The Planck temperature
2859:8.7 × 10 m or 9.2 × 10
2783:grand unification epoch
2708:noncommutative geometry
2677:approximate lower limit
2666:Relationship to gravity
2486:does not extend to the
1642:fine-structure constant
1633:{\displaystyle \alpha }
792:George Johnstone Stoney
206:reduced Planck constant
83:. They are a system of
7993:AdS/CFT correspondence
7328:centimetre–gram–second
7092:Systems of measurement
6969:Choquet-Bruhat, Yvonne
6658:Foundations of Physics
6577:Kiefer, Claus (2012).
6544:10.3842/SIGMA.2014.074
5855:10.1103/PhysRev.97.511
4847:Pavšic, Matej (2001).
3909:locally flat spacetime
3901:gravitoelectromagnetic
3783:gravitational constant
3772:Gravitational constant
3378:
3177:
3161:large extra dimensions
3093:
2881:3 × 10 kg or 1.5 × 10
2601:
2552:
2454:
2359:
2210:
2083:
1969:
1871:
1733:
1634:
1614:
1510:
1412:
1357:
1258:
1176:
1101:
1012:
992:
991:{\displaystyle \hbar }
972:
929:
900:
880:
860:
844:
839:
834:
782:History and definition
662:
563:
411:
196:gravitational constant
8219:Bosonic string theory
8131:Semiclassical gravity
8028:Holographic principle
8018:Gravitational anomaly
7850:Biblical and Talmudic
7316:metre–kilogram–second
7006:Stavrov, Iva (2020).
6050:"Quantum Black Holes"
4869:10.1007/0-306-47136-1
4818:"Blackbody Radiation"
4386:Space Science Reviews
4013:in terms of its mass
3625:naturally appears in
3456:Examples include the
3451:nondimensionalization
3379:
3178:
3132:is comparable to its
3094:
3021:Analysis of the units
2950:Cosmological constant
2885:(only counting stars)
2728:Further information:
2660:non-renormalizability
2589:uncertainty principle
2455:
2360:
2211:
2084:
1970:
1872:
1734:
1635:
1615:
1511:
1413:
1358:
1259:
1177:
1102:
1013:
993:
973:
930:
901:
881:
861:
714:corresponding to
663:
575:and equally valid in
564:
419:can be expressed as:
412:
8252:Loop quantum gravity
8111:Quantum field theory
6506:Hossenfelder, Sabine
6207:Hossenfelder, Sabine
6104:on 14 February 2019.
5386:University of Oregon
4769:Deza, Michel Marie;
4182:Dimensional analysis
4128:reduced Planck units
4020:and the area of its
3905:gravitational fields
3596:electrostatic fields
3586:operator applied to
3282:
3270:Planck unit of force
3167:
3134:Schwarzschild radius
3047:
2704:loop quantum gravity
2656:quantum field theory
2635:of the Planck mass,
2541:puts it succinctly:
2390:
2241:
2114:
1996:
1902:
1764:
1686:
1624:
1582:
1422:
1395:
1296:
1214:
1139:
1057:
1002:
982:
955:
910:
890:
870:
850:
827:black-body radiation
594:
425:
263:
152:quantum field theory
130:intervals of around
8121:Bunch–Davies vacuum
8113:in curved spacetime
8013:Faddeev–Popov ghost
6906:1983PhRvL..51...87S
6799:2012BASI...40....1A
6744:2021Univ....7..403J
6714:(3 February 2021).
6681:2002FoPh...32.1891G
6626:1993FoPhL...6..561D
6536:2014SIGMA..10..074H
6468:2003PhRvD..68d5001J
6407:1995ApJ...441..144B
6313:2020PhRvL.124x1301W
6248:10.12942/lrr-2013-2
6239:2013LRR....16....2H
6071:2005SciAm.292e..48C
6058:Scientific American
6046:Giddings, Steven B.
6003:2009PhT....62e...8M
5889:1958NCim....7..215R
5847:1955PhRv...97..511W
5775:2017AmJPh..85..865F
5678:2010AmJPh..78..925A
5561:2011GReGr..43.2555B
5350:2008SchpJ...3.7117R
5286:1975CMaPh..43..199H
5185:1964PhRv..135..849M
5146:2001PhT....54f..12W
5099:The Road to Reality
5052:(second ed.).
5027:1981IrAJ...15..152R
4579:on 12 December 2020
4449:1983QJRAS..24...24B
4399:1980SSRv...27..109W
4296:2005PhT....58j..12W
4113:Friedmann equations
3553:theoretical physics
3434:Hawking evaporation
3430:thermal equilibrium
3003:age of the universe
2992:≈ 10 s ≈ 10 (km/s)/
2941:temperature of the
2812:observable universe
2804:
2479:orders of magnitude
2320:
2283:
1781:
1703:
1658:
1348:
1284:Planck temperature
1023:
636:
623:
571:Both equations are
361:
345:
241:, for example, the
175:universal constants
8306:Group field theory
8234:Superstring theory
8189:Firewall (physics)
7322:metre–tonne–second
7118:UK imperial system
6634:10.1007/BF00662806
6355:www.cosmic-ray.org
6044:Carr, Bernard J.;
5897:10.1007/BF02744199
5871:(1 January 1958).
5716:Starts with a Bang
5468:. Pantheon Books.
5422:The Early Universe
5376:Schombert, James.
5294:10.1007/BF02345020
4407:10.1007/bf00212237
4321:Misner, Charles W.
4239:General relativity
4215:For example, both
3779:general relativity
3627:Poisson's equation
3576:inverse-square law
3394:Planck temperature
3374:
3231:The Planck energy
3173:
3089:
2816:Approximate number
2802:
2787:inflationary epoch
2736:Big Bang cosmology
2614:physical cosmology
2496:general relativity
2450:
2355:
2306:
2269:
2206:
2079:
1965:
1867:
1767:
1729:
1689:
1656:
1630:
1610:
1506:
1408:
1353:
1334:
1254:
1172:
1097:
1021:
1008:
988:
968:
925:
896:
876:
856:
823:Wien approximation
658:
624:
611:
559:
542:
513:
479:
407:
347:
331:
243:SI base quantities
219:Boltzmann constant
156:general relativity
50:physical constants
38:physical cosmology
18:Planck temperature
8394:
8393:
8351:Eternal inflation
8346:Quantum cosmology
8334:
8333:
8126:Hawking radiation
8091:Liouville gravity
7945:
7944:
7833:
7832:
7297:
7296:
7288:Foot–pound–second
7228:
7227:
7205:Heaviside–Lorentz
7019:978-1-4704-6313-7
6984:978-0-19-155226-7
6842:978-0-7167-2220-5
6665:(12): 1891–1901.
6588:978-0-191-62885-6
6446:Physical Review D
6440:(4 August 2003).
6209:(December 2013).
6184:978-981-12-1982-5
6136:978-3-319-33851-4
6011:10.1063/1.3141952
5969:on 25 April 2019.
5783:10.1119/1.4994804
5686:10.1119/1.3439650
5615:978-0-521-66280-2
5545:(10): 2555–2560.
5513:978-0-19-282147-8
5432:978-0-201-62674-2
5382:HC 441: Cosmology
5332:"Quantum gravity"
5155:10.1063/1.1387576
5109:978-0-679-45443-4
5067:978-0-691-14034-6
4878:978-0-7923-7006-2
4747:978-0-521-83143-7
4304:10.1063/1.2138392
4203:Explanatory notes
4197:Zero-point energy
3875:in Planck units).
3611:rationalized form
3551:is ubiquitous in
3419:thermal radiation
3369:
3345:
3325:
3322:
3308:
3292:
3176:{\displaystyle G}
3087:
3086:
2999:
2998:
2777:is assumed to be
2712:causal set theory
2688:quantum mechanics
2633:energy equivalent
2525:, which exist as
2475:
2474:
2448:
2447:
2421:
2418:
2400:
2353:
2321:
2313:
2302:
2284:
2276:
2266:
2251:
2204:
2184:
2180:
2170:
2152:
2149:
2139:
2124:
2077:
2076:
2050:
2047:
2019:
2006:
1963:
1962:
1936:
1933:
1912:
1865:
1864:
1824:
1810:
1774:
1727:
1696:
1602:
1499:
1489:
1462:
1432:
1405:
1382:
1381:
1351:
1350:
1341:
1306:
1252:
1251:
1224:
1170:
1169:
1149:
1095:
1094:
1067:
1011:{\displaystyle h}
966:
899:{\displaystyle c}
879:{\displaystyle h}
859:{\displaystyle G}
653:
554:
541:
538:
512:
509:
478:
475:
443:
440:
402:
362:
354:
338:
328:
309:
16:(Redirected from
8458:
8423:
8422:
8411:
8410:
8402:
8383:
8382:
8207:
8169:Bekenstein bound
7986:Central concepts
7972:
7965:
7958:
7949:
7412:Mesures usuelles
7306:
7237:
7103:
7085:
7078:
7071:
7062:
7032:
7031:
7003:
6997:
6996:
6965:
6959:
6958:
6956:
6954:
6924:
6918:
6917:
6894:Phys. Rev. Lett.
6885:
6879:
6878:
6876:
6861:
6855:
6854:
6828:
6822:
6821:
6792:
6764:
6758:
6757:
6755:
6737:
6707:
6701:
6700:
6674:
6652:
6646:
6645:
6607:
6601:
6600:
6574:
6568:
6567:
6565:
6563:
6529:
6502:
6496:
6495:
6461:
6433:
6427:
6426:
6400:
6398:astro-ph/9410067
6377:
6371:
6370:
6368:
6366:
6347:
6341:
6340:
6306:
6291:Phys. Rev. Lett.
6285:
6279:
6278:
6268:
6250:
6232:
6203:
6197:
6196:
6163:Schwarz, John H.
6159:
6153:
6152:
6150:
6148:
6112:
6106:
6105:
6103:
6097:. Archived from
6054:
6041:
6035:
6034:
6032:
6030:
5981:Mermin, N. David
5977:
5971:
5970:
5965:. Archived from
5939:Gorelik, Gennady
5935:
5929:
5928:
5926:
5924:
5877:Il Nuovo Cimento
5865:
5859:
5858:
5825:
5819:
5818:
5812:
5810:
5768:
5742:
5736:
5735:
5733:
5731:
5710:(26 June 2019).
5704:
5698:
5697:
5671:
5649:
5640:
5639:
5637:
5635:
5602:Callender, Craig
5590:
5581:
5580:
5554:
5532:
5526:
5525:
5502:(1st ed.).
5494:Tipler, Frank J.
5486:
5480:
5479:
5458:
5449:
5448:
5446:
5444:
5416:
5410:
5401:
5399:
5397:
5373:
5364:
5363:
5361:
5324:
5318:
5317:
5315:
5313:
5260:
5254:
5253:
5251:
5249:
5227:
5221:
5220:
5218:
5203:
5197:
5196:
5166:
5160:
5159:
5157:
5120:
5114:
5113:
5090:
5084:
5083:
5040:
5031:
5030:
5010:
5004:
5003:
5001:
4999:
4980:
4974:
4973:
4971:
4969:
4950:
4944:
4943:
4941:
4939:
4933:
4912:
4901:
4895:
4894:
4892:
4890:
4862:
4844:
4838:
4837:
4835:
4833:
4813:
4807:
4806:
4804:
4802:
4766:
4760:
4759:
4734:Zwiebach, Barton
4730:
4715:
4714:
4712:
4710:
4691:
4682:
4681:
4679:
4677:
4658:
4652:
4651:
4649:
4647:
4628:
4622:
4621:
4619:
4617:
4598:
4589:
4588:
4586:
4584:
4578:
4567:
4556:
4545:
4544:in this article.
4515:
4513:
4511:
4484:
4473:
4472:
4470:
4468:
4431:(1 March 1983).
4425:
4419:
4418:
4377:
4371:
4370:
4364:
4356:
4329:Wheeler, John A.
4317:
4308:
4307:
4270:
4250:
4236:
4230:
4213:
4165:
4158:
4139:
4138:
4137:
4125:
4120:
4099:
4095:
4093:
4092:
4091:
4086:
4083:
4072:
4067:
4056:
4041:
4005:
3998:
3996:
3995:
3990:
3987:
3983:
3948:
3946:
3945:
3941:
3936:
3933:
3917:electromagnetism
3895:
3887:Poisson equation
3881:
3874:
3870:
3856:
3835:
3834:
3832:
3831:
3830:
3825:
3822:
3807:
3799:
3795:
3764:
3728:
3714:
3706:
3704:
3703:
3702:
3697:
3694:
3687:
3685:
3684:
3683:
3678:
3675:
3668:
3666:
3665:
3664:
3659:
3656:
3646:
3639:
3624:
3605:
3566:
3550:
3534:
3513:, which becomes
3512:
3487:
3474:, which becomes
3473:
3416:
3414:
3410:
3383:
3381:
3380:
3375:
3373:
3367:
3366:
3365:
3346:
3341:
3340:
3331:
3326:
3324:
3323:
3320:
3314:
3310:
3309:
3306:
3299:
3294:
3293:
3290:
3258:
3250:
3243:observed in 1991
3222:
3217:
3199:
3191:The Planck time
3182:
3180:
3179:
3174:
3123:
3115:
3112:
3110:
3106:
3098:
3096:
3095:
3090:
3088:
3085:
3084:
3075:
3067:
3066:
3061:
3060:
3059:
3038:
2989:
2988:
2965:
2964:
2892:Eddington number
2818:of Planck units
2805:
2645:
2643:
2640:
2630:
2625:
2610:particle physics
2598:
2471:
2466:
2459:
2457:
2456:
2451:
2449:
2446:
2438:
2437:
2428:
2427:
2422:
2420:
2419:
2416:
2407:
2402:
2401:
2398:
2376:
2371:
2364:
2362:
2361:
2356:
2354:
2352:
2351:
2350:
2337:
2336:
2327:
2322:
2319:
2314:
2311:
2305:
2304:
2303:
2300:
2290:
2285:
2282:
2277:
2274:
2268:
2267:
2264:
2258:
2253:
2252:
2249:
2227:
2222:
2215:
2213:
2212:
2207:
2205:
2200:
2199:
2190:
2185:
2183:
2182:
2181:
2178:
2172:
2171:
2168:
2158:
2153:
2151:
2150:
2147:
2141:
2140:
2137:
2131:
2126:
2125:
2122:
2100:
2095:
2088:
2086:
2085:
2080:
2078:
2072:
2071:
2070:
2057:
2056:
2051:
2049:
2048:
2045:
2036:
2031:
2030:
2021:
2020:
2017:
2008:
2007:
2004:
1982:
1974:
1972:
1971:
1966:
1964:
1958:
1957:
1956:
1943:
1942:
1937:
1935:
1934:
1931:
1922:
1914:
1913:
1910:
1888:
1883:
1876:
1874:
1873:
1868:
1866:
1863:
1862:
1853:
1852:
1851:
1832:
1831:
1826:
1825:
1817:
1815:
1811:
1809:
1808:
1799:
1791:
1780:
1775:
1772:
1750:
1745:
1738:
1736:
1735:
1730:
1728:
1726:
1725:
1716:
1708:
1702:
1697:
1694:
1662:Derived unit of
1659:
1639:
1637:
1636:
1631:
1619:
1617:
1616:
1611:
1603:
1601:
1596:
1591:
1586:
1566:
1564:
1558:
1555:
1553:
1515:
1513:
1512:
1507:
1497:
1490:
1487:
1485:
1484:
1463:
1455:
1454:
1439:
1434:
1433:
1430:
1417:
1415:
1414:
1409:
1407:
1406:
1403:
1390:Coulomb constant
1378:
1375:
1373:
1369:
1362:
1360:
1359:
1354:
1352:
1349:
1347:
1342:
1339:
1329:
1328:
1327:
1314:
1313:
1308:
1307:
1304:
1279:
1276:
1274:
1270:
1263:
1261:
1260:
1255:
1253:
1250:
1249:
1240:
1232:
1231:
1226:
1225:
1222:
1197:
1194:
1192:
1188:
1181:
1179:
1178:
1173:
1171:
1165:
1157:
1156:
1151:
1150:
1147:
1122:
1119:
1117:
1113:
1106:
1104:
1103:
1098:
1096:
1093:
1092:
1083:
1075:
1074:
1069:
1068:
1065:
1024:
1017:
1015:
1014:
1009:
997:
995:
994:
989:
977:
975:
974:
969:
967:
959:
934:
932:
931:
926:
924:
923:
922:
905:
903:
902:
897:
885:
883:
882:
877:
865:
863:
862:
857:
837:
773:
772:
760:
756:
752:
751:
739:
730:
729:
724:
711:
710:
702:
692:
682:
681:
674:) is valid with
673:
667:
665:
664:
659:
654:
652:
651:
650:
637:
632:
619:
609:
604:
582:
568:
566:
565:
560:
555:
553:
552:
547:
543:
540:
539:
536:
527:
519:
518:
514:
511:
510:
507:
501:
500:
491:
484:
480:
477:
476:
473:
467:
466:
457:
449:
444:
442:
441:
438:
429:
416:
414:
413:
408:
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:
8466:
8465:
8461:
8460:
8459:
8457:
8456:
8455:
8451:Quantum gravity
8431:
8430:
8429:
8417:
8405:
8397:
8395:
8390:
8389:
8367:
8356:FRW/CFT duality
8330:
8279:
8261:
8238:
8198:
8140:
8112:
8105:
8062:
7981:
7979:Quantum gravity
7976:
7946:
7941:
7915:
7889:
7829:
7773:
7727:
7646:
7535:
7338:
7293:
7266:
7224:
7181:
7139:
7094:
7089:
7041:
7036:
7035:
7020:
7005:
7004:
7000:
6985:
6967:
6966:
6962:
6952:
6950:
6943:
6926:
6925:
6921:
6887:
6886:
6882:
6863:
6862:
6858:
6843:
6830:
6829:
6825:
6818:
6766:
6765:
6761:
6709:
6708:
6704:
6654:
6653:
6649:
6609:
6608:
6604:
6589:
6579:Quantum Gravity
6576:
6575:
6571:
6561:
6559:
6508:(9 July 2014).
6504:
6503:
6499:
6435:
6434:
6430:
6379:
6378:
6374:
6364:
6362:
6349:
6348:
6344:
6287:
6286:
6282:
6205:
6204:
6200:
6185:
6161:
6160:
6156:
6146:
6144:
6137:
6114:
6113:
6109:
6101:
6052:
6043:
6042:
6038:
6028:
6026:
5979:
5978:
5974:
5955:
5937:
5936:
5932:
5922:
5920:
5867:
5866:
5862:
5834:Physical Review
5827:
5826:
5822:
5808:
5806:
5759:(11): 865–869.
5744:
5743:
5739:
5729:
5727:
5706:
5705:
5701:
5651:
5650:
5643:
5633:
5631:
5616:
5592:
5591:
5584:
5534:
5533:
5529:
5514:
5490:Barrow, John D.
5488:
5487:
5483:
5476:
5462:Barrow, John D.
5460:
5459:
5452:
5442:
5440:
5433:
5418:
5417:
5413:
5395:
5393:
5375:
5374:
5367:
5326:
5325:
5321:
5311:
5309:
5262:
5261:
5257:
5247:
5245:
5229:
5228:
5224:
5205:
5204:
5200:
5172:Physical Review
5168:
5167:
5163:
5122:
5121:
5117:
5110:
5092:
5091:
5087:
5068:
5042:
5041:
5034:
5012:
5011:
5007:
4997:
4995:
4982:
4981:
4977:
4967:
4965:
4952:
4951:
4947:
4937:
4935:
4931:
4925:
4917:. p. 953.
4910:
4903:
4902:
4898:
4888:
4886:
4879:
4846:
4845:
4841:
4831:
4829:
4815:
4814:
4810:
4800:
4798:
4791:
4783:. p. 602.
4768:
4767:
4763:
4748:
4732:
4731:
4718:
4708:
4706:
4693:
4692:
4685:
4675:
4673:
4660:
4659:
4655:
4645:
4643:
4630:
4629:
4625:
4615:
4613:
4600:
4599:
4592:
4582:
4580:
4576:
4565:
4558:
4557:
4548:
4518:Planck constant
4509:
4507:
4486:
4485:
4476:
4466:
4464:
4427:
4426:
4422:
4379:
4378:
4374:
4357:
4345:
4319:
4318:
4311:
4272:
4271:
4264:
4259:
4254:
4253:
4237:
4233:
4221:Barton Zwiebach
4214:
4210:
4205:
4173:
4164:
4156:
4154:
4147:
4141:
4135:
4133:
4131:
4118:
4116:
4097:
4089:
4087:
4084:
4081:
4080:
4078:
4065:
4063:
4054:
4047:
4039:
4037:
4031:
4029:
4019:
4003:
3991:
3988:
3981:
3979:
3978:
3976:
3970:
3959:Normalizes the
3955:
3939:
3937:
3934:
3931:
3930:
3928:
3893:
3879:
3868:
3866:
3858:
3851:
3843:
3828:
3826:
3823:
3820:
3819:
3817:
3812:
3805:
3797:
3793:
3774:
3762:
3755:
3737:
3726:
3721:
3712:
3700:
3698:
3695:
3692:
3691:
3689:
3681:
3679:
3676:
3673:
3672:
3670:
3662:
3660:
3657:
3654:
3653:
3651:
3644:
3637:
3622:
3603:
3590:. For example,
3564:
3548:
3541:
3524:
3514:
3502:
3492:
3475:
3460:
3446:
3427:
3412:
3408:
3406:
3404:
3396:
3357:
3332:
3315:
3301:
3300:
3285:
3280:
3279:
3272:
3257:
3248:
3246:
3237:
3229:
3215:
3213:
3198:
3195:
3192:
3189:
3165:
3164:
3155:The strings of
3121:
3113:
3108:
3104:
3102:
3101:It is equal to
3076:
3068:
3050:
3045:
3044:
3039:, is a unit of
3037:
3034:
3031:
3028:
3023:
2987:
2984:
2983:
2982:
2974:Hubble constant
2963:
2960:
2959:
2958:
2940:
2936:
2917:
2910:
2886:
2878:
2856:
2837:
2817:
2809:
2795:
2760:
2732:
2726:
2720:
2692:quantum gravity
2668:
2648:quantum effects
2641:
2638:
2636:
2623:
2621:
2606:
2596:
2569:incommensurable
2565:electric charge
2515:anthropocentric
2511:
2492:quantum gravity
2464:
2462:
2439:
2429:
2411:
2393:
2388:
2387:
2369:
2367:
2342:
2338:
2328:
2295:
2291:
2259:
2244:
2239:
2238:
2220:
2218:
2191:
2173:
2163:
2162:
2142:
2132:
2117:
2112:
2111:
2093:
2091:
2062:
2058:
2040:
2022:
2012:
1999:
1994:
1993:
1977:
1948:
1944:
1926:
1905:
1900:
1899:
1881:
1879:
1854:
1843:
1833:
1800:
1792:
1786:
1785:
1762:
1761:
1743:
1741:
1717:
1709:
1684:
1683:
1650:
1622:
1621:
1580:
1579:
1562:
1560:
1556:
1551:
1549:
1543:
1536:
1473:
1446:
1425:
1420:
1419:
1398:
1393:
1392:
1376:
1371:
1367:
1365:
1330:
1319:
1315:
1299:
1294:
1293:
1277:
1272:
1268:
1266:
1241:
1233:
1217:
1212:
1211:
1195:
1190:
1186:
1184:
1158:
1142:
1137:
1136:
1120:
1115:
1111:
1109:
1084:
1076:
1060:
1055:
1054:
1000:
999:
980:
979:
953:
952:
913:
908:
907:
888:
887:
868:
867:
848:
847:
808:electron charge
786:The concept of
784:
763:
762:
758:
754:
750:
737:
733:
732:
722:
718:
717:
708:
704:
700:
694:
690:
684:
679:
675:
671:
642:
638:
610:
597:
592:
591:
580:
531:
521:
520:
502:
492:
485:
468:
458:
451:
450:
433:
423:
422:
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:
8464:
8462:
8454:
8453:
8448:
8443:
8433:
8432:
8428:
8427:
8415:
8392:
8391:
8388:
8387:
8373:
8372:
8369:
8368:
8366:
8365:
8364:
8363:
8358:
8353:
8342:
8340:
8336:
8335:
8332:
8331:
8329:
8328:
8326:Twistor theory
8323:
8318:
8313:
8308:
8303:
8298:
8293:
8287:
8285:
8281:
8280:
8278:
8277:
8271:
8269:
8263:
8262:
8260:
8259:
8254:
8248:
8246:
8240:
8239:
8237:
8236:
8231:
8226:
8221:
8215:
8213:
8204:
8200:
8199:
8197:
8196:
8191:
8186:
8181:
8176:
8171:
8166:
8161:
8156:
8150:
8148:
8142:
8141:
8139:
8138:
8133:
8128:
8123:
8117:
8115:
8107:
8106:
8104:
8103:
8098:
8093:
8088:
8083:
8078:
8072:
8070:
8064:
8063:
8061:
8060:
8055:
8050:
8045:
8040:
8035:
8030:
8025:
8020:
8015:
8010:
8005:
8000:
7995:
7989:
7987:
7983:
7982:
7977:
7975:
7974:
7967:
7960:
7952:
7943:
7942:
7940:
7939:
7934:
7929:
7927:Absolute scale
7923:
7921:
7917:
7916:
7914:
7913:
7908:
7903:
7897:
7895:
7891:
7890:
7888:
7887:
7882:
7877:
7872:
7867:
7862:
7857:
7852:
7847:
7841:
7839:
7835:
7834:
7831:
7830:
7828:
7827:
7822:
7817:
7812:
7807:
7802:
7797:
7792:
7787:
7781:
7779:
7775:
7774:
7772:
7771:
7766:
7761:
7756:
7751:
7746:
7741:
7735:
7733:
7729:
7728:
7726:
7725:
7720:
7715:
7710:
7705:
7700:
7695:
7690:
7685:
7680:
7675:
7670:
7665:
7660:
7654:
7652:
7648:
7647:
7645:
7644:
7639:
7634:
7629:
7624:
7619:
7614:
7609:
7604:
7599:
7594:
7589:
7584:
7579:
7574:
7569:
7564:
7559:
7554:
7549:
7543:
7541:
7537:
7536:
7534:
7533:
7528:
7523:
7518:
7513:
7508:
7503:
7498:
7493:
7488:
7483:
7478:
7473:
7468:
7463:
7458:
7453:
7448:
7443:
7438:
7433:
7432:
7431:
7421:
7416:
7415:
7414:
7409:
7399:
7394:
7389:
7388:
7387:
7382:
7372:
7367:
7362:
7357:
7352:
7346:
7344:
7340:
7339:
7337:
7336:
7331:
7325:
7319:
7312:
7310:
7303:
7299:
7298:
7295:
7294:
7292:
7291:
7285:
7280:
7274:
7272:
7268:
7267:
7265:
7264:
7259:
7254:
7249:
7243:
7241:
7234:
7230:
7229:
7226:
7225:
7223:
7222:
7217:
7212:
7207:
7202:
7197:
7191:
7189:
7183:
7182:
7180:
7179:
7173:
7168:
7163:
7158:
7153:
7147:
7145:
7141:
7140:
7138:
7137:
7136:
7135:
7125:
7120:
7115:
7109:
7107:
7100:
7096:
7095:
7090:
7088:
7087:
7080:
7073:
7065:
7059:
7058:
7052:
7040:
7039:External links
7037:
7034:
7033:
7018:
6998:
6983:
6960:
6941:
6919:
6889:Sorkin, Rafael
6880:
6865:Shor, Peter W.
6856:
6841:
6823:
6816:
6759:
6710:Jowsey, Aden;
6702:
6672:hep-th/0210109
6647:
6620:(6): 561–570.
6602:
6587:
6569:
6497:
6436:Judes, Simon;
6428:
6415:10.1086/175344
6372:
6342:
6297:(24): 241301.
6280:
6198:
6183:
6154:
6135:
6107:
6036:
5972:
5953:
5930:
5883:(2): 215–221.
5860:
5841:(2): 511–536.
5829:Wheeler, J. A.
5820:
5737:
5699:
5662:(9): 925–932.
5641:
5614:
5582:
5527:
5512:
5481:
5474:
5450:
5431:
5411:
5365:
5328:Rovelli, Carlo
5319:
5280:(3): 199–220.
5264:Hawking, S. W.
5255:
5222:
5216:hep-ph/0207124
5198:
5161:
5124:Wilczek, Frank
5115:
5108:
5094:Penrose, Roger
5085:
5066:
5032:
5005:
4975:
4945:
4924:978-3540347620
4923:
4896:
4877:
4839:
4816:Elert, Glenn.
4808:
4790:978-3662528433
4789:
4761:
4746:
4716:
4683:
4653:
4623:
4590:
4546:
4474:
4420:
4372:
4343:
4325:Thorne, Kip S.
4309:
4274:Wilczek, Frank
4261:
4260:
4258:
4255:
4252:
4251:
4241:predicts that
4231:
4207:
4206:
4204:
4201:
4200:
4199:
4194:
4189:
4184:
4179:
4172:
4169:
4168:
4167:
4162:
4152:
4145:
4060:
4059:
4058:
4052:
4045:
4035:
4027:
4017:
4000:
3966:
3957:
3953:
3925:charge density
3890:
3876:
3862:
3847:
3773:
3770:
3760:
3753:
3719:
3540:
3537:
3520:
3498:
3490:Dirac equation
3445:
3442:
3425:
3402:
3395:
3392:
3372:
3364:
3360:
3356:
3353:
3349:
3344:
3339:
3335:
3329:
3318:
3313:
3304:
3297:
3288:
3271:
3268:
3255:
3235:
3228:
3225:
3196:
3193:
3188:
3185:
3172:
3118:standard error
3083:
3079:
3074:
3071:
3064:
3058:
3053:
3035:
3032:
3027:
3024:
3022:
3019:
2997:
2996:
2990:
2985:
2976:
2970:
2969:
2966:
2961:
2952:
2946:
2945:
2937:
2934:
2928:
2922:
2921:
2920:9.9 × 10 kg⋅m
2918:
2915:
2908:
2902:
2896:
2895:
2879:
2876:
2870:
2864:
2863:
2857:
2854:
2848:
2842:
2841:
2838:
2835:
2829:
2823:
2822:
2819:
2814:
2793:
2775:Standard Model
2758:
2722:Main article:
2719:
2716:
2667:
2664:
2605:
2602:
2510:
2507:
2473:
2472:
2460:
2445:
2442:
2436:
2432:
2425:
2414:
2410:
2405:
2396:
2385:
2378:
2377:
2365:
2349:
2345:
2341:
2335:
2331:
2325:
2318:
2309:
2298:
2294:
2288:
2281:
2272:
2262:
2256:
2247:
2236:
2229:
2228:
2216:
2203:
2198:
2194:
2188:
2176:
2166:
2161:
2156:
2145:
2135:
2129:
2120:
2109:
2102:
2101:
2089:
2075:
2069:
2065:
2061:
2054:
2043:
2039:
2034:
2029:
2025:
2015:
2011:
2002:
1991:
1984:
1983:
1975:
1961:
1955:
1951:
1947:
1940:
1929:
1925:
1920:
1917:
1908:
1897:
1890:
1889:
1877:
1861:
1857:
1850:
1846:
1842:
1839:
1836:
1829:
1823:
1820:
1814:
1807:
1803:
1798:
1795:
1789:
1784:
1779:
1770:
1759:
1752:
1751:
1739:
1724:
1720:
1715:
1712:
1706:
1701:
1692:
1681:
1674:
1673:
1666:
1663:
1649:
1646:
1629:
1609:
1606:
1600:
1595:
1590:
1541:
1534:
1505:
1502:
1496:
1493:
1483:
1480:
1476:
1472:
1469:
1466:
1461:
1458:
1453:
1449:
1445:
1442:
1437:
1428:
1401:
1380:
1379:
1363:
1346:
1337:
1333:
1326:
1322:
1318:
1311:
1302:
1291:
1285:
1281:
1280:
1264:
1248:
1244:
1239:
1236:
1229:
1220:
1209:
1203:
1199:
1198:
1182:
1168:
1164:
1161:
1154:
1145:
1134:
1128:
1124:
1123:
1107:
1091:
1087:
1082:
1079:
1072:
1063:
1052:
1046:
1045:Planck length
1042:
1041:
1034:
1031:
1028:
1007:
987:
965:
962:
921:
916:
895:
875:
855:
783:
780:
776:Paul S. Wesson
748:
698:
688:
657:
649:
645:
641:
635:
631:
627:
622:
618:
614:
607:
603:
600:
558:
551:
546:
534:
530:
524:
517:
505:
499:
495:
488:
483:
471:
465:
461:
454:
447:
436:
432:
406:
399:
395:
388:
384:
378:
374:
366:
359:
350:
343:
334:
324:
317:
313:
306:
302:
295:
291:
285:
281:
274:
271:
268:
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:
8463:
8452:
8449:
8447:
8444:
8442:
8441:Natural units
8439:
8438:
8436:
8426:
8421:
8416:
8414:
8409:
8404:
8400:
8386:
8378:
8375:
8374:
8370:
8362:
8359:
8357:
8354:
8352:
8349:
8348:
8347:
8344:
8343:
8341:
8337:
8327:
8324:
8322:
8319:
8317:
8314:
8312:
8309:
8307:
8304:
8302:
8301:Dual graviton
8299:
8297:
8294:
8292:
8289:
8288:
8286:
8282:
8276:
8273:
8272:
8270:
8268:
8264:
8258:
8255:
8253:
8250:
8249:
8247:
8245:
8241:
8235:
8232:
8230:
8227:
8225:
8222:
8220:
8217:
8216:
8214:
8212:
8211:String theory
8208:
8205:
8201:
8195:
8192:
8190:
8187:
8185:
8182:
8180:
8177:
8175:
8172:
8170:
8167:
8165:
8162:
8160:
8157:
8155:
8152:
8151:
8149:
8147:
8143:
8137:
8134:
8132:
8129:
8127:
8124:
8122:
8119:
8118:
8116:
8114:
8108:
8102:
8099:
8097:
8094:
8092:
8089:
8087:
8084:
8082:
8079:
8077:
8074:
8073:
8071:
8069:
8065:
8059:
8056:
8054:
8051:
8049:
8046:
8044:
8041:
8039:
8036:
8034:
8031:
8029:
8026:
8024:
8021:
8019:
8016:
8014:
8011:
8009:
8006:
8004:
8001:
7999:
7996:
7994:
7991:
7990:
7988:
7984:
7980:
7973:
7968:
7966:
7961:
7959:
7954:
7953:
7950:
7938:
7935:
7933:
7930:
7928:
7925:
7924:
7922:
7918:
7912:
7909:
7907:
7904:
7902:
7899:
7898:
7896:
7894:List articles
7892:
7886:
7883:
7881:
7878:
7876:
7873:
7871:
7868:
7866:
7863:
7861:
7858:
7856:
7853:
7851:
7848:
7846:
7843:
7842:
7840:
7836:
7826:
7823:
7821:
7818:
7816:
7813:
7811:
7808:
7806:
7803:
7801:
7798:
7796:
7793:
7791:
7788:
7786:
7783:
7782:
7780:
7778:South America
7776:
7770:
7767:
7765:
7762:
7760:
7757:
7755:
7752:
7750:
7747:
7745:
7742:
7740:
7737:
7736:
7734:
7732:North America
7730:
7724:
7721:
7719:
7716:
7714:
7713:South African
7711:
7709:
7706:
7704:
7701:
7699:
7696:
7694:
7691:
7689:
7686:
7684:
7681:
7679:
7676:
7674:
7671:
7669:
7666:
7664:
7661:
7659:
7656:
7655:
7653:
7649:
7643:
7640:
7638:
7635:
7633:
7630:
7628:
7625:
7623:
7620:
7618:
7615:
7613:
7610:
7608:
7605:
7603:
7600:
7598:
7595:
7593:
7590:
7588:
7585:
7583:
7580:
7578:
7575:
7573:
7570:
7568:
7565:
7563:
7560:
7558:
7555:
7553:
7550:
7548:
7545:
7544:
7542:
7538:
7532:
7529:
7527:
7524:
7522:
7519:
7517:
7514:
7512:
7509:
7507:
7504:
7502:
7499:
7497:
7494:
7492:
7489:
7487:
7484:
7482:
7479:
7477:
7474:
7472:
7469:
7467:
7464:
7462:
7461:Luxembourgian
7459:
7457:
7454:
7452:
7449:
7447:
7444:
7442:
7439:
7437:
7434:
7430:
7427:
7426:
7425:
7422:
7420:
7417:
7413:
7410:
7408:
7405:
7404:
7403:
7400:
7398:
7395:
7393:
7390:
7386:
7383:
7381:
7378:
7377:
7376:
7373:
7371:
7368:
7366:
7363:
7361:
7358:
7356:
7353:
7351:
7348:
7347:
7345:
7341:
7335:
7334:gravitational
7332:
7329:
7326:
7323:
7320:
7317:
7314:
7313:
7311:
7307:
7304:
7300:
7289:
7286:
7284:
7281:
7279:
7276:
7275:
7273:
7269:
7263:
7260:
7258:
7255:
7253:
7250:
7248:
7245:
7244:
7242:
7238:
7235:
7231:
7221:
7218:
7216:
7213:
7211:
7208:
7206:
7203:
7201:
7198:
7196:
7193:
7192:
7190:
7188:
7184:
7177:
7174:
7172:
7169:
7167:
7164:
7162:
7159:
7157:
7154:
7152:
7151:Apothecaries'
7149:
7148:
7146:
7142:
7134:
7131:
7130:
7129:
7126:
7124:
7121:
7119:
7116:
7114:
7111:
7110:
7108:
7104:
7101:
7097:
7093:
7086:
7081:
7079:
7074:
7072:
7067:
7066:
7063:
7056:
7053:
7050:
7046:
7043:
7042:
7038:
7029:
7025:
7021:
7015:
7011:
7010:
7002:
6999:
6994:
6990:
6986:
6980:
6976:
6975:
6970:
6964:
6961:
6948:
6944:
6942:9780792336709
6938:
6934:
6930:
6923:
6920:
6915:
6911:
6907:
6903:
6899:
6896:
6895:
6890:
6884:
6881:
6875:
6870:
6866:
6860:
6857:
6853:
6851:
6844:
6838:
6834:
6827:
6824:
6820:
6815:
6808:
6804:
6800:
6796:
6791:
6786:
6782:
6778:
6774:
6770:
6763:
6760:
6754:
6749:
6745:
6741:
6736:
6731:
6727:
6723:
6722:
6717:
6713:
6706:
6703:
6698:
6694:
6690:
6686:
6682:
6678:
6673:
6668:
6664:
6660:
6659:
6651:
6648:
6643:
6639:
6635:
6631:
6627:
6623:
6619:
6615:
6614:
6606:
6603:
6598:
6594:
6590:
6584:
6580:
6573:
6570:
6557:
6553:
6549:
6545:
6541:
6537:
6533:
6528:
6523:
6519:
6515:
6511:
6507:
6501:
6498:
6493:
6489:
6485:
6481:
6477:
6473:
6469:
6465:
6460:
6459:gr-qc/0205067
6455:
6452:(4): 045001.
6451:
6447:
6443:
6439:
6432:
6429:
6424:
6420:
6416:
6412:
6408:
6404:
6399:
6394:
6390:
6386:
6385:
6376:
6373:
6360:
6356:
6352:
6346:
6343:
6338:
6334:
6330:
6326:
6322:
6318:
6314:
6310:
6305:
6300:
6296:
6293:
6292:
6284:
6281:
6276:
6272:
6267:
6262:
6258:
6254:
6249:
6244:
6240:
6236:
6231:
6226:
6222:
6218:
6217:
6212:
6208:
6202:
6199:
6194:
6190:
6186:
6180:
6176:
6172:
6168:
6164:
6158:
6155:
6142:
6138:
6132:
6128:
6124:
6120:
6119:
6111:
6108:
6100:
6096:
6092:
6088:
6084:
6080:
6076:
6072:
6068:
6064:
6060:
6059:
6051:
6047:
6040:
6037:
6024:
6020:
6016:
6012:
6008:
6004:
6000:
5996:
5992:
5991:
5990:Physics Today
5986:
5982:
5976:
5973:
5968:
5964:
5960:
5956:
5954:0-8176-3479-7
5950:
5946:
5945:
5940:
5934:
5931:
5918:
5914:
5910:
5906:
5902:
5898:
5894:
5890:
5886:
5882:
5878:
5874:
5870:
5864:
5861:
5856:
5852:
5848:
5844:
5840:
5836:
5835:
5830:
5824:
5821:
5817:
5804:
5800:
5796:
5792:
5788:
5784:
5780:
5776:
5772:
5767:
5762:
5758:
5754:
5753:
5748:
5741:
5738:
5725:
5721:
5717:
5713:
5709:
5708:Siegel, Ethan
5703:
5700:
5695:
5691:
5687:
5683:
5679:
5675:
5670:
5665:
5661:
5657:
5656:
5648:
5646:
5642:
5629:
5625:
5621:
5617:
5611:
5607:
5603:
5599:
5595:
5589:
5587:
5583:
5578:
5574:
5570:
5566:
5562:
5558:
5553:
5548:
5544:
5540:
5539:
5531:
5528:
5523:
5519:
5515:
5509:
5505:
5501:
5500:
5495:
5491:
5485:
5482:
5477:
5475:0-375-42221-8
5471:
5467:
5463:
5457:
5455:
5451:
5438:
5434:
5428:
5424:
5423:
5415:
5412:
5409:
5405:
5391:
5387:
5383:
5379:
5372:
5370:
5366:
5360:
5355:
5351:
5347:
5343:
5339:
5338:
5333:
5329:
5323:
5320:
5307:
5303:
5299:
5295:
5291:
5287:
5283:
5279:
5275:
5274:
5269:
5265:
5259:
5256:
5243:
5239:
5238:
5233:
5226:
5223:
5217:
5212:
5208:
5202:
5199:
5194:
5190:
5186:
5182:
5178:
5174:
5173:
5165:
5162:
5156:
5151:
5147:
5143:
5139:
5135:
5134:
5133:Physics Today
5129:
5125:
5119:
5116:
5111:
5105:
5101:
5100:
5095:
5089:
5086:
5082:
5077:
5073:
5069:
5063:
5059:
5055:
5051:
5050:
5045:
5039:
5037:
5033:
5028:
5024:
5020:
5016:
5009:
5006:
4993:
4989:
4985:
4979:
4976:
4963:
4959:
4955:
4949:
4946:
4930:
4926:
4920:
4916:
4909:
4908:
4900:
4897:
4884:
4880:
4874:
4870:
4866:
4861:
4860:gr-qc/0610061
4856:
4852:
4851:
4843:
4840:
4827:
4823:
4819:
4812:
4809:
4796:
4792:
4786:
4782:
4778:
4777:
4772:
4765:
4762:
4757:
4753:
4749:
4743:
4739:
4735:
4729:
4727:
4725:
4723:
4721:
4717:
4704:
4700:
4696:
4690:
4688:
4684:
4671:
4667:
4663:
4657:
4654:
4641:
4637:
4633:
4627:
4624:
4611:
4607:
4603:
4597:
4595:
4591:
4575:
4571:
4564:
4563:
4555:
4553:
4551:
4547:
4543:
4542:
4537:
4536:
4531:
4527:
4523:
4519:
4505:
4501:
4498:(in German).
4497:
4493:
4489:
4483:
4481:
4479:
4475:
4462:
4458:
4454:
4450:
4446:
4442:
4438:
4434:
4430:
4429:Barrow, J. D.
4424:
4421:
4416:
4412:
4408:
4404:
4400:
4396:
4392:
4388:
4387:
4382:
4381:Wesson, P. S.
4376:
4373:
4368:
4362:
4354:
4350:
4346:
4344:0-7167-0334-3
4340:
4336:
4335:
4330:
4326:
4322:
4316:
4314:
4310:
4305:
4301:
4297:
4293:
4289:
4285:
4281:
4280:
4279:Physics Today
4275:
4269:
4267:
4263:
4256:
4248:
4244:
4240:
4235:
4232:
4228:
4227:
4222:
4218:
4217:Frank Wilczek
4212:
4209:
4202:
4198:
4195:
4193:
4190:
4188:
4185:
4183:
4180:
4178:
4175:
4174:
4170:
4161:
4151:
4144:
4129:
4126:are known as
4123:
4114:
4110:
4106:
4102:
4076:
4070:
4061:
4051:
4044:
4034:
4026:
4023:
4022:event horizon
4016:
4012:
4008:
4001:
3994:
3986:
3974:
3969:
3965:
3962:
3958:
3952:
3944:
3926:
3922:
3918:
3915:equation) of
3914:
3913:Lorentz force
3910:
3906:
3902:
3898:
3891:
3888:
3884:
3877:
3873:
3865:
3861:
3857:(rather than
3855:
3850:
3846:
3841:
3838:
3837:
3815:
3810:
3804:Normalizing 4
3803:
3802:
3801:
3791:
3787:
3784:
3780:
3771:
3769:
3766:
3759:
3752:
3748:
3744:
3740:
3734:
3732:
3725:
3718:
3710:
3649:
3642:
3635:
3630:
3628:
3620:
3616:
3612:
3608:
3601:
3597:
3593:
3592:gravitational
3589:
3585:
3581:
3577:
3573:
3569:
3562:
3558:
3554:
3545:
3538:
3536:
3532:
3528:
3523:
3518:
3510:
3506:
3501:
3496:
3491:
3486:
3482:
3478:
3471:
3467:
3463:
3459:
3454:
3452:
3443:
3441:
3439:
3435:
3431:
3424:
3420:
3401:
3393:
3391:
3388:
3384:
3362:
3358:
3354:
3351:
3347:
3342:
3337:
3333:
3327:
3316:
3311:
3302:
3295:
3286:
3277:
3269:
3267:
3265:
3260:
3254:
3244:
3241:
3234:
3227:Planck energy
3226:
3224:
3221:
3211:
3207:
3204:required for
3203:
3186:
3184:
3170:
3162:
3158:
3157:string theory
3153:
3151:
3146:
3145:event horizon
3142:
3137:
3135:
3131:
3127:
3119:
3099:
3081:
3077:
3072:
3062:
3051:
3042:
3026:Planck length
3025:
3020:
3018:
3016:
3012:
3008:
3004:
2995:
2991:
2981:
2977:
2975:
2972:
2971:
2967:
2957:
2953:
2951:
2948:
2947:
2944:
2938:
2933:
2929:
2927:
2924:
2923:
2919:
2914:
2907:
2903:
2901:
2898:
2897:
2893:
2889:
2884:
2880:
2875:
2871:
2869:
2866:
2865:
2862:
2858:
2853:
2849:
2847:
2844:
2843:
2839:
2834:
2830:
2828:
2825:
2824:
2820:
2815:
2813:
2807:
2806:
2800:
2797:
2792:
2788:
2784:
2780:
2776:
2772:
2771:unified force
2768:
2764:
2757:
2753:
2750:, before the
2749:
2745:
2741:
2737:
2731:
2725:
2717:
2715:
2713:
2709:
2705:
2701:
2697:
2696:string theory
2693:
2689:
2685:
2680:
2678:
2674:
2665:
2663:
2661:
2657:
2653:
2649:
2634:
2629:
2619:
2615:
2611:
2603:
2600:
2592:
2590:
2586:
2580:
2578:
2574:
2573:proton charge
2570:
2566:
2562:
2558:
2551:
2549:
2542:
2540:
2539:Frank Wilczek
2536:
2535:Planck length
2532:
2528:
2524:
2520:
2516:
2508:
2506:
2504:
2503:22 micrograms
2499:
2497:
2493:
2489:
2485:
2480:
2470:
2461:
2443:
2434:
2430:
2423:
2412:
2408:
2403:
2394:
2386:
2383:
2380:
2379:
2375:
2366:
2347:
2343:
2333:
2329:
2323:
2316:
2307:
2296:
2286:
2279:
2270:
2260:
2254:
2245:
2237:
2234:
2231:
2230:
2226:
2217:
2201:
2196:
2192:
2186:
2174:
2164:
2154:
2143:
2133:
2127:
2118:
2110:
2107:
2104:
2103:
2099:
2090:
2073:
2067:
2063:
2052:
2041:
2032:
2027:
2023:
2013:
2009:
2000:
1992:
1989:
1986:
1985:
1981:
1976:
1959:
1953:
1949:
1938:
1927:
1918:
1915:
1906:
1898:
1895:
1892:
1891:
1887:
1878:
1859:
1855:
1848:
1840:
1827:
1821:
1818:
1812:
1805:
1801:
1796:
1787:
1782:
1777:
1768:
1760:
1757:
1754:
1753:
1749:
1740:
1722:
1718:
1713:
1704:
1699:
1690:
1682:
1679:
1676:
1675:
1671:
1667:
1664:
1661:
1660:
1654:
1648:Derived units
1647:
1645:
1643:
1627:
1607:
1604:
1598:
1593:
1588:
1577:
1572:
1570:
1547:
1540:
1533:
1529:
1525:
1521:
1516:
1503:
1500:
1494:
1491:
1481:
1478:
1474:
1470:
1467:
1464:
1459:
1451:
1447:
1443:
1440:
1435:
1426:
1399:
1391:
1387:
1364:
1344:
1335:
1331:
1324:
1320:
1309:
1300:
1292:
1289:
1286:
1283:
1282:
1265:
1246:
1242:
1237:
1227:
1218:
1210:
1207:
1204:
1201:
1200:
1183:
1166:
1162:
1152:
1143:
1135:
1132:
1129:
1126:
1125:
1108:
1089:
1085:
1080:
1070:
1061:
1053:
1050:
1047:
1044:
1043:
1039:
1035:
1032:
1029:
1026:
1025:
1019:
1005:
963:
960:
950:
946:
942:
938:
914:
893:
873:
853:
843:
838:
836:
830:
828:
824:
820:
816:
812:
809:
805:
801:
797:
793:
789:
788:natural units
781:
779:
777:
770:
766:
747:
743:
736:
728:
721:
715:
707:
697:
687:
678:
668:
655:
647:
639:
633:
629:
625:
620:
616:
612:
605:
601:
598:
589:
586:
578:
574:
569:
556:
549:
544:
532:
528:
522:
515:
503:
497:
493:
486:
481:
469:
463:
459:
452:
445:
434:
430:
420:
417:
404:
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:
8376:
8339:Applications
8229:Supergravity
8136:Unruh effect
8043:Quantum foam
8038:Planck units
8037:
8033:IR/UV mixing
8008:Causal patch
7875:Mesopotamian
7769:Puerto Rican
7209:
7166:Astronomical
7008:
7001:
6973:
6963:
6951:. Retrieved
6932:
6922:
6900:(2): 87–90.
6897:
6892:
6883:
6859:
6846:
6832:
6826:
6813:
6810:
6772:
6768:
6762:
6725:
6719:
6712:Visser, Matt
6705:
6662:
6656:
6650:
6617:
6611:
6605:
6578:
6572:
6560:. Retrieved
6517:
6513:
6500:
6449:
6445:
6438:Visser, Matt
6431:
6388:
6382:
6375:
6363:. Retrieved
6354:
6345:
6294:
6289:
6283:
6220:
6214:
6201:
6166:
6157:
6145:. Retrieved
6117:
6110:
6099:the original
6065:(5): 48–55.
6062:
6056:
6048:(May 2005).
6039:
6027:. Retrieved
5994:
5988:
5983:(May 2009).
5975:
5967:the original
5943:
5933:
5921:. Retrieved
5880:
5876:
5863:
5838:
5832:
5823:
5814:
5807:. Retrieved
5756:
5750:
5740:
5728:. Retrieved
5715:
5702:
5659:
5653:
5632:. Retrieved
5605:
5542:
5536:
5530:
5497:
5484:
5465:
5441:. Retrieved
5421:
5414:
5394:. Retrieved
5381:
5341:
5337:Scholarpedia
5335:
5322:
5310:. Retrieved
5277:
5271:
5258:
5246:. Retrieved
5237:CERN Courier
5235:
5225:
5201:
5176:
5170:
5164:
5140:(6): 12–13.
5137:
5131:
5118:
5098:
5088:
5079:
5058:172, 434–435
5048:
5044:Zee, Anthony
5018:
5014:
5008:
4996:. Retrieved
4987:
4978:
4966:. Retrieved
4957:
4948:
4936:. Retrieved
4906:
4899:
4887:. Retrieved
4849:
4842:
4830:. Retrieved
4821:
4811:
4799:. Retrieved
4775:
4764:
4737:
4707:. Retrieved
4698:
4674:. Retrieved
4665:
4656:
4644:. Retrieved
4635:
4626:
4614:. Retrieved
4605:
4581:. Retrieved
4574:the original
4561:
4540:
4534:
4529:
4525:
4521:
4508:. Retrieved
4499:
4495:
4465:. Retrieved
4440:
4436:
4423:
4390:
4384:
4375:
4337:. New York.
4333:
4283:
4277:
4234:
4224:
4211:
4159:
4149:
4142:
4127:
4121:
4100:
4074:
4068:
4049:
4042:
4032:
4024:
4014:
4006:
4002:Eliminates 4
3992:
3984:
3967:
3963:
3950:
3942:
3921:mass density
3896:
3892:Eliminates 4
3882:
3878:Eliminates 4
3871:
3863:
3859:
3853:
3848:
3844:
3813:
3808:
3789:
3785:
3775:
3767:
3757:
3750:
3746:
3742:
3738:
3735:
3730:
3724:rationalized
3723:
3716:
3647:
3640:
3633:
3631:
3614:
3606:
3599:
3588:flux density
3567:
3560:
3547:The factor 4
3546:
3542:
3530:
3526:
3521:
3516:
3508:
3504:
3499:
3494:
3484:
3480:
3476:
3469:
3465:
3461:
3455:
3447:
3437:
3422:
3399:
3397:
3389:
3385:
3273:
3261:
3252:
3232:
3230:
3190:
3154:
3138:
3100:
3029:
3014:
3006:
3000:
2979:
2955:
2931:
2912:
2905:
2883:solar masses
2873:
2851:
2832:
2821:Equivalents
2810:present-day
2798:
2790:
2755:
2743:
2740:Planck epoch
2739:
2733:
2718:In cosmology
2681:
2669:
2618:energy scale
2607:
2604:Planck scale
2594:
2581:
2553:
2544:
2512:
2509:Significance
2500:
2488:Planck epoch
2476:
2382:acceleration
1978:6.5249
1668:Approximate
1651:
1576:Stoney units
1574:Compared to
1573:
1568:
1545:
1538:
1531:
1527:
1523:
1519:
1517:
1383:
1202:Planck time
1127:Planck mass
998:rather than
845:
841:
832:
810:
803:
799:
796:Stoney units
785:
768:
764:
745:
741:
734:
726:
719:
713:
705:
695:
685:
676:
669:
590:
576:
570:
421:
418:
259:
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:
8296:Causal sets
8146:Black holes
7739:Costa Rican
7703:Seychellois
7612:Singaporean
7407:Traditional
7262:Metrication
7200:Geometrised
7156:Avoirdupois
6728:(11): 403.
6365:21 December
5344:(5): 7117.
5056:. pp.
4889:31 December
4832:22 February
4801:9 September
4771:Deza, Elena
4583:31 December
4502:: 440–480.
4488:Planck, Max
4334:Gravitation
4226:Gravitation
4177:cGh physics
3927:, and with
3580:Gauss's law
3187:Planck time
3043:defined as:
2926:Temperature
2872:approx. 10
2861:light-years
2808:Property of
2752:time passed
2646:) at which
2577:proton mass
2548:quintillion
1672:equivalent
1665:Expression
1288:temperature
1033:Expression
949:temperature
120:10 GeV
8446:Max Planck
8435:Categories
8361:Multiverse
8203:Approaches
8081:CGHS model
8068:Toy models
8003:CA-duality
7825:Venezuelan
7810:Paraguayan
7764:Nicaraguan
7642:Vietnamese
7617:Sri Lankan
7602:Philippine
7562:Indonesian
7486:Portuguese
7380:Winchester
7283:Comparison
7233:Background
7171:Electrical
7028:1202475208
6953:16 January
6874:1807.04363
6735:2102.01831
6304:2005.11572
5997:(5): 8–9.
5766:1705.09749
5594:Baez, John
5404:Planck era
5248:4 November
5207:Witten, Ed
4994:. May 2024
4964:. May 2024
4705:. May 2024
4672:. May 2024
4642:. May 2024
4612:. May 2024
4393:(2): 117.
4257:References
4111:, and the
3949:replacing
3923:replacing
3584:divergence
3582:, and the
3559:of radius
3488:, and the
2888:10 protons
2831:8.08 × 10
2744:Planck era
2644:10 kg
2585:Heisenberg
2559:, because
2527:base units
1193:10 kg
1030:Dimension
815:Max Planck
806:, and the
235:base units
188:in vacuum,
142:of around
118:of around
101:Max Planck
93:free space
8377:See also:
8316:Spin foam
8096:RST model
7820:Uruguayan
7805:Colombian
7795:Brazilian
7785:Argentine
7723:Tanzanian
7693:Mauritian
7663:Ethiopian
7627:Taiwanese
7597:Pakistani
7577:Mongolian
7552:Cambodian
7471:Norwegian
7441:Icelandic
7436:Hungarian
7429:Byzantine
7385:Exchequer
7133:Hong Kong
6993:317496332
6807:810438317
6790:1203.3827
6697:118154613
6642:120924238
6597:785233016
6527:1403.2080
6492:119094398
6484:0556-2821
6423:119092012
6337:218870394
6257:2367-3613
6230:1203.6191
6193:245575026
6019:0031-9228
5913:123012079
5905:1827-6121
5869:Regge, T.
5799:119022491
5791:0002-9505
5669:1001.1205
5624:924701824
5552:1105.3105
5076:659549695
4457:0035-8738
4415:120784299
4361:cite book
4290:: 12–13.
4103:from the
3885:from the
3415:10 K
3355:×
3348:≈
3111:10 m
3070:ℏ
3052:ℓ
2930:1.9 × 10
2904:1.8 × 10
2850:5.4 × 10
2441:ℏ
2340:ℏ
2293:ℏ
2246:ρ
2160:ℏ
2060:ℏ
2038:ℏ
1946:ℏ
1924:ℏ
1838:ℏ
1794:ℏ
1711:ℏ
1628:α
1605:≈
1599:α
1492:≈
1479:−
1471:×
1465:≈
1457:ℏ
1448:ϵ
1444:π
1374:10 K
1317:ℏ
1275:10 s
1235:ℏ
1160:ℏ
1118:10 m
1078:ℏ
986:ℏ
964:π
253:example,
173:The four
144:10 m
136:10 s
124:10 J
110:The term
8224:M-theory
8184:ER = EPR
8023:Graviton
7906:Obsolete
7901:Humorous
7855:Egyptian
7815:Peruvian
7790:Bolivian
7754:Honduran
7718:Tunisian
7698:Moroccan
7688:Malagasy
7673:Eritrean
7668:Egyptian
7658:Algerian
7587:Nepalese
7567:Japanese
7501:Scottish
7491:Romanian
7392:Estonian
7302:Historic
7278:Overview
7247:Overview
7144:Specific
6971:(2009).
6947:Archived
6721:Universe
6562:16 April
6556:Archived
6552:14373748
6359:Archived
6329:32639827
6275:28179841
6223:(1): 2.
6147:22 March
6141:Archived
6095:10872062
6087:15882021
6029:22 March
6023:Archived
5963:24011430
5923:22 March
5917:Archived
5803:Archived
5724:Archived
5694:55181581
5634:20 March
5628:Archived
5596:(2001).
5577:55125081
5522:87028148
5496:(1986).
5464:(2002).
5443:10 April
5437:Archived
5408:Universe
5396:20 March
5390:Archived
5330:(2008).
5312:20 March
5306:Archived
5302:55539246
5266:(1975).
5242:Archived
5126:(2001).
5096:(2005).
5046:(2010).
4929:Archived
4915:Springer
4883:Archived
4826:Archived
4795:Archived
4781:Springer
4773:(2016).
4756:58568857
4736:(2004).
4504:Archived
4490:(1899).
4467:16 April
4461:Archived
4331:(1973).
4171:See also
4062:Setting
3842:becomes
3438:decrease
3251:10
3218:10
2846:Diameter
2748:Big Bang
2700:M-theory
2626:10
2484:Big Bang
2467:10
2372:10
2223:10
2096:10
1894:momentum
1884:10
1746:10
1620:, where
1468:1.875546
644:′
634:′
621:′
602:′
168:Big Bang
116:energies
8425:Science
8413:Physics
8399:Portals
7937:Modulor
7911:Unusual
7880:Persian
7838:Ancient
7800:Chilean
7759:Mexican
7749:Haitian
7678:Guinean
7582:Myanmar
7521:Swedish
7516:Spanish
7506:Serbian
7496:Russian
7476:Ottoman
7466:Maltese
7456:Latvian
7451:Italian
7397:Finnish
7375:English
7355:Cypriot
7350:Cornish
7257:History
7252:Outline
7187:Natural
7128:Chinese
7106:General
7099:Current
7051:(NIST).
6902:Bibcode
6795:Bibcode
6779:, NASA
6740:Bibcode
6677:Bibcode
6622:Bibcode
6532:Bibcode
6464:Bibcode
6403:Bibcode
6391:: 144.
6309:Bibcode
6266:5255898
6235:Bibcode
6067:Bibcode
5999:Bibcode
5885:Bibcode
5843:Bibcode
5809:9 April
5771:Bibcode
5730:26 June
5674:Bibcode
5557:Bibcode
5346:Bibcode
5282:Bibcode
5181:Bibcode
5142:Bibcode
5023:Bibcode
5021:: 152.
4445:Bibcode
4395:Bibcode
4292:Bibcode
4155:)/2 = 2
4132:√
4094:
4079:
3997:
3977:
3947:
3929:
3919:, with
3899:in the
3833:
3818:
3705:
3690:
3686:
3671:
3667:
3652:
3200:is the
3114:
2968:≈ 10 m
2939:2.725 K
2900:Density
2773:of the
2684:gravity
2652:gravity
2620:around
2529:in the
2233:density
1640:is the
1557:
1488: C
1377:
1278:
1196:
1121:
1040:units)
1036:Value (
140:lengths
8284:Others
7932:N-body
7870:Indian
7845:Arabic
7708:Somali
7683:Libyan
7651:Africa
7622:Syrian
7572:Korean
7557:Indian
7547:Afghan
7511:Slovak
7481:Polish
7419:German
7402:French
7365:Danish
7343:Europe
7309:Metric
7240:Metric
7220:Stoney
7210:Planck
7195:Atomic
7026:
7016:
6991:
6981:
6939:
6839:
6805:
6695:
6640:
6595:
6585:
6550:
6520:: 74.
6490:
6482:
6421:
6335:
6327:
6273:
6263:
6255:
6191:
6181:
6133:
6093:
6085:
6017:
5961:
5951:
5911:
5903:
5797:
5789:
5720:Forbes
5692:
5622:
5612:
5575:
5520:
5510:
5472:
5429:
5300:
5106:
5074:
5064:
4998:18 May
4968:18 May
4938:31 May
4921:
4875:
4787:
4754:
4744:
4709:18 May
4676:18 May
4646:18 May
4616:18 May
4510:23 May
4455:
4443:: 24.
4413:
4353:585119
4351:
4341:
4286:(10).
3557:sphere
3368:
3352:1.2103
3210:vacuum
3126:proton
3041:length
2738:, the
2710:, and
2523:second
2463:5.5608
2368:5.1550
2219:1.2103
2108:(LMT)
2092:1.9561
1990:(LMT)
1988:energy
1980:kg⋅m/s
1896:(LMT)
1880:4.2217
1756:volume
1742:2.6121
1498:
1049:length
947:, and
937:length
906:, and
819:action
703:, and
217:, the
204:, the
194:, the
184:, the
89:nature
70:, and
44:are a
7920:Other
7885:Roman
7865:Hindu
7860:Greek
7744:Cuban
7632:Tatar
7592:Omani
7531:Welsh
7526:Swiss
7446:Irish
7424:Greek
7370:Dutch
7360:Czech
7330:(CGS)
7324:(MTS)
7318:(MKS)
7290:(FPS)
7271:UK/US
6869:arXiv
6785:arXiv
6783:: 5.
6775:(1).
6730:arXiv
6693:S2CID
6667:arXiv
6638:S2CID
6548:S2CID
6522:arXiv
6488:S2CID
6454:arXiv
6419:S2CID
6393:arXiv
6333:S2CID
6299:arXiv
6225:arXiv
6189:S2CID
6102:(PDF)
6091:S2CID
6053:(PDF)
5909:S2CID
5795:S2CID
5761:arXiv
5690:S2CID
5664:arXiv
5600:. In
5573:S2CID
5547:arXiv
5298:S2CID
5211:arXiv
4932:(PDF)
4911:(PDF)
4855:arXiv
4577:(PDF)
4566:(PDF)
4411:S2CID
3643:(or 8
3636:but 4
3468:) + (
3407:1.416
3276:force
3206:light
3103:1.616
2978:≈ 10
2954:≈ 10
2637:2.176
2519:metre
2384:(LT)
2235:(LM)
2106:force
1366:1.416
1267:5.391
1185:2.176
1110:1.616
1027:Name
247:metre
208:, and
7637:Thai
7607:Pegu
7540:Asia
7178:(US)
7161:Troy
7024:OCLC
7014:ISBN
6989:OCLC
6979:ISBN
6955:2018
6937:ISBN
6837:ISBN
6803:OCLC
6593:OCLC
6583:ISBN
6564:2022
6480:ISSN
6367:2016
6325:PMID
6271:PMID
6253:ISSN
6179:ISBN
6149:2022
6131:ISBN
6083:PMID
6031:2022
6015:ISSN
5959:OCLC
5949:ISBN
5925:2022
5901:ISSN
5811:2022
5787:ISSN
5732:2019
5636:2022
5620:OCLC
5610:ISBN
5518:LCCN
5508:ISBN
5470:ISBN
5445:2010
5427:ISBN
5398:2022
5314:2022
5250:2021
5104:ISBN
5081:out.
5072:OCLC
5062:ISBN
5000:2024
4992:NIST
4970:2024
4962:NIST
4940:2020
4919:ISBN
4891:2019
4873:ISBN
4834:2021
4803:2020
4785:ISBN
4752:OCLC
4742:ISBN
4711:2024
4703:NIST
4678:2024
4670:NIST
4648:2024
4640:NIST
4618:2024
4610:NIST
4585:2019
4538:and
4528:and
4512:2020
4469:2022
4453:ISSN
4367:link
4349:OCLC
4339:ISBN
4219:and
3867:= −4
3669:(or
3594:and
3572:flux
3563:is 4
3411:(16)
3214:5.39
3202:time
3107:(18)
2868:Mass
2763:time
2698:and
2622:1.22
2612:and
2567:are
2563:and
2561:mass
2521:and
2374:kg/m
1758:(L)
1680:(L)
1678:area
1608:11.7
1559:and
1544:and
1530:and
1495:11.7
1370:(16)
1290:(Θ)
1271:(60)
1208:(T)
1206:time
1189:(24)
1133:(M)
1131:mass
1114:(18)
1051:(L)
945:mass
941:time
825:for
154:and
138:and
128:time
36:and
6910:doi
6748:doi
6685:doi
6630:doi
6540:doi
6472:doi
6411:doi
6389:441
6317:doi
6295:124
6261:PMC
6243:doi
6171:doi
6123:doi
6075:doi
6063:292
6007:doi
5893:doi
5851:doi
5779:doi
5682:doi
5565:doi
5354:doi
5290:doi
5189:doi
5177:135
5150:doi
4865:doi
4403:doi
4300:doi
4148:= (
4124:= 1
4071:= 1
4048:= (
3971:of
3907:or
3852:= −
3836:):
3763:= 1
3741:= 4
3533:= 0
3511:= 0
3495:iħγ
3464:= (
3409:784
3405:is
3247:2.5
3105:255
2994:Mpc
2827:Age
2796:).
2742:or
2734:In
2650:of
2608:In
2587:'s
2469:m/s
1561:2.2
1550:1.6
1368:784
1269:247
1187:434
1112:255
774:",
771:= 1
731:or
577:any
122:or
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