Abstract
In this paper we present a new unified theory of electromagnetic and gravitational interactions. By considering a four-dimensional spacetime as a hypersurface embedded in a five-dimensional bulk spacetime, we derive the complete set of field equations in the four-dimensional spacetime from the fivedimensional Einstein field equation. Besides the Einstein field equation in the four-dimensional spacetime, an electromagnetic field equation is obtained: ∇a F ab - ξR b a A a = -4πJ b with ξ = -2, where F ab is the antisymmetric electromagnetic field tensor defined by the potential vector A a, R ab is the Ricci curvature tensor of the hypersurface, and J a is the electric current density vector. The electromagnetic field equation differs from the Einstein–Maxwell equation by a curvature-coupled term ξR b a A a, whose presence addresses the problem of incompatibility of the Einstein–Maxwell equation with a universe containing a uniformly distributed net charge, as discussed in a previous paper by the author [L.-X. Li, Gen. Relativ. Gravit. 48, 28 (2016)]. Hence, the new unified theory is physically different from Kaluza–Klein theory and its variants in which the Einstein–Maxwell equation is derived. In the four-dimensional Einstein field equation derived in the new theory, the source term includes the stress-energy tensor of electromagnetic fields as well as the stress-energy tensor of other unidentified matter. Under certain conditions the unidentified matter can be interpreted as a cosmological constant in the four-dimensional spacetime. We argue that, the electromagnetic field equation and hence the unified theory presented in this paper can be tested in an environment with a high mass density, e.g., inside a neutron star or a white dwarf, and in the early epoch of the universe.
Similar content being viewed by others
References
A. Einstein, Zur allgemeinen Relativitätstheorie, Seitsber. Preuss. Akad. Wiss. Berlin, 1915, p. 778
A. Einstein, Die Feldgleichungen der Gravitation, Seitsber. Preuss. Akad. Wiss. Berlin, 1915, p. 844
H. F. M. Goenner, On the history of unified field theories, Living Rev. Relativity 7, 2 (2004)
A. Einstein, A generalized theory of gravitation, Rev. Mod. Phys. 20, 35 (1948)
A. Einstein, The Meaning of Relativity, 5th Ed., Including the Relativistic Theory of the Non-Symmetric Field, Princeton University Press, Princeton, 1955
H. Weyl, Gravitation und Elektrizität, Seitsber. Preuss. Akad. Wiss. Berlin, 1918, p. 465
A. Eddington, A generalisation of Weyl’s theory of the electromagnetic and gravitational fields, Proc. R. Soc. Ser. A 99, 104 (1921)
E. Schrödinger, The final affine field laws I, Proc. Royal Irish Acad. A 51, 163 (1947)
G. Nordström, Über die Moglichkeit, das electromagnetische Feld und das Gravitationsfeld zu vereinigen, Phys. Z. 15, 504 (1914)
T. Kaluza, Zum Unitätsproblem der Physik, Sitzungsber. Preuss. Akad. Wiss, 1921, p. 966
O. Klein, Quantentheorie und fünfdimensionale Relativitätstheorie, Z. Phys. 37, 895 (1926)
O. Klein, The atomicity of electricity as a quantum theory law, Nature 118, 516 (1926)
D. Bailin and A. Love, Kaluza–Klein theories, Rep. Prog. Phys. 50, 1087 (1987)
J. M. Overduin and P. S. Wesson, Kaluza–Klein gravity, Phys. Rep. 283, 303 (1997)
N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, The hierarchy problem and new dimensions at a millimeter, Phys. Lett. B 429, 263 (1998)
I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, New dimensions at a millimeter to a fermi and superstrings at a TeV, Phys. Lett. B 436, 257 (1998)
L. Randall and R. Sundrum, Large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83, 3370 (1999)
L. Randall and R. Sundrum, An alternative to compactification, Phys. Rev. Lett. 83, 4690 (1999)
R. M. Wald, General Relativity, University of Chicago Press, Chicago, 1984
S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press, Cambridge, 1973
S. Carroll, Spacetime and Geometry: An Introduction to General Relativity, Addison-Wesley, New York, 2003
L.-X. Li, Electrodynamics on cosmological scales, Gen. Relativ. Gravit. 48, 28 (2016)
S. Weinberg, Gravitation and Cosmology, John Wiley and Sons, New York, 1972
C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, W. H. Freeman, New York, 1973
A. Einstein, Die Grundlage der allgemeinen Relativitätstheorie, Ann. Phys. 354, 769 (1916)
R. L. Arnowitt, S. Deser, and C. W. Misner, in: Gravitation: An Introduction to Current Research, Ed. L. Witten, John Wiley and Sons, Inc., New York, 1962
According to Campbell’s theorem, any analytic n- dimensional Riemannian space can be locally embedded in an (n+1)-dimensional Ricci-flat space 43, 44. Hence, consideration of an n-dimensional spacetime embedded in an (n+1)-dimensional spacetime does not seem to put much constraint on the properties of the n-dimensional spacetime.
In this paper tensor space on a manifoldMwill generally be denoted by T (M), regardless of the type of tensor (scalar, vector, dual vector, or tensor of any type).
The Lagrangian in Eq. (65) does not contain any derivatives of N so we do not interpret N as a matter field.
Note that all abab; 2, abab, 2, ab, and are proportional to N -2.
E. C. G. Stueckelberg, Théorie de la radiation de photon de masse arbitrairement petite, Helv. Phys. Acta 30, 209 (1957)
M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Westview Press, New York, 1995
C. Liang and B. Zhou, An Introduction to Differential Geometry and General Relativity, Vol. I, Science Press, Beijing, 2006
J. Binney and S. Tremaine, Galactic Dynamics, Princeton University Press, Princeton, 1987
G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. L. Bennett, et al., Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Cosmological parameter results, Astrophys. J. Supp. 208, 19 (2013)
P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont, et al. (Planck Collaboration), Planck 2015 results. XIII. Cosmological parameters, arXiv: 1502.01589, 2015
The idea of interpreting the extra geometric terms in a four-dimensional Einstein field equation derived from 5D gravity as representing induced matter in a fourdimensional spacetime has been extensively investigated by Wesson and his collaborators (see 14, 45 and references therein). They proposed that the extra geometric terms are the stress-energy tensors of the induced matter and regarded the fifth dimension as being associated with the rest mass of particles instead of a real space dimension. However, in their theory, they did not derive the field equations of matter and electromagnetic fields.
T. Shiromizu, K. Maeda, and M. Sasaki, The Einstein equations on the 3-brane world, Phys. Rev. D 62, 024012 (2000)
R. M. Wald, Black hole in a uniform magnetic field, Phys. Rev. D 10, 1680 (1974)
Strictly, when an electromagnetic field is present the spacetime cannot be exactly Ricci-flat, since the stressenergy tensor of the electromagnetic field will make Rab = 0. However, if the electromagnetic field is weak its effect on the spacetime curvature can be ignored and the spacetime can be approximately Ricci-flat if the mass density of other matter is sufficiently low.
M. S. Turner and L. M. Widrow, Inflation-produced, large-scale magnetic fields, Phys. Rev. D 37, 2743 (1988)
A. S. Goldhaber and M. M. Nieto, Photon and graviton mass limits, Rev. Mod. Phys. 82, 939 (2010)
J. E. Campbell, A Course of Differential Geometry, Clarendon Press, Oxford, 1926
C. Romero, R. Tavakol, and R. Zalaletdinov, The embedding of general relativity in five dimensions, Gen. Relativ. Gravit. 28, 365, 1996
P. S. Wesson, The status of modern five-dimensional gravity (A short review: Why physics needs the fifth dimension), Int. J. Mod. Phys. D 24, 1530001 (2015)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, LX. A new unified theory of electromagnetic and gravitational interactions. Front. Phys. 11, 110402 (2016). https://doi.org/10.1007/s11467-016-0588-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11467-016-0588-z