The Small Deformation Strain Tensor as a Fundamental Metric Tensor

Affiliation(s)

Instituto de Investigaciones Eléctricas, División de Energías Alternas, Mexico City, México.

Instituto de Investigaciones Eléctricas, División de Energías Alternas, Mexico City, México.

ABSTRACT

In the general theory of relativity, the fundamental metric tensor plays a special role, which has its physical basis in the peculiar aspects of gravitation. The fundamental property of gravitational fields provides the possibility of establishing an analogy between the motion in a gravitational field and the motion in any external field considered as a noninertial system of reference. Thus, the properties of the motion in a noninertial frame are the same as those in an inertial system in the presence of a gravitational field. In other words, a noninertial frame of reference is equivalent to a certain gravitational field. This is known as the principle of equivalence. From the mathematical viewpoint, the same special role can be played by the small deformation strain tensor, which describes the geometrical properties of any region deformed because of the effect of some external agent. It can be proved that, from that tensor, all the mathematical structures needed in the general theory of relativity can be constructed.

In the general theory of relativity, the fundamental metric tensor plays a special role, which has its physical basis in the peculiar aspects of gravitation. The fundamental property of gravitational fields provides the possibility of establishing an analogy between the motion in a gravitational field and the motion in any external field considered as a noninertial system of reference. Thus, the properties of the motion in a noninertial frame are the same as those in an inertial system in the presence of a gravitational field. In other words, a noninertial frame of reference is equivalent to a certain gravitational field. This is known as the principle of equivalence. From the mathematical viewpoint, the same special role can be played by the small deformation strain tensor, which describes the geometrical properties of any region deformed because of the effect of some external agent. It can be proved that, from that tensor, all the mathematical structures needed in the general theory of relativity can be constructed.

Cite this paper

Palacios, A. (2015) The Small Deformation Strain Tensor as a Fundamental Metric Tensor.*Journal of High Energy Physics, Gravitation and Cosmology*, **1**, 35-47. doi: 10.4236/jhepgc.2015.11004.

Palacios, A. (2015) The Small Deformation Strain Tensor as a Fundamental Metric Tensor.

References

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[2] Mc Connell, A.J. (1931) Applications of Tensor Analysis. Dover Publications, Inc., New York.

[3] Einstein, A. (1923) The Principle of Relativity. Dover Publications, Inc., Mineola, New York.

[4] Eddington, A.S. (1923) The Mathematical Theory of Relativity. Chelsea Publishing Company, New York.

[5] Landau, L.D. and Lifshitz, E.M. (1962) The Classical Theory of Fields. Addison-Wesley Publishing Company, Inc., Boston.

[6] Bergman, P.G. (1942) Introduction to the Theory of Relativity. Prentice-Hall, Inc., Upper Saddle River.

[1] Fierros Palacios, A. (2006) The Hamilton-Type Principle in Fluid Dynamics. Fundamentals and Applications to Magnetohydrodynamics, Thermodynamics, and Astrophysics. Springer-Verlag, Wien.

[2] Mc Connell, A.J. (1931) Applications of Tensor Analysis. Dover Publications, Inc., New York.

[3] Einstein, A. (1923) The Principle of Relativity. Dover Publications, Inc., Mineola, New York.

[4] Eddington, A.S. (1923) The Mathematical Theory of Relativity. Chelsea Publishing Company, New York.

[5] Landau, L.D. and Lifshitz, E.M. (1962) The Classical Theory of Fields. Addison-Wesley Publishing Company, Inc., Boston.

[6] Bergman, P.G. (1942) Introduction to the Theory of Relativity. Prentice-Hall, Inc., Upper Saddle River.