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ModPhy1/Unit1/GeneralRelativity/

8/21/04

 

1.      Tensors

2.      Einstein Tensor

3.      Matter Tensor

Einsteinian Gravitation

 

The Einsteinian concept of gravitation asserts that the distribution of matter in space and time is related to the geometrical curvature of spacetime as follows:

 

Einstein’s Gravitational Field Equation

* = Einstein’s geometrical tensor

* = matter tensor       

 

 

The units of measurement for this equation have been chosen so that the speed of light equals one (c = 1) and Newton’s universal gravitational constant equals one (G = 1). Therefore, everything in this equation is measured in units of distance (meters). The conversion factors for conventional units are given below:

 

Conversion Factors for MKS Units

Physical Quantity

MKS unit

Conversion Factor

Time

Seconds

1 = c = 2.99792458x108 m/s

Mass

Kilogram

1 = G/c2 = 0.7425x10-27 m/kg

Energy

Joules

1 = G/c4 = 0.826x10-44 m/J

 

Einstein’s gravitational field equation is deceptively simple because it is a tensor equation. The Einstein tensor is a symmetrical, second rank tensor uniquely determined by the geometry of spacetime whose divergence is identically zero. The matter tensor *contains the stress (force per unit area), the momentum density (momentum per unit volume), and the energy density (energy or mass per unit volume). Therefore, it is sometimes called the stress-momentum-energy-density tensor. The fact that its divergence is equal to zero results in the law of conservation of energy, the law of conservation of momentum, and Newton’s second law all combined into one tensor equation.

 

Therefore, Einstein’s gravitational field equation not only explains gravity in terms of geometry but also explains the laws of conservation of mass-energy and momentum as an inevitable consequence of that geometry.

Questions:

Q1.      Which are true concerning Einstein’s gravitational field equation? (A) It is a scalar equation. (B) It relates certain properties of the curvature of spacetime with the properties of the electromagnetic fields in space and time. (C) It contains within itself the law of conservation of energy, the law of conservation of momentum, and Newton’s second law. (D) More than one of these. (E) None of these.
C 524

 

  1. Tensors – The properties of tensors and tensor algebra.
  2. Einstein Tensor – The Einstein tensor and other geometrical tensors.
  3. Matter Tensor – The stress-momentum-energy-density tensor and its conservation laws.

ModPhy1/Unit1/GeneralRelativity/