Generalizing Equations

To get the appropriate equations in general relativity from the corresponding equations in special relativity we adopt a principle of simplicity.

Rule 1

If a physical quantity can be defined as a Cartesian tensor (flat space tensor) in Special Relativity then it can be a definition in General Relativity by defining it exactly the same way in a local Cartesian coordinate system (i.e. use geodesic coordinates). Then to get the quantity in any other coordinate system requires using the tensor transformation equation.

Rule 2

Any Cartesian tensor equation can be converted to an equation valid in General Relativity in any coordinate system, simply by replacing Special Relativity derivatives with General Relativity derivatives and we replace the Minkowski metric with the nonflat GR metric.

Ex. - In a Cartesian coordinate system in Special Relativity Maxwell's Equations have the form

 (11.1)
 (11.2)

To generalize we adopt this pattern and use General Relativity covariant derivatives for the Special Relativity partial derivatives.

 (11.3)
 (11.4)

In generalizing we want to make simple GR equations with no complicated terms involving the curvature tensor, which theoretically could be present since, in the flat space of Special Relativity, they would automatically be zero. As an example, consider the Special Relativity equation for the conservation of the energy momentum tensor .

 (11.5)

The simplest generalization of this equation without a curvature term would be

 (11.6)

Theoretically the following equation also works as well since it has the correct reduced state:

We choose (11.6) because it's the simplest representation in curved space. To experimentally test each equation to see which one fits the data we would have to be in a strong gravity environment. Since the Earth is not in a strong gravity environment this is not possible. From applications to dense stars and cosmological situations we can check for consistency with various strong gravity data. So far the simplicity principle for choosing GR equations has not been contradicted by strong field astronomical data.

As a further example of the ideas of generalization we consider moving particles in Special Relativity. We will generalize the equation for the path of a moving massive particle under the influence of an external force.

Let the SR velocity of the particle be given as

 (11.7)

where  is the proper time for the particle defined in terms of the following equation:

 (11.8)

The 4-D version of Newton's Laws in SR takes the form

 (11.9)

with

 (11.10)

being the relativistic 4-momentum.  is the rest mass of the particle and  is the 4-force acting on it.

We now write down the General Relativity equivalents of these equations.

The velocity equation stays the same as (11.7)

 (11.11)

The following relation must now define the proper time

 (11.12) .

Newton’s 2nd Law equation now becomes

 (11.13)

where the absolute derivative is introduced since  is an indirect function of .

The defining momentum equation takes the same form as before:

 (11.14)

As in Special Relativity we assume that the most correct clock is one that measures it’s own proper time. This time will, in general, be substantially different from the coordinate time that comes into the components of the particular choice of metric for the spacetime manifold.

We now take a closer look at (11.13) and compare it to the usual Newton’s Law result. This equation can be manipulated as follows:

Expanding this out using the definition of the absolute derivative as follows

 (11.15)

we see that we have

 (11.16)

This equation is the GR counterpart of the 3-D Newtonian equation

 (11.17)

When we impose the free particle condition, , the solutions of this equation give rise to straight-line paths for the particle.

In General Relativity the free particle condition  gives back the geodesic equation:

 (11.18)

Free particles do not necessarily follow straight lines through space as they do in the standard Newtonian case if the affine connections are nonzero. A particle responds to a geodesic in the curved spacetime of the gravitating body. It follows the shortest path consistent with the curvature of the spacetime no matter how curved that path is.

In Newtonian Laws the third law stipulates that when a gravitational force acts on a body, then that body also acts with an equal, and oppositely directed, force back on the source.

Interaction between two large bodies (i.e. extended in 3-D space) in GR is not so simple. The concept of force is done away with in favor of motion along curved paths. Each body in GR should produce its own curvature of spacetime for the other body to move in. To avoid the complicated problems with extended body interactions in GR, it is standard to introduce the concept of a Test Particle. A Test Particle is a particle that has a PASSIVE gravitational mass but approximately no ACTIVE gravitational mass. Hence we can ignore the Test Particle’s effect on spacetime. A planet moving around the Sun can, to a very good approximation, act as a Test Particle relative to the Sun’s gravitational field.

Note: In gravitational theory three types of masses can be distinguished.

Type 1: Inertial mass  The mass that enters into Newton’s 2nd Law

Type 2: Active mass  The mass that produces a gravitational field.

Type 3: Passive mass  The mass that gets acted upon by a gravitational field.

The Equivalence Principle says that Type 1 and Type 2 masses must be equal. The Type 2 and Type 3 masses have always been assumed to be equal since the role of source and receiver can be switched around symmetrically in nonquantum gravitational theory.