Section 29

# The Solution Of The Problem Of Gravitation On The Basis Of General Relativity

March 6, 2022

We began with a Galileian domain where there is no gravitational field relative to the Galileian reference-body `K`.

Special Relativity (SR) explains the behaviour of measuring-rods and clocks with reference to `K` and the behaviour of “isolated” material points.

• Isolated points move uniformly and in straight lines.

## Gauss coordinates

Now let us refer this domain to a random Gauss co-ordinate system or to a “mollusk” as reference-body K'.

• Then with respect to K’ there is a gravitational field G of a particular kind.

The behaviour of measuring-rods and clocks and also of freely-moving material points with reference to `K'` can be known simply by mathematical transformation of spacetime as an effect of the gravitational field `G`.

With this, we introduce a hypothesis: the influence of the gravitational field on measuring-rods, clocks and freely-moving material points continues to take place according to the same laws, even when the prevailing gravitational field is not derivable from the Galileian special case, simply by means of a transformation of co-ordinates.

The next step is to investigate the space-time behaviour of the gravitational field `G`, which was derived from the Galileian special case simply by the transformation of the co-ordinates.

• This behaviour is formulated in a law, which is always valid, no matter how the reference-body (mollusk) used in the description may be chosen.
• This law is not yet the general law of the gravitational field, since the gravitational field under consideration is of a special kind.

In order to find out the general law-of-field of gravitation, we still need to generalize this new law of gravity=

1. The required generalisation must likewise satisfy GR
1. Only the inertial mass or its energy (as per Section 15) are important for its effect in exciting a field
2. Gravitational field and matter together must satisfy the law of the conservation of energy (and of impulse)

Finally, GR lets us determine the influence of the gravitational field on processes without a gravitational field which is the case of Special Relativity

The gravitation of GR removes the defect of:

• classical mechanics explained in Section 21
• the wrong interpretation of the empirical law of the equality of inertial and gravitational mass

But it explains astronomical observations which classical mechanics is powerless in.

GR absorbs the Newtonian theory if we confine gravitational fields to be:

• regarded as being weak, and
• in which all masses move with respect to the coordinate system with velocities which are small compared with the speed of light

Thus, GR absorbs Newton’s theory without needing* the inverse square law where gravitation between two objects depends on the square of the distance between them.

*Superphysics note: This is because General Relativity has its version of the inverse square law in the metric tensor

If we increase the calculation’s accuracy, deviations from Newton’s theory appear. These deviations are small and escape the test of observation.

According to Newton’s theory, a planet moves around the sun in an ellipse. This ellipse would permanently maintain its position with respect to the fixed stars, if we disregard:

• the motion of the fixed stars themselves and
• the action of the other planets

However, this is not true for Mercury, lying nearest the sun.

Since Leverrier’s time, Mercury’s ellipse was found not stationary with respect to the sun. Instead, it rotates slowly as a “precession”=

• in the plane of the orbit and
• in the sense of the orbital motion.

This rotation was 43 seconds of arc per century.

GR has found that the ellipse of every planet around the sun must rotate in the way indicated above.

All planets, except Mercury, rotates in a very small manner to be detected. But in the case of Mercury, it amounts to 43 seconds of arc per century – this agrees with observation.*

*Superphysics note: The precession of Mercury is caused by its core, not by spacetime coordinates

Apart from this one, GR allows only two deductions which can be tested by observation=

• the curvature of light rays by the sun’s gravitational field [gravitational lensing] and
• a displacement of the spectral lines of light reaching us from large stars, as compared with the corresponding lines for light produced in an analogous manner terrestrially (i.e. by the same kind of molecule * ).

These deductions from the theory will be confirmed also.