Including Gravity: The Move to the General Theory of Relativity

Facets of the Newtonian theory of gravity:

Here M1, M2, are the masses of the two objects while r12 is the distance between their centers.

The force between these two bodies is attractive with a direction along the line connecting their centers and has a magnitude which falls off inversely as the square of the separation between the bodies:

The basic fact about gravity is that the force is attractive and very long range and that its strength is proportional to the product of the mass of the source and object acted upon. Another important fact is that when an object is dropped in a gravitational field, it necessarily accelerates unless there are additional forces around to counteract such acceleration.

The acceleration of a freely falling object of mass m in the field of an object of mass M is given by the following relation:

Note that the acceleration is independent of the mass of the falling object! This implies a kind of universality of the property of free fall: all objects at the same distance from the center of the source have the same acceleration. Near the surface of the earth this value is 9.8 m/s2 .

Problem: Since free fall is ubquitous in a gravitational environment, one realizes immediately that there is no natural place for gravitational effects within the context of the special theory of relativity.

The Principle of Equivalence

In 1911 Einstein introduced the principle of equivalence. This principle was a stepping stone in his path toward a comprehensive theory which would incorporate both the special theory and gravitation.

The Principle of Equivalence: In a small region it is not possible to distinguish the results of any experiment performed in a permanent gravitational field from an analogous experiment performed in a uniformly accelerated frame of reference in gravity-free space.

This principle has been tested directly and indirectly in several key experiments since 1911. Among these we mention:

• The Eddington expedition of 1919, which observed the bending of light as it grazes the edge of the Sun

• The Pound-Rebka and Pound-Snider Experiments measured the shift in the frequency of electromagnetic radiation falling in a gravitational field

The Principle of Equivalence implies that the shift in frequency relative to the emitted frequency is proportional to H:

Thus, there is a blue shift of light falling in a gravitational field. When the source of light is emitted at the bottom of the shaft and climbs up to the to, an analogous application of the principle gives the same result as above except with the opposite sign. That is, when the light climbs in a gravitational field, it is red-shifted.

Implication of the Principle of Equivalence

Here we show that clocks run at different rates depending on their position in a permanent gravitational field.

We make use of the famous energy relation first written by Max Planck in 1900:

E = h f

where h is Planck's constant and f is the frequency of the radiation.

We also make use of the conservation of energy:

Efinal = Einitial +/- work done

Consider the following thought experiment, analogous to the Pound-Rebka/Snider experiments, but here with a different purpose.

Using the conservation of energy we have:

Energy of photon at top = Energy of photon at

bottom - work done on photon by gravity

The only thing that remains is to figure out the work done by gravity on the climbing photon. Generally, work done is defined to be the force which acts multiplied by the incremental distance over which it acts. Here the force is that due to gravity and is Fgravity = mass * g. Now we make use of Einstein's famous E = m c2 to write for the work done:

Work done by gravity = - Fgravity * h

= - (Ebottom/c2) * g * h
Then, the energy accounting becomes:

Etop = Ebottom -(Ebottom/c2) * g * h

hftop= hfbottom( 1 - g*h/c2)

Cancelling out the common factor of h, we find that the frequency received at the top is lower than that sent at the bottom.

Now let's translate this into a statement about time. Since frequency = number of complete oscillations per cycle and calling the time for one cycle T, we have f = 1/T. Hence, we have

1/Ttop= (1/Tbottom)( 1 - g*h/c2)
The inversion of the above relation leads to
Ttop = Tbottom / ( 1 - g*h/c2)

Since the factor ( 1 - g*h/c2) is smaller than 1.0, it follows that the time for one cycle of a photon emitted at the bottom is measured at the top to be longer than it would be if the tick occurred on a clock at rest at the top.

Thus clocks on the surface of the earth tick slower than clocks at rest out in space orbit.

Minkowski spacetime and gravitational redshift

Recall that for Minkowski spacetime, the interval between events is:

ds2 = c2dt2 - dx2 -dy2 -dz2

no matter where we measure events. The above result shows that when gravitation is included, it does matter where you are in relation to the gravitating source when you make comparisons about clocks. Thus, on the face of it Minkowski spacetime is incompatible with gravity.

How to fix this?

Einstein's answer was to geometrize gravity so that its effects were 'embedded' in the fabric of the geometric structure of space and time.

One simple modification of the Minkowski spacetime model which would be in accord with the redshift effect would be to take as the spacetime model, even in the prescence of gravity, an organization of the set of events in which the spacetime interval between events is:

ds2 = c2(1 - 2gh/c2)dt2 - dx2 -dy2 -dz2

The factor of 2 gh/c2 must be present in order to get the correct result for the bending of the light near the sun.

We shall see that this is but the first step toward the final form of the theory. Einstein realized that in order to make the geometric conception complete and consistent with experience that both space and time need to be influenced by gravity, not just time...