Previous:  Space-time Curvature

Introduction

In special relativity, there is a well-known effect referred to as time dilation.  In the context of uniformly moving frames of reference, an observer notes a clock in a moving frame of reference as ticking more slowly than a clock in his own frame of reference.  This phenomenon can be used to explain many odd phenomena, for example the ability of particles called muons, created in the upper atmosphere of the earth, to reach the surface of the earth.

Vertical Motion of a Light Beam 

In a gravitational field, time dilation effects are also observed.  To examine how these effects arise, we consider the vertical motion of a beam of light in a gravitational field.  Using the Principle of Equivalence, we can determine the consequences by considering the vertical motion of the light waves in a frame of reference that is being accelerated upwards with an acceleration equal to that of the local gravitational field.  The effects observed in the accelerated frame will be the same as those observed in the gravitational field.  The accelerated frame is shown in the following diagram.

This diagram shows a frame of reference being accelerated upwards with acceleration g.  Thus, in the time T that it takes the light photons to travel from the top to the bottom of the frame of reference, the frame acquires a speed 

To a very good approximation, the time required for the photons to travel from the top to the bottom of the frame is 

Where c is the speed of light.  Therefore 

Now, an observer in the frame of reference has acquired a velocity v in the time required for the photons to reach the floor of the frame.  In these circumstances, the observer will note that the frequency of the light has changed slightly – an effect known as the Doppler Effect.

The Doppler Effect 

The Doppler Effect is very familiar to us, especially when we are considering sound waves.  We have all stood at the road side and listened to the siren of an approaching police vehicle or ambulance, and as the vehicle has gone past, we have noticed how the pitch of the siren changes.  This is the Doppler Effect in action, and the change of pitch corresponds to a change in the frequency of the sound waves, depending on whether the source of the waves is approaching us or moving away. 

For light waves observed by someone moving at relative speed v to the source, and provided that v is much less than the speed of light, the Doppler Effect is described by the following equation: 

That is, the relative change in the frequency Δf is proportional to the relative velocity v of the source and observer.  In our frame of reference, therefore, the magnitude of the Doppler Effect is given by

Where we have used the expression derived above for velocity.  From the Principle of Equivalence, if we observe this effect on an accelerating frame of reference as shown in the diagram above, then we also observe it in a gravitational field of the same strength as the accelerating frame. 

From now on, we will consider only the frame of reference subject to the gravitational field.

Doppler Effect in Terms of Gravitational Potential 

The quantity gh that appears on the right-hand side of this equation is equal to the difference in gravitational potential ϕ between the top and bottom of the frame of reference in the gravitational field.  By using integral calculus on the expression above, it can be shown that the frequency of the light wave varies with gravitational potential according to 

Where Δφ is the difference in gravitational potential between the source of the light waves and the point at which they are received, and f₀ is the frequency of the light waves, as observed at the source. 

This is an important result, because it states that the perceived frequency of light waves depends on the gravitational potential in which the observation of frequency is being made.  This effect is known as the gravitational red shift, and it has been verified experimentally (see Wikipedia or Google for details). 

It is important to note the sense in which this applies.  Movement to a stronger gravitational field implies that the gravitational potential becomes more negative.  Hence, considering the difference between the top and bottom of our frame of reference, Δφ is negative and f > f₀.  That is, the frequency at the bottom of the frame of reference (in the stronger gravitational field) is higher than at the top (where the gravitational field is weaker).

Gravitational Time Dilation

The gravitational time dilation effect can be deduced if we consider the time periods of the light waves, instead of the frequencies.  Now, the time period T is related to the frequency f through the relation 

So, if the frequency at the bottom of the frame of reference is greater than at the top, then the time period of the light waves at the bottom is smaller than at the top.  That is, 

We can interpret this as follows.  Consider an observer at the bottom of the frame of reference, in the stronger gravitational field.  If the source of the light waves were also located at the bottom of the frame, then the observer at the bottom would note a frequency f₀, and the time period of the waves would be

When the light waves come from a source at the top of the frame of reference, in the weaker field, the observer (at the bottom) sees a frequency f, with time period 

Therefore, the observer at the bottom sees things occurring more rapidly at the top of the frame of reference (in the weaker gravitational field), because 

This is the gravitational time dilation phenomenon that we seek.  Time runs more slowly in a frame of reference in a strong gravitational field, compared with what is observed (from the frame of reference in the stronger field) in lesser gravitational fields. 

As ever, it should be noted that what we have presented here is a non-rigorous account of gravitational time dilation.  The full theory is much more complex than this.  Note that the effects of gravitational time dilation have been verified by experiment.  See Wikipedia or do a search on Google for details.

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