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Curvature of a Light Beam

The curvature of space time is an aspect of general relativity that many people will have heard of in some form or another.  In this section we will use the Principle of Equivalence (see the previous section) to examine why it is that we need to think in terms of curved space-time in a gravitational field. 

We will consider a simple thought experiment in an accelerating frame of reference, and then use the Principle of Equivalence to consider the same experiment in a gravitational field.  Our experiment will involve standing in an accelerating frame of reference, and then shining a torch from one side of the frame of reference to the other, perpendicular to the direction of acceleration.   

This experiment is shown in the following diagram.

To an observer in this accelerating frame of reference, the light beam appears to suffer a deflection downwards, because the vertical position of the frame of reference is changing due to its upward acceleration.  Furthermore, because of the acceleration, the velocity of the frame of reference is changing during the passage of light from one side to the other.  The result is that the light beam, to the observer in the accelerated frame, appears to follow a curved path across the frame of reference. 

Of course, the magnitude of this effect is very small, unless the acceleration of the frame of reference is very high.  Let us investigate this further.  Suppose that the width of the frame of reference is d.  Then the time required for the light beam to travel across the frame of reference is just 

where c is the speed of light.  In this time T, the frame of reference moves upwards through a distance L, given by 

This distance L is the deflection from the horizontal that is observed by an observer in the accelerated frame of reference.  Substituting, we get 

At the surface of the earth, g = 9.8 m s^-2.  Suppose that d = 5 m.  Since the speed of light c is equal to 3 10⁸ m s^-1, this gives 

d ~ 3 10^-16 m 

This is about the dimension of an atomic nucleus, and so would not be measurable or observable.  The effect is small on the surface of the earth!

Applying the Principle of Equivalence 

Nevertheless, we can invoke the Principle of Equivalence, which tells us that we can replace the local effects of gravity with an appropriate accelerated frame of reference.  Thus, if we observe a deflection of a light beam in an accelerated frame, then we would observe the same deflection in an “equivalent” gravitational field.  This equivalent field is shown in the diagram below, with the effect being observed in a non-accelerating frame of reference. 

 This simple argument shows that the geometry along which light rays propagate in a gravitational field (or in an accelerated laboratory) is not in general flat, but rather it is along a curved space, with the extent of the curvature (which is of course related to the vertical deflection we calculated above) depending on the strength of the gravitational field.

Curvature of Space 

This demonstrates in a (relatively) simple way that when we consider the laws of physics in a gravitational field, we will need to consider them in a curved space.  One of the basic principles of classical physics is that light travels in straight lines, but in a gravitational field it follows a curved path.  This is good evidence that a flat geometry will not be adequate for describing the passage of light or other electromagnetic waves in a gravitational field.  In other words, our familiar "flat" space-time will need to be replaced by a curved space time.

Note that strictly this argument only demonstrates the curvature of the spatial part of space-time – the temporal part is curved too, but I think you will agree that this is a much more difficult concept to envisage, and as such is best left to the experts!

Magnitude of Space-time Curvature 

To take the concept of space-time curvature further requires the use of very sophisticated maths.  However, let us finish by showing that the curvature can in some circumstances be very significant.  Let us consider the gravitational field at the surface of an "intermediate-mass" black hole.  For such a black hole, the radius R is about 1,000 km and the mass is about 1,000 times greater than that of the sun, about 2 10³³ kg.  The gravitational field at the surface is then given by 

Which evaluates to about 10¹¹ m s^-2.  If we now re-evaluate the deflection across our frame of reference of width 5 m, we find that 

d ~ 4 10^-6 m 

which is a measurable quantity.  Now suppose that we consider the deflection across the diameter of the black hole, 1,000 km.  This will be approximately 

d ~ 0.7 m 

which is a very appreciable deflection, and would get substantially larger if we considered longer length scales or more massive black holes.

Note that this calculation is only approximate, because we are assuming that the gravitational field across the diameter of the black hole can be represented by a single equivalent accelerated frame, and this is not the case.  Nevertheless, the calculation does hint that, while space-time curvature at the surface of the earth (and indeed the surface of the sun) might be a very small effect, when we deal with black holes and other small massive objects, it can be very significant indeed.

Next:  Gravitational Time Dilation