The equivalence principle and the deflection of light

When Einstein developed his theory of general relativity, one starting point was the so-called equivalence principle. Roughly, it states that an observer in an elevator cannot tell whether he and the elevator are floating in space, far away from all sources of gravity, or whether the elevator is in free fall in a gravitational field. In particular, the laws of physics in an elevator in free fall are those of gravity-free space, in other words: the laws of special relativity.

Even if you don't know much about general relativity, there are ways of deriving some of its main predictions directly from the equivalence principle. One such prediction is that light is influenced by gravity!

Light and the elevator: the inside view

Consider a falling elevator, more precisely: an enclosed room like an elevator cabin but, in contrast with ordinary elevators, in free fall. Assume that, at the start of our experiment, the elevator is still at rest. At this exact moment in time, we shoot a brief pulse of laser light into the elevator, using a little hole in the elevator's side. Our laser is aimed in exactly the right way for the light to travel horizontally as it enters the elevator. What will the subsequent motion of our light pulse look like?

If you are an observer inside the elevator, then it is easy to find an answer to that question. The elevator is in free fall, and so are you. According to the equivalence principle, the laws of physics in your immediate neighbourhood are those of special relativity: Light travels at a constant speed along straight lines. Light entering the the cabin horizontally will continue to travel in the horizontal direction. If there is a second hole in the elevator wall, straight across from the first, then the light will leave the cabin through that hole:

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In the animation, the laser pulse is the red dot. Light speed has been slowed down artificially so you can follow the dot's progress; to see that it is travelling along a straight line, the dot has been made to leave a trace.

Elevator and light: the outside view

With this knowledge of how light travels as seen by an observer inside the cabin, we can deduce what an outside observer will see.

Initially, the light travels horizontally - on this, both observers agree given that, at the start of our experiment, they are at rest relative to each other. Also, both observers agree that the light pulse enters through one hole, and leaves through the other. But for the outside observer, the elevator starts to fall at the precise moment that the light pulse enters it. Whatever the details of the light pulse's motion, it will require a certain time to traverse the elevator compartment. In that time, the cabin will have moved downward - and so will the second hole, through which the light pulse later makes its exit. Consequently, from the point of view of the outside observer, the light pulse cannot travel along a straight line - after all, a straight line which is initially horizontal remains horizontal and at the same height! Instead, the outside observer will see something like this:

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The time it takes for light to traverse the cabin is so short that the speed reached by the falling cabin (as judged by the outside observer) is much, much smaller than that of light. Under these circumstances, we need not consult relativity - without much loss of accuracy, we can describe the cabin's fall and the parabola which is the orbit of the laser pulse with the help of classical mechanics.

As the speed of light in this animation is much lower than in reality, the parabola can readily be seen. For light moving at the proper speed of light, you would not distinguish this parabola from a straight line. Assuming that the two holes in the elevator walls are 1.5 meter (1.64 yards) apart, the light will fall by no more than a tenth of a trillionth of a millimeter as it traverses the cabin - a fraction of the diameter of an atomic nucleus.

The big picture

Our argument does not rely on the cabin being really, physically present. At no point do the cabin and the light interact. We must conclude that our argument holds even in the absence of a falling cabin - quite generally, light is deflected by gravitation.

Still, there is a fundamental restriction implicit in our use of the equivalence principle: Only in a small region of space, and over a brief time period can we use the laws of special relativity and end up with a good approximation. Only in a small falling elevator can we assume that light propagates at constant speed, along a straight line.

On the other hand, if we actually want to measure the deflection of light, we will have to look at the big picture - not at the local, but at the global deflection of light, for instance at light from a distant star passing close to the Sun before it reaches an observer here on Earth.

Is it possible to generalize our argument, in order to deduce the global deflection of light? One natural generalization would appear to be the following: Let us divide the region of space traversed by the light into thin strips. In each strip, we imagine a falling elevator cabin to determine how the light is deflected while crossing that particular strip. Patching the strips together, we obtain the total deflection of light crossing that particular region:

There's a catch, though, or rather: a tacit assumption. With every imaginary falling cabin, we're tracing the behaviour of light in one single strip. But how do we patch these strips together to reconstruct the whole of space? The answer depends on the geometry of space. In fact, one can define the geometry of space as the answer to the question of how infinitesimally small regions of space join together to form a whole.

The magnitude of the deflection of light depends on the patching. In the illustration above, the tacit assumption is that geometry is of the ordinary, Euclidean variety taught in high school. There are other possibilities: Think of a curved surface - for instance a gently sloping hill. You cannot cover this hill with square tiles in a regular chessboard pattern - you will either have to leave gaps between the tiles, or you will have to shape your tiles a bit differently. Similarly, a more complex space geometry might force us to shape our strips into little wedges, which are patched together as in the following illustration:

For every single wedge, an arrow shows which way is down - the direction in which the imaginary elevator is falling. Light is bent downward, but in a sense, so is space itself. The result is that, relative to space, the light ray is not bent as much as in the Euclidean case - the level between the horizontal direction peculiar to the last wedge (the direction at right angles to "down" - not the same as the horizontal of the web-page, unless your computer screen is crooked) is not as large as in Euclidean geometry.

There's also the opposite case, in which the curvature of space increases the magnitude of light deflection, as sketched here:

In this case, space "bends towards the light": The horizontal direction and the downwards direction change from one strip to the next in such a way that, in the end, the deflection of the light relative to the local horizontal direction of the last strip is significantly larger than in the Euclidean case.

In consequence, you can only make statements about the large-scale deflection of light if you know the geometry of space.

In pre-Einsteinian physics, it is eminently plausible to assume that space geometry is Euclidean. If you make the falling-elevator argument in Euclidean space, you can calculate "Newtonian light deflection" for interesting cases like starlight passing the sun. This deflection is a direct consequence of the equivalence principles.

On the other hand, in Einstein's theory of general relativity, the presence of a mass curves the surrounding space in a way similar to the "bending of space towards the light" of our last example above. The resulting angle of deflection for light passing close to the Sun is exactly twice as large as in the Newtonian case - with the equivalence principle and the curvature of space each responsible for half of that value.

The case in which the deflection of light is diminished by the curvature of space is of historical interest only: At the same time that Einstein wrote his first articles on what is now called general relativity, the Finnish physicist Gunnar Nordström looked for different ways of constructing a theory of gravity which was compatible with special relativity. The model known to historians of science as "Nordström's second theory" incorporates the equivalence principle, but space is curved in exactly the right way to cancel out local light deflection.

All in all, the equivalence principle gives us a valuable clue that light is influenced by gravity. In fact, this is how Einstein came to think about the deflection of light as he was working on general relativity. But if we want to be certain whether or not there is large-scale light deflection, we need to go beyond the equivalence principle - and take into account the geometry of space.

[Markus Pössel, AEI]

This spotlight text supplements the chapter General relativity of Elementary Einstein.

For basic information about the equivalence principle, we recommend the spotlight text The elevator, the rocket, and gravity. Related spotlight texts can be found in the category General relativity.

More technical information about the crucial difference between local and global light deflection can be found in the article

J. Ehlers and W. Rindler, "Local and Global Light Bending in Einstein's and other Gravitational Theories" in General Relativity and Gravitation 29 (1997), p. 519-529.