E=mc² and the gravitational frequency shift

One of the basic consequences of general relativity is the so-called gravitational frequency shift of light: Light falling towards a massive body experiences a shift towards higher frequencies (blueshift), while light leaving the vicinity of such a body is shifted towards lower frequencies (redshift).

While the exact formula for this frequency shift can only be derived from general relativity itself, its existence and a good approximate formula for its magnitude follows directly from a combination of the laws of special relativity with the classical notion of gravity, the non-existence of perpetual motion machines, the convertability of energy and laws governing the properties of light - which is what this spotlight text is all about. Here are the physical laws that we need:

Falling bodies

When an object is in free fall in a gravitational field, then from the point of view of an observer who sees it falling and accelerating (for instance an observer at rest on the surface of the earth), it gains energy - more precisely, the object's kinetic energy, the energy associated with its motion, increases as it gathers more and more speed.

Energy is convertible - and so is mass

Energy is versatile - it is possible to turn one form of energy into another. With his famous formula E=mc², Einstein added another form of energy to the varieties that were already known: Even when a material object is at rest with respect to a particular observer, that observer needs to assign to that object an enormous amount of energy, simply because it has a mass.

While there are some restrictions on how energy can be converted from one form to another, let us assume that we can build a machine that can convert the energy associated with a falling body - the energy it has even at rest, because of its mass, and the energy it has because of its motion - completely into light energy. Here is a sketch:

The energy of a body falling into the funnel on the left-hand sight is converted into electromagnetic radiation energy (in short, light energy) which is emitted by the beamer pointing straight up on the right-hand side of the apparatus.

(To make our argument more rigorous, we could assume that only a given fraction of the falling body's energy can be converted to light. However, this does not change the reasoning of the following argument - it just makes it harder to keep track of the different energies involved. Hence, we will stick with the somewhat unrealistic simplifying assumption that all of the body's energy can be converted.)

Properties of light

The only property of light that we need for our argument is one of the foundations of quantum theory: In many settings, light and other forms of electromagnetic radiation behave as if they consisted of myriads of tiny "energy packets" - "particles of light", if you will, which physicists call photons. The energy of each photon is the product of the light's frequency times a universal constant of nature called Planck's constant,

The frequency of the light is directly related to its colour, as sketched in the following image in which photons are symbolized as wiggly arrows (as is usual in physics):

For light of a single frequency ("monochromatic light"), the total energy is the energy of each of the photons (Planck's constant times the frequency) times the total number of photons. If I manage to change the total energy without changing the number of photons, then the frequency of the light must change, as well - there must be a shift towards lower frequencies if the energy has been lowered, or higher frequencies if the energy has increased.

There's no such thing as a free lunch

Energy is a conserved quantity - energy cannot be destroyed, or created from nothing. If the energy of a system changes, that must be because energy has been transferred from or to that system. This is equivalent to saying that there is no "perpetuum mobile (of the first kind)": There is no such thing as a machine, or combination of machines, that goes through a sequence of changes, transfers energy to its environment, and then returns exactly to its original state.

If there were such a machine, we could create energy at will: After each cycle, the machine would be in its original state, so in particular it would have the same energy as before. But during the cycle, energy has been transferred to its environment (for instance, we could hook the machine up to an electrical generator and produce electricity). This additional energy would have been created from nothing.

The necessity of the gravitational frequency shift

Now that we have marshalled all the necessary information, we can show why there must be such a thing as a gravitational frequency shift. Imagine that we build the following machine somewhere on earth:

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The machine drops a matter pellet off a high tower. At the bottom of the tower, there is a contraption which converts the matter (and all its energy) into light of a fixed frequency. After all, we know from special relativity that the original mass of the lead pellet, when it was still at rest at the top of the tower, is just another form of energy. As the lead pellet falls, it gains additional energy.

Photon after photon, that light is sent directly upwards and collected at the top of the tower, where there is a contraption to turn it back into rest mass energy - more precisely, into a pellet of matter.

Evidently, this pellet of matter must have the same mass as the original pellet. If it had a greater mass, our machine would be a perpetuum mobile, since in that case, after the light has arrived at the top of the tower we could use some of the energy to reconstitute the original matter pellet, and extract the remaining energy from the system. That would have restored our machine to its original state, but we would still have been able to extract some energy from it. If the mass were smaller, energy would have vanished, which is just as bad a violation of the laws of physics as the creation of energy from nothing.

But if the mass is the same as originally, something must have happened to the light: Before the fall, the matter pellet had a certain amount of energy - simply because it had a certain rest mass. During the fall, the pellet gained additional energy as it gained more and more speed. The rest mass energy plus the energy gained in the fall are all sent back to the top of the tower in the form of photons. Unless something happens to those photons, the energy arriving at the top of the tower will be larger than the energy corresponding to the original matter pellet's rest mass - it will be that energy plus the energy that the pellet gained during its fall. But we have seen that the energy arriving at the top of the tower must be exactly equal to the original rest mass energy - otherwise, our machine would be a perpetuum mobile.

So what can have happened to the light? It is impossible for photons to simply get lost along the way, so the only possibility is that each photon's energy must have lessened as the photon travelled upwards against the earth's gravitational influence. As we have seen above, a photon's energy is directly proportional to the light frequency. As the light travels upwards, its frequency must decrease. In the language of physics, there is a gravitational redshift (compared with other visible light, red is the light with the lowest frequency).

Corollary: The necessity of the blue-shift

In a similar way, it is possible to show that light falling down must gain energy - it must experience a blueshift. In order to see this, we build an even simpler machine: This machine sends light from the top to the bottom of a tower. At the bottom, there is a mirror - all light that arrives is immediately sent back.

Imagine now that the light does not change when it travels from the tower's top to the base. Then, as the light travels down and up, energy would be destroyed: The light would travel downward, its energy remaining constant. But afterwards, when travelling upwards, it would lose energy (this is the redshift we have derived above).

But a machine that destroys energy is just as impossible as a perpetuum mobile, a machine that creates new energy. Energy can neither be created nor destroyed. Evidently, the light must gain energy during its fall - its frequency must be shifted towards higher frequencies. There must be a gravitational blueshift.

Conclusion

Thus, we have derived the gravitational frequency shift, using Einstein's famous formula, the fact that a falling body will gain energy, and the fact that energy can neither be created nor destroyed: Light falling towards a massive body will get blue-shifted, light climbing out of its gravitational embrace will get red-shifted.

The derivation is not rigorous, but it places the gravitational redshift in a rich context, showing how conservation laws, the action of gravity and the influence of gravity on the frequency are all connected. A more rigorous derivation of the gravitational redshift can be made using the equivalence principle, and of course, the gravitational redshift also follows directly from the full theory of general relativity.

[Markus Pössel, AEI]

This spotlight topic complements the information given in the section General relativity of Elementary Einstein.

Related spotlights on relativity can be found in the category General relativity.