# Einstein’s equivalence principle updated with a touch of quantum | Techilink The core of Einstein’s theory of gravity (general relativity) is the equivalence principle. The equivalence principle says there is no difference between standing still and being subject to the gravity pulling you and accelerating in a vehicle that is free of gravity.

In practice, this means that there is no difference between inertial mass (the mass on which a rocket operates) and gravitational force (the mass on which the Earth pulls). This equivalence has been measured time and time again with no violation ever found. But these tests assumed that quantum mechanics didn’t change the equivalent principle: that assumption is partially incorrect.

## A quantum in your equivalent

In relativity, mass and energy are two sides of the same coin. For very small objects, we have to think about that in terms of quantum mechanics, where a particle can be in a superposition of energy states. A particle in a superposition of energy states has two energies at once until it is measured, after which it has a single fixed energy. An object in a superposition of energetic states can have a superposition of inertial masses. But does it have the same superposition of gravitational masses?

The intention of the equivalence principle says that yes, it should. But the mathematical explanation of the equivalence principle does not take into account the quantum properties of the objects.

Now a few researchers have picked up that thread and started pulling on it. They reformulated the principle of equivalence so that it takes into account how energy can be distributed internally in a quantum object.

Their conclusion is that, although the classical equivalence principle requires that the classical inertial mass and gravitational mass are the same, this is not enough for quantum mechanics. And now we’re getting a little technical. In a quantum system, the researchers also found that the inertial mass and gravitational mass operators must commute. What does that mean?

## Commutation

In physics terms, this means that when two operators commute, we can measure the physical quantity of one and not interfere with the value of the other. To give the most famous example: position and momentum do not commute. When we measure the position of an electron, we lose information about its momentum. If we then measure the momentum of the same electron, we lose information about its position. The same is not true for momentum and energy. If I measure momentum and then measure energy, I don’t lose any information about momentum.

In a sense, the claim that inertial mass and gravitational mass must commute is trivial: if the inertial mass and the gravitational mass are the same, they have the same operators and must commute. If that weren’t true, it would be like saying that measuring an electron’s momentum destroys knowledge about the electron’s momentum. That makes no sense.

Likewise, measuring the mass of a particle does not destroy the knowledge about the mass of the particle. However, if inertial mass and gravity are different, measuring inertial mass makes gravity uncertain.

As a result, classical tests of the equivalence principle can find agreement when the test actually conflicts with the equivalence principle. Additional measurements are needed to confirm that the equivalence holds for quantum objects.

## Sensitive to a difference in mass

Let’s take an example. Physicists sometimes use a Bose Einstein condensate (BEC) to test the equivalence principle. A BEC is a blob of atoms that behaves like a single quantum particle. The blob is split into two equal parts and sent along two different paths to meet again. In one path, the BEC blob is put in a superposition state: the BEC is in two energetic states at the same time. Gravity works on both blobs, but its effect should be different because one blobs has a different internal state.

When the two blobs meet, they interfere, resulting in bits of material that create bright and dark areas on a screen. If everything goes perfectly and the equivalence principle applies, then the dark spots are completely dark and the light spots are all equally bright.

If inertial mass and gravity are different, the interference will not be perfect. The bright spots will not be as bright and the dark spots will have some light.

There are similar differences for different quantum tests of the quantum equivalence principle; the others are very difficult. The researchers examined four different experiments and found that for three of them the current and future experiments would not be sensitive enough. For the only remaining method, an earlier experiment had proved the method’s viability and had shown that the equivalence principle held.

## Not all equivalence principles are equivalent

I should note that I skated over a lot of technical details here. In particular, the equivalence principle can be broken down into a combination of the weak equivalence principle, local Lorentz invariance and local position invariance. Together, these three form Einstein’s equivalence principle. Typically, most experiments test only a subset of these three (and usually only the weak equivalence principle).

Here, however, the researchers are dealing with all three. If experiments can be improved to perform the measurements suggested in this article, then this will be the strongest test of Einstein’s equivalence principle yet.

Nature physics2018, DOI: 10.1038/s41567-018-0197-6. (about DOIs)