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# Equivalent Principles

10 March 2012

I’ve been preparing for an intro physics class Monday, and that means covering Newton’s laws of motion. Since it is an introductory class I don’t discuss the nature of mass too deeply. Essentially I tell my students that there is inertial mass, given by the second law of motion, and a gravitational mass given by Newton’s law of gravity. I then go on to say that since everything near the Earth falls at the same rate, the gravitational mass in the law of gravity must be proportional to the inertial mass in Newton’s second law. That is, the two masses are equivalent, which is the heart of the equivalence principle.

But things are never quite as simple as they seem, and the concept of mass is no exception. In Newtonian physics there are not two types of mass, but three. There is the intertial mass, which determines the acceleration due to an applied force; there is the passive gravitational mass, which interacts with the local external gravitational field; then there is the active gravitational mass, which creates the external gravitational field in which other particles interact. Newton assumed that all three types of mass were one and the same, and it is generally assumed that Newton’s was correct, but nothing in general relativity requires it, and there is (as yet) no experimental evidence to validate it.

When Einstein first proposed the principle of equivalence as a foundation to general relativity, his basic argument was that, without some external point of reference, a free-floating observer far from gravitational sources and a free-falling observer in the gravitational field of a massive body each have the same experience. Likewise an observer standing on the surface of a massive body and an observer which uniformly accelerates at a rate equal to the body’s surface gravity have identical experiences. Thus, the free-float and free-fall frames can be considered equivalent. In the same manner, the uniform acceleration frame and the surface frame are equivalent. This is known as the weak equivalence principle:

All effects of a uniform gravitational field are identical to the effects of a uniform acceleration of the coordinate system.

In order to formulate general relativity in terms of general covariance, Einstein later strengthened this argument to yield what is known as the strong equivalence principle:

The ratio between the inertial mass of a particle and its gravitational mass is a universal constant.

It is this latter principle which was experimentally validated by the classic Eotvos experiment, which determined that objects fall at the same rate regardless of their material consistency.

The strong equivalence principle does not require that all masses are equal. It only requires that an object’s inertial and passive masses are proportional. Although the equivalence principle says nothing about active mass, conservation of momentum does. If you apply conservation of momentum to two gravitationally interacting objects, you find that momentum is only conserved is if the active mass of an object is proportional to its inertial and passive masses. Thus in order to relate all three masses, we need not only the equivalence principle, but also the conservation of energy-momentum.

The constants of proportionality can be wrapped into the gravitational constant, so it would seem we can simply follow Newton, set all three types of mass equal to each other and be done with it. There is, however, a catch. Although we can arbitrarily set the magnitudes of active and passive mass equal to each other, it is possible for them to be opposite in sign. In other words, if there was some weird type of matter that gravitationally repelled other masses, the equivalence principle and conservation of momentum would still hold true. The equivalence principle has been tested between regular matter, which requires all three masses to be the same. Since ordinary matter is mutually attractive we can say that Newton’s assertion is correct for matter.

But what about anti-matter? No one has been able to test this assumption, so we can’t say for certain. It is possible that active mass is negative for antimatter, which would mean it falls upward in a gravitational field. If that is the case, then although general relativity would still apply to regular matter, it wouldn’t apply to matter + antimatter. Since general relativity is a powerful and experimentally validated theory, it is generally assumed that Newton’s assertion would hold for anti-matter as well. But the only way to know for sure is to test it.

Recently we’ve been able to create usable quantities of anti-hydrogen, which will finally give us the chance to put antimatter to the test. It’s generally thought that antimatter will fall downward just like regular matter, but if it doesn’t, it will be time for some new ideas for gravity.