# Where is Strong Equivalence Principle stronger than Weak Equivalence Principle?

Where is Strong Equivalence Principle stronger than Weak Equivalence Principle? In my note, the two equivalence principles are stated as follows

Weak Equivalence Principle:

Gravitational and inertial masses are equal.

Strong Equivalence Principle:

There is no observable distinction between the local effects of gravity and acceleration.

I think the weak one will imply that the acceleration at a point and the gravitational field at that point are equal, thus we can remove the gravitational effect by using an accelerating frame. But, what does this differ from the strong version?

Is the change from at a point to local playing the role?

I'm not sure your note is correct in its statement of the two forms of the equivalence principle. They key distinction is this: the weak equivalence principle applies only to non-gravitational experiments, that is, experiments where gravitational interactions between the objects are not important. The strong equivalence principle extends this to include experiments where the objects may have strong self-gravity or important gravitational interactions.

For a concrete example, the Eöt-Wash group has carried out experiments to compare gravitational to inertial mass of bodies of different compositions. This tests the weak equivalence principle: does (say) nuclear binding energy fall the same way as proton rest mass? This was tested by comparing copper and uranium objects. An analogous test of the strong equivalence principle asks, does gravitational binding energy fall the same way as (say) proton rest mass? This is hard to test in the laboratory, because the gravitational binding energy of a laboratory-scale mass is minuscule. But lunar laser ranging allows us to compare how the Earth and the Moon fall - the Earth has considerably more gravitational binding energy per unit mass than the Moon. It turns out they fall the same way, to experimental accuracy, so the SEP looks solid.

How could the SEP be violated? Well, you could have some additional scalar field that coupled to gravity. This would be stronger in more strongly self-gravitating bodies and would affect how they fell.

The Eöt-Wash group's explanation: http://www.npl.washington.edu/eotwash/EquivalencePrinciple

A discussion of current tests of relativity, including both versions of the equivalence principle: http://relativity.livingreviews.org/Articles/lrr-2006-3/fulltext.html

I think one of the motivations for the strong equivalence principle is to see why gravity bends light. So I think the way to show why the weak equivalence principle does not imply the strong equivalence principle would be to construct a theory where the weak equivalence principle holds, but light is not affected by gravity. I think you could try special relativity with gravity, but it just does gravity similar to how it does electromagnetism.

My guess would be to define a potential by $\partial^2 A = -j$, and put in a force law that says a mass will feel a force given by $F^{\mu\nu} j_\nu$. (Note by $j$, $F$, and $A$, I mean not electrodynamics quantities, but their gravitational analogues.) You would add these forces into your usual forces of special relativity, and treat the light as massless so it doesn't feel this force. Then gravitional mass will equal inertial mass, but the light won't be affected by gravity. I am not sure the theory I described makes sense, but I think you get the idea.

# Edit

I looked it up and there isn't a good way to make the mass current $j$ a four vector, since mass transforms non-trivially under boosts unlike charge. But hopefully I have given some intuition as to how a theory could violate the strong equivalence principle but not violate the weak equivalence principle.

• you do not make a "mass current", you make a "mass-density current" in exactly the same way that you make a "charge-density current". mass-density and charge-density transform in the same way. Jun 12, 2017 at 9:14