Is a scalar field incompatible with the Principle of Equivalence?

[Bibipandi]

As I know, Einstein initially tried describe the gravitational interaction as mediated by a scalar field, but he later gave up this idea because it is incompatible with the Principle of Equivalence.I don't know how this idea is incompatible with the Principle of Equivalence. Please help me. Thanks

 

[atyy]

Scalar field theories are consistent with special relativity and the principle of equivalence. They are ruled out by experiment (wrong prediction for perihelion of mercury).

http://arxiv.org/abs/gr-qc/0405030
http://arxiv.org/abs/gr-qc/0611100

 

[haushofer]

Did you try to write down the gravitational potential as a scalar field in a special-relativistic setting?

 

[hamster143]

As I know, Einstein initially tried describe the gravitational interaction as mediated by a scalar field, but he later gave up this idea because it is incompatible with the Principle of Equivalence.I don't know how this idea is incompatible with the Principle of Equivalence. Please help me. Thanks

I really don't think he did ... He didn't approach the problem like that at all. The whole expression, "to describe the gravitational interaction as mediated by a scalar field", is based on a point of view that is deeply rooted in quantum field theory and dates to 1960's at the earliest.

 

[JustinLevy]

There was classical field theory way before there was quantum field theory. They don't mean "scalar" as in spin-0.

Newton's theory had a scalar gravitational potential. Many tried to make this relativistic, and quickly ran into problems with theory not matching experiment or behaving "reasonably" (orbits weren't stable, and other problems). Nordstom came very "close". Einstein most likely played with these himself to understand what was going on. Ultimately it was a tensor theory that worked.

Sure, they didn't look at it as a "spin-2 field" while working towards the finished goal, but it clearly was a tensor theory and not a scalar theory that was being fleshed out in the end.

 

[Bibipandi]

Thanks for your answers. However, would you please explain more detail about how the scalar field is not consistent with the Principle of Equivalence?

 

[haushofer]

I think this is the article you're looking for:

http://arxiv.org/PS_cache/gr-qc/pdf/0506/0506063v1.pdf

"The roots of scalar-tensor theory: an approximate history" by Brans. At page four he explicitly writes down a gravitational theory with the gravitational potential as a Lorentz scalar (so it's in the (0,0)-representation of the Poincare algebra) and shows that a local acceleration depends on a modified inertial mass. This contradicts the fact that the inertial mass and gravitational mass are the same (I believe this is what people call the Weak Equivalence Principle, but I'm bad with names).

 

[haushofer]

There was classical field theory way before there was quantum field theory. They don't mean "scalar" as in spin-0.

Newton's theory had a scalar gravitational potential. Many tried to make this relativistic, and quickly ran into problems with theory not matching experiment or behaving "reasonably" (orbits weren't stable, and other problems). Nordstom came very "close". Einstein most likely played with these himself to understand what was going on. Ultimately it was a tensor theory that worked.

Sure, they didn't look at it as a "spin-2 field" while working towards the finished goal, but it clearly was a tensor theory and not a scalar theory that was being fleshed out in the end.

Yes, but as this article by Brans shows: the natural approach for people who know SR is ofcourse to describe gravity by a Lorentz scalar, and it is this approach which violates the equivalence principle.

I can't remember if this is a route which Einstein took before constructing his GR, but it's quite plausible I would say.

 

[Bibipandi]

Thanks for your suggestion. As I understand, the principle of equivalence states that: in small enough regions of spacetime, the laws of physics reduce to those of special relativity; it is impossible to detect the existence of a gravitational field by means of local experiments. The scalar field describes only the gravitational potential (as stated in Newton's theory of gravitation), not the electromagnetic field (described by tensor field), so the scalar field doesn't satisfy the equivalence principle which requires all kinds of experiment (involving electromagnetic experiments). Am I right?

 

[JustinLevy]

Yes, but as this article by Brans shows: the natural approach for people who know SR is ofcourse to describe gravity by a Lorentz scalar, and it is this approach which violates the equivalence principle.

Yes, there were problems regarding the most straight forward descriptions with a scalar (ie, just swapping the derivatives in Newton's theory with the d'Alembertian operator).
But no, that doesn't mean scalar theories generically violate the equivalence principle.

As I already mentioned, there was a theory proposed by Nordstrom before GR that was a scalar theory and did not violate the equivalence principle. Einstein considered it the only real contender to the theory he was working on at the time.

You can read more about it in the introduction in one of the papers atyy linked
http://arxiv.org/abs/gr-qc/0405030
or wiki has an article about Nordstrom's theory as well
http://en.wikipedia.org/wiki/Nordström's_theory_of_gravitation

Thanks for your suggestion. As I understand, the principle of equivalence states that: in small enough regions of spacetime, the laws of physics reduce to those of special relativity; it is impossible to detect the existence of a gravitational field by means of local experiments. The scalar field describes only the gravitational potential (as stated in Newton's theory of gravitation), not the electromagnetic field (described by tensor field), so the scalar field doesn't satisfy the equivalence principle which requires all kinds of experiment (involving electromagnetic experiments). Am I right?

Not quite.
One of the most famous "gravity + EM" tests is the Pound-Rebka "gravitational redshift of light" test. Nordstrom's theory correctly predicted that. Furthermore, his theory did not violate the equivalence principle as you state it there.

However you are correct in that his theory predicted that light does not "bend", while Einstein predicted that it does. I cannot speak to whether or not all scalar theories will have this result, but probably so considering the EM contribution to the stress-energy tensor is traceless.

The point however, is that the equivalence principle was very useful in guiding the development of gravitational theories (and the first attempts at a scalar relativistic gravitational theory did run into problems with the equivalence principle), but you cannot rule out scalar theories based just on the equivalence principle.

 

[haushofer]

Nice article, JustinLevy! Thanks!

 

[atyy]

A few other things to conosider - local versus global light bending, and whether the version of the EP considered excludes or includes gravitationally bound objects.

http://www.aei.mpg.de/einsteinOnline/en/spotlights/equivalence_deflection/index.html

http://relativity.livingreviews.org/Articles/lrr-2006-3/ , section 3.6