Electromagnetic General Relativity

[The_Duck]

I wonder if maybe this is a question that a lot of people have asked; if so, I apologize; a quick search didn't seem to turn up any relevant results.

Why isn't there an equivalent of general relativity for electromagnetism? Coulomb's law, at least, has the same form as Newton's law for gravity. And, in my very limited understanding of the matter, I don't see what there is in the motivation for general relativity that doesn't also apply to electric forces. For example, a person weightless in an elevator can't tell whether the Earth has vanished or he is in free fall. Similarly, an electron shouldn't be able to tell whether it's free-falling in an electric field or floating in a field-free region. What gives? Is magnetism the difference?

 

[dx]

Similarly, an electron shouldn't be able to tell whether it's free-falling in an electric field or floating in a field-free region.

It certainly can. If you have two charged particles of unequal charge in a closed box, then an applied electric field will accelerate them both differently. The key thing about gravity is that all bodies are given the same acceleration.

 

[tiny-tim]

Why isn't there an equivalent of general relativity for electromagnetism?

Hi The_Duck!

From the PF Library on geodesic deviation …

Electromagnetic comparison:

By comparison, the world-line deviation equation between world-lines followed by two charged particles with the same charge/mass ratio freely moving (in flat Minkowski spacetime) in an electromagnetic field is:

$$\frac{D^2\,\delta x^{\alpha}}{D\tau^2}\ =\ \frac{q}{m}\,F^{\alpha}_{\ \mu\,;\,\beta}\,V^{\mu}\,\delta x^{\beta}$$

where $q$ is charge, $m$ is mass, and $F$ is the electromagnetic tensor

 

[ZikZak]

As dx has said, electric fields accelerate different particles differenty. Gravitational fields never do. There can never be an Equivalence Principle for EM. An experimenter in a closed box can easily detect an externally applied EM field.

The electromagnetic potential energy is not electrically charged, yielding a linear field equation. The gravitational field DOES generate its own gravity, making the Einstein equations nonlinear.

Magnetism is NOT a difference between EM and gravity. Magnetism is electricity + relativity. Similar effects occur gravitationally and are sometimes referred to as "gravitomagnetism." In that language, gravitomagnetism is responsible, for instance, for the Earth apparently responding gravitationally to the instantaneous position of the Sun rather than its retarded position, even though gravitational effects must travel at c.

 

[The_Duck]

Thanks for the very informative replies. The wikipedia page on gravitomagnetism is really interesting. Presumably you can derive equations for gravitational waves from the gravitomagnetic equations just like you can get electromagnetic waves out of Maxwell's equations?

I'm still not completely satisfied, though. Suppose the matter in the universe universe consisted entirely of electrons. Then all bodies would accelerate equally in an electric field, right? Could you conduct an experiment in such a universe to distinguish acceleration from electric fields?

 

[atyy]

Thanks for the very informative replies. The wikipedia page on gravitomagnetism is really interesting. Presumably you can derive equations for gravitational waves from the gravitomagnetic equations just like you can get electromagnetic waves out of Maxwell's equations?

Yes.

I'm still not completely satisfied, though. Suppose the matter in the universe universe consisted entirely of electrons. Then all bodies would accelerate equally in an electric field, right? Could you conduct an experiment in such a universe to distinguish acceleration from electric fields?

mg=kma (gravitational mass proportional to inertial mass by universal constant k), a=g/k

qE=ma, a=qE/m, since q/m is a universal constant in your universe, it looks like the answer is no (neglecting gravity between electrons).

But it has nothing to do with magnetism. It has to do with whether q/m is a universal constant.

 

[Naty1]

Coulomb's law, at least, has the same form as Newton's law for gravity.

True, but each is a simplification...valid only at non relativistic speeds...

 

[The_Duck]

mg=kma (gravitational mass proportional to inertial mass by universal constant k), a=g/k

qE=ma, a=qE/m, since q/m is a universal constant in your universe, it looks like the answer is no (neglecting gravity between electrons).

But it has nothing to do with magnetism. It has to do with whether q/m is a universal constant.

OK. So shouldn't the same arguments that apply to gravity apply to electric forces in this electron-only universe? In fact, you shouldn't really be able to distinguish a gravitational and an electric force. Suppose the mass to charge ratio is

m/e = a.

Then the force between two electrons will be

ke²/r² + Gm²/r² = k(m/a)²/r² + Gm²/r² = (G + k/a²) m²/r²

So there's just one force, really. There's no way to distinguish two distinct properties "mass" and "charge" of the electrons in this universe. Can the general relativity arguments be applied to this merged in this universe? If so, surely the physics doesn't change just because we've moved to universe with less variety of particles. So the GR arguments should apply to electric forces in this universe, too.

 

[tiny-tim]

OK. So shouldn't the same arguments that apply to gravity apply to electric forces in this electron-only universe? In fact, you shouldn't really be able to distinguish a gravitational and an electric force.

There is a difference between the geodesic deviation equations:

$$\frac{D^2\,\delta x^{\alpha}}{D\tau^2}\ =\ -\,R^{\alpha}_{\ \mu\beta\sigma}\,V^{\mu}\,V^{\sigma}\,\delta x^{\beta}$$

and

$$\frac{D^2\,\delta x^{\alpha}}{D\tau^2}\ =\ \frac{q}{m}\,F^{\alpha}_{\ \mu\,;\,\beta}\,V^{\mu}\,\delta x^{\beta}$$

R is more complicated than F_;, and gravity uses V (velocity) twice, while electromangetism uses it only once.

The 1/r² similarity is only an approximation.

 

[The_Duck]

Perhaps I should stop here and go take a general relativity course, but what I'm wondering is where in the derivation of those equations do gravity and electricity diverge, and why?

 

[ZikZak]

Perhaps I should stop here and go take a general relativity course, but what I'm wondering is where in the derivation of those equations do gravity and electricity diverge, and why?

You can attempt to construct a vector theory of gravity, but such a theory will predict that gravitational waves carry negative energy. Since that is ridiculous, we tend to prefer GR.

 

[atyy]

There is a difference between the geodesic deviation equations:

$$\frac{D^2\,\delta x^{\alpha}}{D\tau^2}\ =\ -\,R^{\alpha}_{\ \mu\beta\sigma}\,V^{\mu}\,V^{\sigma}\,\delta x^{\beta}$$

and

$$\frac{D^2\,\delta x^{\alpha}}{D\tau^2}\ =\ \frac{q}{m}\,F^{\alpha}_{\ \mu\,;\,\beta}\,V^{\mu}\,\delta x^{\beta}$$

R is more complicated than F_;, and gravity uses V (velocity) twice, while electromangetism uses it only once.

The 1/r² similarity is only an approximation.

But if you can detect geodesic deviation, then the equivalence principle doesn't apply even to gravity, since that's a nonlocal experiment. So if we restrict ourselves to local experiments, and The_Duck's universe, wouldn't the equivalence principle apply even for electromagnetism?

 

[atyy]

So the GR arguments should apply to electric forces in this universe, too.

I'm tempted to agree, but what about this issue? If a charge is accelerated with respect to an inertial frame by an electric field, it will radiate. If there is no electric field, and we simply choose an accelerated frame of reference, I'd imagine there's no radiation, since the charge is stationary with respect to an inertial frame.

 

[atyy]

So the GR arguments should apply to electric forces in this universe, too.

According to this article, the answer is yes. "In electrodynamics, the Larmor theorem has played an important role in the description of the motion of charged spinning particles. Explicitly, for slowly varying fields and to linear order in v/c and field strength, the electromagnetic field can be replaced by an accelerated system with translational acceleration aL = −qE/m and rotational frequency ωL = qB/(2mc). For all charged particles with the same q/m, the electromagnetic forces are locally the same as inertial forces; this is reminiscent of the principle of equivalence." http://arxiv.org/abs/gr-qc/0011014

I'm confused why radiation is not an issue, unless that is taken care of by specifying slowly varying fields and linear order in v/c

 

[The_Duck]

Does an charged particle falling in a gravitational field radiate? If so, then by equivalence, I would expect that a stationary charged particle would appear to radiate when viewed from an accelerating frame.

 

[George Jones]

I haven't really been following this thread, but maybe comments 11, 12, 18 and 20 in

http://blogs.discovermagazine.com/cosmicvariance/2009/06/02/susskind-lectures-on-general-relativity/

are worth looking at. Sean is Sean Carroll.

 

[belliott4488]

Does an charged particle falling in a gravitational field radiate? If so, then by equivalence, I would expect that a stationary charged particle would appear to radiate when viewed from an accelerating frame.

Yikes! This has always struck me as weird, but I believe the answer is 'yes'. I remember being astounded in grad school by the statement that photon number is not frame-invariant, i.e. two observers will detect different numbers of photons being emitted by a charged particle, depending on their motion relative to it, specifically on their relative acceleration.

Don't ask me to explain how this can be, however ...

 

[atyy]

Does an charged particle falling in a gravitational field radiate? If so, then by equivalence, I would expect that a stationary charged particle would appear to radiate when viewed from an accelerating frame.

http://relativity.livingreviews.org/Articles/lrr-2004-6/

In GR, the equivalence principle does not apply to electrically charged particles - they will not fall at the same rate as electrically uncharged particles. However, this is not a violation of the EP because the electrically charged particle is not free falling, being affected by its own electric radiation. A gravitationally charged particle (ie. an electrically uncharged point mass) gives off gravitational radiation, but it still obeys the equivalence principle as it moves on a geodesic of the spacetime modified by the gravitational radiation. In short, gravitationally charged electrically uncharged particles obey the EP, but gravitationally and electrically charged particles do not.

I naively expect electrically uncharged point masses to obey the EP in GEM, since GEM is an approximation of GR. By the analogy between GEM and EM, and with the additional restriction that all electrically charged particles have the same q/m ratio, I also naively expect electrically charged particles to obey some sort of "EM-EP" in EM. But what happens with the radiation? In GR, gravitational radiation isn't a problem for the EP, so maybe it's also not a problem in GEM? If so, then by analogy with GEM, maybe EM radiation wouldn't be a problem for the "EM-EP"?

Actually, I think I'm thinking too much. In gravity itself, the EP has to hold for identical particles - the whole point of the EP is that different masses fall the same, it's not interesting that identical masses fall the same.

 

[thehangedman]

Read up on a theory proposed by Kaluza. The essential problem is that in order for a "force theory" to work as the result of geometry the space needs enough degrees of freedom to account for all the input variables the effect motion resulting from that force. In the case of gravity, thanks to the equivalence principle, the only factors that effect motion in a gravitational field are the motion of the particle in question. Namely, it's position and velocity. The degrees of freedom needed for position and velocity are just the three space dimensions and one time. In the case of EM, there is also the charge of the particle (actually, the charge / mass ratio). So, there isn't enough "room" in just the 4 space-time variables to make a geometric theory of EM work. Kaluza put together a theory where by he added a fictional 5th dimension. He showed that if you set the curvature of the 5th dimensional space to zero (along with some other minor assumptions) that you not only get out current GR style gravity but you also get the equations of EM. The "momentum" of a particle in the 5th dimension is just it's charge / mass ratio.

So, in summary, GR on just space-time doesn't allow for a working theory that would include EM. There aren't enough degrees of freedom to make that possible. You need to somehow include additional degrees of freedom into the theory, and one such mechanism proposed early on was to add an additional 5th dimension. BTW, this line of thought eventually leads to String Theory...