Why Not Develop Relativistic Gravitational Theory Analogy to Electromagnetism?

[MichPod]

How can it be easily shown that it's a bad idea to develop relativistic gravitational theory in full analogy with electromagnetism? I.e. why not to introduce 4-potential for gravitation in analogy with 4-potential in covariant form of Maxwell equations and get something fully analogous to Maxwell equations for gravitation in the end? On the surface gravitational force according to Newton looks exactly as electrostatic force (except that we have only single sign if charge which attract), so are there theoretical considerations why making relativistic gravitational theory this way is bad?

 

[BvU]

Well, for one thing there is no repulsive variety of the force. And second there is no equivalent for magnetic force once charge moves ...

 

[stevendaryl]

How can it be easily shown that it's a bad idea to develop relativistic gravitational theory in full analogy with electromagnetism? I.e. why not to introduce 4-potential for gravitation in analogy with 4-potential in covariant form of Maxwell equations and get something fully analogous to Maxwell equations for gravitation in the end? On the surface gravitational force according to Newton looks exactly as electrostatic force (except that we have only single sign if charge which attract), so are there theoretical consideration why making relativistic gravitational theory this way is bad?

As @BvU says, the big difference is that for electromagnetism, like charges repel. So there would need to be a sign change in the equations. You might think that that wouldn't be a big deal, but ...

The celebrated defect of a theory of gravitation modeled after Maxwell electromagnetism was first pointed out by Maxwell himself (Maxwell 1864, 571). In such a theory, due to the change of signs, the energy density of the gravitational field is negative and becomes more negative as the field becomes stronger. In order not to introduce net negative energies into the theory, one must then suppose that space, in the absence of gravitational forces, must contain a positive energy density sufficiently great to offset the negative energy of any possible field strength. Maxwell professed himself baffled by the question of how a medium could possesses such properties and renounced further work on the problem. As it turns out it was Einstein’s foe, Abraham, shortly after his exchange with Einstein, who refined Maxwell’s concern into a more telling objection. In a lecture of October 19, 1912, he reviewed his own gravitation theory based on Einstein’s idea of using the speed of light as a gravitational potential. (Abraham 1912e) He first reflected (pp. 193–94), however, on a gravitation theory modeled after Maxwell electromagnetism. In such a theory, a mass, set into oscillation, would emit waves analogous to light waves. However, because of the change of sign, the energy flow would not be away from the mass but towards it, so that the energy of oscillation would increase. In other words such an oscillating mass would have no stable equilibrium. Similar difficulties were reported by him for gravitation theories of Maxwellian form due to H.A. Lorentz and R. Gans.

https://www.pitt.edu/~jdnorton/papers/Nordstroem.pdf
page 15

 

[MichPod]

In such a theory, due to the change of signs, the energy density of the gravitational field is negative and becomes more negative as the field becomes stronger.

Hmm... this taken alone looks pretty normal, after all, the attraction is assotiated with negative energy. Why would that scare Maxwell?
And is it not right in the end that the graviational field has negative energy?

 

[stevendaryl]

Hmm... this taken alone looks pretty normal, after all, the attraction is assotiated with negative energy. Why would that scare Maxwell?

As the quote says, if the energy associated with the field is negative, it means that the energy goes down as the gravitational field becomes stronger. That would lead to a runaway situation as the gravitational field goes to infinity.

In electromagnetism, the field associated with the electromagnetic field is positive, not negative. If you have two parallel charged plates, one positive and the other negative, then the electric field between the plates is $E \propto Q$. The energy density is $\rho \propto E^2 \propto Q^2$. The total energy is proportional to the volume between the plates: $U \propto \rho V \propto Q^2 A d$, where $d$ is the distance between the plates. So even though the electromagnetic field energy is positive, it can be lowered by decreasing $d$.

 

[stevendaryl]

As the quote says, if the energy associated with the field is negative, it means that the energy goes down as the gravitational field becomes stronger. That would lead to a runaway situation as the gravitational field goes to infinity.

In electromagnetism, the field associated with the electromagnetic field is positive, not negative. If you have two parallel charged plates, one positive and the other negative, then the electric field between the plates is $E \propto Q$. The energy density is $\rho \propto E^2 \propto Q^2$. The total energy is proportional to the volume between the plates: $U \propto \rho V \propto Q^2 A d$, where $d$ is the distance between the plates. So even though the electromagnetic field energy is positive, it can be lowered by decreasing $d$.

I'm not sure, off the top of my head, what is the relationship in electromagnetism, between the potential energy for a configuration of charges and the energy stored in the electromagnetic field for that configuration of charges. Maybe for two particles a distance $d$ apart, the potential energy $U(d)$ is the difference between the energy stored in the field in the limit as $d \rightarrow \infty$ and the energy in the field at finite $d$. For point-particles, that would be a subtraction of two infinities, so I'm not sure how it works.

 

[Jonathan Scott]

And second there is no equivalent for magnetic force once charge moves ...

Gravity does have an equivalent of magnetic forces. Look up "gravitomagnetism". A gravitomagnetic field causes effects as if space is locally rotating, in contrast to the usual gravitational field, which causes effects as of local linear acceleration.

In some cases, gravity can be described as being similar to electromagnetism. That model is usually called "Gravitoelectromagnetism" or GEM. However, that model does not accurately reproduce the results of General Relativity, which involves curved space and tensor terms, which vary with reference frame in a different way from the Lorentz transformation of the vector terms in Special Relativity.

 

[Daverz]

You can heuristically develop the linearized equations for gravity in analogy with electromagnetism. Ohanion does this in his book Gravitation and Spacetime, as does Zee in his Nutshell book. As Jonathan Scott mentions, you can decompose this into fields analogous to E & B fields. Maybe someone knows of an online resource that works this out.

Sorry, for some reason I thought Zee covered this.

 

[robphy]

Two comments:

• From a past comment of mine from 2016 [making reference to an older comment of mine from 2004]...
  
  (I just bolded the relevant keywords)
  https://www.physicsforums.com/threa...gnetism-a-useful-analogy.869784/#post-5461001
  

  While GEM is an approximation to GR,
  there is a Maxwell-like formulation of GR called the "Quasi-Maxwellian equations".
  
  Here's a really old post of mine (from 2004) quoting from Hawking&Ellis
  https://www.physicsforums.com/threads/gravitomagnetism-and-gr.54932/#post-388636
  
  [Still] On my to-do list... How are these (GEM and Quasi-Maxwell) related?
  What are the Quasi-Maxwell equations telling us?
  Could they be useful for solving the initial-value problem analytically and numerically?
  Could they be "quantized"?
  
  Possibly useful... but I haven't looked at them in detail...
  http://arxiv.org/abs/1302.7248 "The Quasi-Maxwellian Equations of General Relativity: Applications to the Perturbation Theory" (Novello, et al)
  https://arxiv.org/abs/1207.0465 "Gravito-electromagnetic analogies" (Costa & Natario)

• One other feature that hasn't been mentioned so far is that
  • gravitons are massless spin 2 ... related to $C_{[ab][cd]}$... and some vague memory of a claim that "even spin" is attractive(*)
  • photons are massless spin 1 ... related to $F_{[ab]}$
    
    (*) A google search found this. [I haven't read it.]
    https://journals.aps.org/prd/abstract/10.1103/PhysRevD.33.2475
    "Attraction/repulsion between like charges and the spin of the classical mediating field" by Jagannathan & Singh