Special relativity and gravitation

[tom.stoer]

It is well-known that Newtonian's law of gravitation

$$\Phi(\vec{r}) = -\frac{Gm}{|\vec{r} - \vec{a}|}$$

for the gravitational potential $\Phi$ generated by a mass $m$ located at $\vec{a}$ is in several ways inconsistent with special relativity.

What one could do is to modify the definition of $\Phi$ in several ways

$$m \to E/c^2$$
$$d(\vec{r},\vec{a}) = |\vec{r} - \vec{a}| \to d_\text{SRT}(\vec{r},\vec{a})$$
...

In the 2nd modification one would have to introduce a definition of "distance" respecting Lorentz covariance. So for a gravitating mass at $\vec{a}(t)$ and a test mass at $\vec{r}(t)$ the relative velocity of the two masses should be calculated w.r.t. the relativistic velocity addition formula.

Another reasonable approach is to introduce a "mass-" or "energy-density" $\rho$ and a retarded potential

$$\Phi(\vec{r},t) \sim \int d^3r^\prime \frac{\rho(\vec{r}^\prime,t_r)}{|\vec{r}-\vec{r}^\prime|}$$
$$t_r = t - \frac{|\vec{r}-\vec{r}^\prime|}{c}$$

I am sure that Einstein worked on these ideas after having published his famous paper in 1905.

My question is whether there are articles where these ideas are discussed and where the various reasons why the fail are shown.

Does the construction not make sense theoretically? (for electrodynamics it does) Or does it fail phenomenologically? (e.g. for deflection of light, perihelion precession)

 

[Bill_K]

The Newtonian gravitational potential $\Phi$ obeys Laplace's Equation. To make it relativistic, you can say it is the non-relativistic limit of a quantity $\phi$ obeying the wave equation. You have three possibilities:

a) $\phi$ is a scalar which couples to mass
b) $\phi$ is a vector which couples to energy-momentum
c) $\phi$ is a tensor which couples to stress-energy

These theories differ in the ultrarelativistic limit, and especially how they couple to light. Only (c) gives the correct light deflection.

 

[tom.stoer]

Has Einstein investigated $\Phi$? Has he investigated a) and b) before studying c)?

 

[WannabeNewton]

He did study (a). There are many GR textbooks which discuss scalar and vector theories of gravity on flat background and why they are inadequate. See e.g. chapter 3 of Padmanabhan or chapter 2 of Straumann.

 

[stevendaryl]

Has Einstein investigated $\Phi$? Has he investigated a) and b) before studying c)?

According to this Wikipedia page:
http://en.wikipedia.org/wiki/Nordström's_theory_of_gravitation
Nordstrom's scalar theory of gravity was a big influence on Einstein's eventual theory of GR.

He actually had two scalar theories, a linear one and a nonlinear one. It's the latter that is most like GR, which is also nonlinear.

Nordstrom's scalar theories were described in terms of differential equations, with the D'alembertian operator $\Box$ replacing the operator $\nabla^2$ appearing in Poisson's equation. It would be interesting to know what differential equation your $\Phi$ satisfies (instead of the integral equation that you wrote down). Your equation is linear (that is, $\Phi$ doesn't act as a source, only matter fields do), so it might be related to Nordstrom's original linear theory.

 

[BruceW]

Nordstrom's scalar theories were not 'correct' though, right? (Although I'm sure he did contribute a lot to some of the original ideas of general relativity).

His first theory had $\Box \Phi = 4\pi \rho$. Is this in fact correct in the weak-field limit? If this is true, then I guess tom.stoer's idea about a retarded potential would be correct in the weak-field limit. hmmm... I'm not sure what rho would need to be, in this case... (p.s. I am new to general relativity, in case you hadn't noticed).

edit: did I just repeat stevendaryl's point? sorry.

 

[WannabeNewton]

His first theory had $\Box \Phi = 4\pi \rho$. Is this in fact correct in the weak-field limit? If this is true, then I guess tom.stoer's idea about a retarded potential would be correct in the weak-field limit. hmmm... I'm not sure what rho would need to be, in this case...

It depends on the choice of gauge. Certain gauge choices will make it look like the scalar potential $h_{00}$ is a radiative mode but this is unphysical. A gauge free treatment shows that in the weak field limit the scalar potential obeys a Poisson equation instead of a wave equation. This is in agreement with Newtonian theory. A set of retarded solutions would be needed for truly dynamical modes that aren't an artifact of gauge. In linearized GR these correspond to $h_×$ and $h_+$ in the transverse traceless gauge. But for general gauge choices all components $h_{\mu\nu}$ will be given in terms of retarded solutions because of the gauge artifact.

 

[BruceW]

very interesting, thanks Newton! Also, I just found this, which is very helpful and tom.stoer might find it very helpful too http://web.mit.edu/edbert/GR/gr6.pdf It's from MIT physics department.

 

[haushofer]

I believe 'gravity and strings' by Ortin has some treatment on this :)

 

[tom.stoer]

Thanks to all for the comments and references!

 

[tom.stoer]

Reading about Nordström and other alternative theories I come to the conclusion that there are several models which are consistent mathematically but fail phenomenologically due to wrong predictions (no or wrong light deflection, wrong perihelion precession, ...).

 

[PeterDonis]

There are many GR textbooks which discuss scalar and vector theories of gravity on flat background and why they are inadequate. See e.g. chapter 3 of Padmanabhan or chapter 2 of Straumann.

Or MTW, Chapter 7 and exercises 7.1, 7.2, and 7.3.

 

[stevendaryl]

Reading about Nordström and other alternative theories I come to the conclusion that there are several models which are consistent mathematically but fail phenomenologically due to wrong predictions (no or wrong light deflection, wrong perihelion precession, ...).

In addition to getting the equations governing the scalar potential right, there's a secondary issue of getting the right force law (how matter responds to the scalar potential). The simplest generalization from the Newtonian case is:

$\dot{u}_\alpha = -\frac{\partial}{\partial x^\alpha} \Phi$

where $u$ is the 4-velocity, and $\dot{}$ means differentiation with respect to proper time.

Wikipedia says that this was the Abraham force law, which was known to be wrong. But they don't explain what was wrong with it. Was it inconsistent somehow, or disagreed with experiment, or what?

 

[stevendaryl]

In addition to getting the equations governing the scalar potential right, there's a secondary issue of getting the right force law (how matter responds to the scalar potential).

The beauty of using Lagrangians is that the same interaction term automatically tells you how the fields affect matter and how matter affects the fields.

 

[Matterwave]

Reading about Nordström and other alternative theories I come to the conclusion that there are several models which are consistent mathematically but fail phenomenologically due to wrong predictions (no or wrong light deflection, wrong perihelion precession, ...).

One prevailing difficulty with gravity that the naive scalar, vector, and tensor treatments don't deal with is its non-linearity. In order to have the correct theory of gravity, you must have non-linearity because ultimately gravity is non-linear, as evidenced by the fact that Einstein's general relativity is non-linear.

Misner Thorne and Wheeler actually has the reader do the scalar and vector parts of the problem in chapter 7. IIRC the scalar theory produces no deflection of light, while the vector theory predicts too much perihelion shift.

They show that the tensor theory (rank 2 symmetric tensor field on a background Minkowskian metric), formulated to be linear because the gravitational field couples only to the stress energy tensor of "other" things, is internally inconsistent. The equation of motion of the particles do not respect conservation of the stress-energy tensor (which is required by the field equations). One is lead to the particularly undesirable consequence that this gravitational field cannot actually affect material particles at all, due to the requirement of stress-energy conservation imposed by the field equations.

This tensor theory can be fixed by adding in the stress-energy tensor of the gravitational field itself. This leads to an infinite series of terms (because each iteration produces a new term in the stress-energy tensor) which converge in the right limit to give you general relativity. In other words, by adding in all the higher order stress-energy terms, you are left with a theory in which you cannot, even in principle, ever investigate the background Minkowski metric. The background is lost and so it's basically equivalent to general relativity.

 

[bcrowell]

One prevailing difficulty with gravity that the naive scalar, vector, and tensor treatments don't deal with is its non-linearity. In order to have the correct theory of gravity, you must have non-linearity because ultimately gravity is non-linear, as evidenced by the fact that Einstein's general relativity is non-linear.

If the point is to find a natural path leading from SR to GR, then you can't really invoke GR for the insight that gravity is nonlinear. But there's a pretty straightforward argument to that effect that only requires knowledge of SR. SR says that mass and energy are equivalent. In Newtonian gravity, a gravitational field has energy (which is negative and proportional to the square of the field). Therefore we would naturally expect that a relativistic theory of gravity would be one in which the energy in gravitational fields would produce gravitational fields. Now by the time you get done constructing GR, in its standard formulation as the Einstein field equations, you find that the energy in gravitational fields isn't actually included as a term in the stress-energy tensor. The nonlinearity is included in a different way. But I don't think that means that the naive argument was completely invalid -- it was naive, but led in the right direction.

 

[Matterwave]

If the point is to find a natural path leading from SR to GR, then you can't really invoke GR for the insight that gravity is nonlinear. But there's a pretty straightforward argument to that effect that only requires knowledge of SR. SR says that mass and energy are equivalent. In Newtonian gravity, a gravitational field has energy (which is negative and proportional to the square of the field). Therefore we would naturally expect that a relativistic theory of gravity would be one in which the energy in gravitational fields would produce gravitational fields. Now by the time you get done constructing GR, in its standard formulation as the Einstein field equations, you find that the energy in gravitational fields isn't actually included as a term in the stress-energy tensor. The nonlinearity is included in a different way. But I don't think that means that the naive argument was completely invalid -- it was naive, but led in the right direction.

If we were back in 1905-1915 then it would be true that we couldn't use GR to constrain the types of theories that we want, but given a valid theory of gravity already exists, I don't think it's invalid to use this new theory to put some constraints on whatever reformulations of gravity we want to construct are. At least it doesn't hurt to use GR to give us some insights into what reformulations we can make.

 

[tom.stoer]

In addition to getting the equations governing the scalar potential right, there's a secondary issue of getting the right force law (how matter responds to the scalar potential). The simplest generalization from the Newtonian case is:

$\dot{u}_\alpha = -\frac{\partial}{\partial x^\alpha} \Phi$

where $u$ is the 4-velocity, and $\dot{}$ means differentiation with respect to proper time.

Wikipedia says that this was the Abraham force law, which was known to be wrong. But they don't explain what was wrong with it. Was it inconsistent somehow, or disagreed with experiment, or what?

I don't find this in Wikipedia. Can you please post a link?

According to Bill's post and MTW chapter 7 it fails to reproduce bending of light rays.

 

[stevendaryl]

I don't find this in Wikipedia. Can you please post a link?

At http://en.wikipedia.org/wiki/Nordström's_theory_of_gravitation#Development_of_the_theories

The relevant sentence is
"This force law had earlier been proposed by Abraham, and Nordström knew that it wouldn't work."

According to Bill's post and MTW chapter 7 it fails to reproduce bending of light rays.

That applies to the nonlinear scalar theory. But it seems that the "wrong" force law for the linear version was rejected earlier.

 

[tom.stoer]

That applies to the nonlinear scalar theory. But it seems that the "wrong" force law for the linear version was rejected earlier.

Hm; Bill says that a scalar gravitational potential must couple to a scalar source, i.e. rest mass (not energy); therefore any scalar gravitational potential cannot correctly reproduce light deflection b/c the rest mass of the photon is zero (is this reasoning too simple or stupid to be true?)

 

[stevendaryl]

Hm; Bill says that a scalar gravitational potential must couple to a scalar source, i.e. rest mass (not energy); therefore any scalar gravitational potential cannot correctly reproduce light deflection b/c the rest mass of the photon is zero (is this reasoning too simple or stupid to be true?)

No, you're right. I'm saying that the "wrong" force law for the linear theory was rejected for some more basic reason than that, which Wikipedia doesn't give any details about.