Are there any alternative interpretations to GR?

[Ilja]

Note, this rules out a closed universe, thus you can't handle a universe with high energy density.

If one removes one point from the closed universe, it becomes $\mathbb{R}^3\times \mathbb{R}$, so this is not really a problem. Harmonic coordinates would exist there (I have seen them but forgotten where).

The more serious point is that such solutions would not define anymore a homogeneous universe. Thus, the interpretation favors a flat universe as the only homogeneous one.

But there is no inability to handle universes with high energy density. There may be no such homogeneous solutions, but so what, it means they become inhomogeneous.

 

[Dale]

But what would be the point of having forces? The advantage?

None. But that is specifically what the OP is asking for.

I just want to make clear to him/her that these interpretations do not meet the listed desiderata so that they don't mistakenly assume that the rest of us were "holding back". I don't think these interpretations are what the OP is searching for.

 

[haushofer]

Seems clear enough. Any solution you get by applying the spin 2 field method satisfies EFE everywhere, but not all solutions of the EFE against any manifold are realized. You only find solutions with R⁴ topology.

But what I don't understand is that in GR, the EFE don't constrain the topology. So somehow you depart from Fierz-Pauli theory on Minkowski-spacetime, hence with certain restrictions on the topology, and ending up with field equations which don't constrain the topology anymore.

To put it differently: exactly what and where is the difference between the usual EFE and the equations obtained from the Fierz-Pauli iteration?

 

[Ilja]

Fierz-Pauli theory, as any theory on Minkowski spacetime, is not a generally covariant theory. It is Lorentz-covariant, of course, but does not have a hole problem. The field equations of GR may be presented in such a way using a coordinate condition. Once iterations do not get rid of coordinate conditions - if the equations have no hole problem, their iterations will not have one - it seems quite clear that these iterations can give EFE only in some particular coordinate condition. While I have not checked it, the most common coordinate condition is the harmonic one, so I expect them to be used here too. At least they have an important property one would need to connect with a Minkowski background, namely that linear combinations of harmonic coordinates are harmonic too.

In this case, the equations are restricted to the particular part covered by the coordinates preferred by the used coordinate condition. If this is the harmonic condition, the mathematics of the equations would be the same as in the ether interpretation mentioned above.

This does not mean that there are no differences. The ether interpretation enforces that one time coordinate - the one interpreted as the true time by the interpretation - has to be a global time-like coordinate. If a theory starts on a Minkowski background, thus, with an Einstein-causal theory, whatever the iteration, it would have to remain Einstein-causal. At least I'm not aware of an iteration mechanism which would allow to transform lower-than-c trajectories into faster-than-c. Now, each of the original time-like Minkowski coordinates would, in this case, do the job. But this is, nonetheless, a more restrictive condition. One can imagine solutions which allow for one time-like preferred coordinate, but not for a whole Minkowski metric containing the lightcone of the physical metric everywhere.

Logunov's RTG, a theory with massive graviton, which also uses the harmonic coordinates, has a similar problem of correspondence between the background Minkowski metric and the physical metric. It imposes an explicit causality condition, but it has no connection to the equations. So they have solutions which somewhere start to violate this causality condition. This is possible in the ether interpretation too - but the condition that preferred time is time-like circumvents this. It translates into the condition that the ether density $\rho=g^{00}\sqrt{-g}$ has to be positive. And that condensed matter theory equations start to fail if the density becomes negative is quite obvious. It simply means that the boundaries of the condensed matter approximation have been reached. I think, a nice way to get rid of causal loops in the ether interpretation.

 

[PAllen]

But what I don't understand is that in GR, the EFE don't constrain the topology. So somehow you depart from Fierz-Pauli theory on Minkowski-spacetime, hence with certain restrictions on the topology, and ending up with field equations which don't constrain the topology anymore.

To put it differently: exactly what and where is the difference between the usual EFE and the equations obtained from the Fierz-Pauli iteration?

See my next post (#34) for the resolution to this question.

 

[haushofer]

See my next post (#34) for the resolution to this question.

Thanks. But how is this possible if you end up with exactly the same equations as conventional GR? Shouldn't the solutions then also be the same?

I guess I should dive more deeply into how this iteration is precisely performed, but I'm still puzzled by this topology-issue.

 

[martinbn]

Thanks. But how is this possible if you end up with exactly the same equations as conventional GR? Shouldn't the solutions then also be the same?

I haven't seen the analysis of how the field on flat background approach is equvalent to orthodox GR, but my guess is that for the convergence of the series involved the result (that they are equvalent) will hold only locally. Then there is no topology issue, the solutions are locally the same.

 

[PAllen]

Thanks. But how is this possible if you end up with exactly the same equations as conventional GR? Shouldn't the solutions then also be the same?

I guess I should dive more deeply into how this iteration is precisely performed, but I'm still puzzled by this topology-issue.

It is equivalent to ending up with a solution with specified coordinate conditions. The EFE cannot be solved except in the context of specific coordinates. However, in general, that solution you end up with is incomplete because many topologies can't be covered with one coordinate patch (and even some that can, cannot be covered with a particular type of coordinates - e.g. those 'similar to Minkowski coordinates'). What is done in classical GR is then to study the issue of completeness and perform extension with new coordinates to cover the whole manifold.

 

[Ilja]

And to require some completeness of the metric, of course, requires some metaphysical interpretation. Somehow the clock time has to be a true, "proper", time. If it would be, as in the Lorentz ether, simply describe clock showings distorted by some ether, what would be the justification for proposing completeness of this metric?

 

[haushofer]

It is equivalent to ending up with a solution with specified coordinate conditions.

Ah, OK, because the starting point (Fierz Pauli) is not general covariant but only Lorentz-covariant. That is the restriction on the coordinates you mention, I assume?

 

[Dale]

And to require some completeness of the metric, of course, requires some metaphysical interpretation.

I don't agree. The topology and geodesic completeness is experimentally measurable. So in principle you can test it experimentally. It is physics, not just metaphysics.

 

[Ilja]

I don't agree. The topology and geodesic completeness is experimentally measurable. So in principle you can test it experimentally. It is physics, not just metaphysics.

In these fundamental aspects it is hard to draw the line between physical and metaphysical. The problem is that to predict something for a real experiment you need always more than some single principle. Usually you need even more than one physical theory. Say, purely formal the Einstein equations of GR taken alone predict nothing. Because any difference $G_{mn} - T^{obs}_{mn}$ can be named the energy-momentum tensor of some dark matter. So you need some assumptions about matter too, else the equation alone gives nothing.

Similarly, I would not wonder if one could falsify some theories which include trivial topology together with something else, so that a theory with nontrivial topology and the same something else remains viable. But without something else?

Given this general situation, it is important to get rid of the classical positivist rejection of anything metaphysical. A principle, which, taken alone, does not lead to any observable predictions, can be named metaphysical - but this should not be misinterpreted as a justification to reject it. Taken together with other, similarly metaphysical, principles, it may be possible to derive physical predictions.

At least a way to test completeness, taken alone, I cannot see at all, I would even doubt that one can test it at all. All what seems possible would be to falsify particular theories which claim that a given particular metric is "complete" even if the metric is incomplete as a metric. For topology, the chances seem better, if we would see a lot of things which look like copies of our own history, that would be a case.

 

[Dale]

You are trying to make this more complicated than it is.

If you have two theories, where one predicts a geodesically incomplete spacetime and the other predicts a geodesically complete spacetime, then all you have to do is take a clock or a ruler to the region of spacetime which is excluded from the geodesically incomplete theory.

There is no metaphysics beyond the minimal interpretation required. This is a testable difference.

 

[Dale]

Since this thread has wandered far off topic from force based interpretations to aether based interpretations, it is now closed.