I Are there any alternative interpretations to GR?

[olgerm]

Are there any interpretation to general relativity that described gravity as field (which do not have to be vectorfield, but may have 10 components) and physical-space as classical euclidean space?
Can it be mathematically proven that such interpretation can not exist?

I am not talking about approximation ,but an interpretation ,which where exactly in line with the general relativity.

 

[PAllen]

First, if you were going to use a flat spacetime background, it would have to be Minkowski spacetime not Euclidean. Secondly, many interesting GR solutions have non-trivial topology, so a flat background could not be ordinary Minkowski spactime. However, given Minkowski spacetime with appropriate topology, you could do this. None of the observables would be related to the background (flat) metric, but it could still be used as the basis for interpretation. There are authors who take this approach.

 

[olgerm]

First, if you were going to use a flat spacetime background, it would have to be Minkowski spacetime not Euclidean.

Would it be possible to describe spatial coordinates in Euclidean space and time separately (like in classical physics)? Time dilation could then be explained by forces ,that gravity-field causes. For example: 2 pointmasses are orbiting circulary around their center of mass so that their z-coordinate is constant.
$T=\frac{\omega}{2 \cdot \pi}$
$\omega=\sqrt{\frac{a}{r}}$
$a=\frac{m \cdot G}{4\cdot r^2}$
so $T=\sqrt{\frac{m \cdot G}{r^3}}\cdot \frac{1}{4\cdot \pi}$
Now let's watch these pointmasses from frame of reference which is moving with velocity v in direction of z-axis. Period of orbiting must be $\sqrt{1-\frac{v^2}{c^2}}$ times bigger than in last frame of reference, according to SR. So gravitational force must be $1-\frac{v^2}{c^2}$ times smaller, than in last frame of reference.

Since force may depend of velocity in this interpretation, this stipulation may be met.

 

[PAllen]

No, because there is invariant differential aging in SR with no gravity, and GR must approximate SR everywhere, locally.

Also, you ignored the points I made about topology.

 

[olgerm]

No, because there is invariant differential aging in SR with no gravity, and GR must approximate SR everywhere, locally.

Locally means in area ,which volume approaches to 0?
which conditions, more specify, must be met locally to satisfy SR?

 

[PAllen]

Locally means in a small volume, for a short time.

The metric must locally approach (in appropriate coordinates) (+1,-1,-1,-1) or (-1,+1,+1,+1). This ensures that locally, the Lorentz transform applies between local inertial frames.

 

[olgerm]

Locally means in a small volume, for a short time.

In force-based interpretation locally must gravityfield be constant and homogeneous.

This ensures that locally, the Lorentz transform applies between local inertial frames.

Could not it be ensured by forces in force-baced interpretation?

 

[Dale]

Could not it explained by forces in another interpretation?

I can't think of a force that would lead to time dilation.

 

[olgerm]

I can't think of a force that would lead to time dilation.

I do not have any general overview of such force in force-based interpretation, but it does not mean, that such force can not exist.
In special case of 2 pointmasses having a circular orbit I know a condition that the force must satisfy.
In first frame of reference (notated without ´) masscentre of pointmasses is in rest.
Second frame of reference (notated with ´) is moving with speed v.
In both interpretations:

$a=\frac{F}{m}$
$\omega=\sqrt \frac{a}{r}$
$T=\frac{2 \cdot \pi}{\omega}$
so $T=2 \cdot \pi \cdot \sqrt\frac{F}{m \cdot r}$​

in standard interpretation:

$\frac{_Δt}{_Δt´}=\sqrt{1-\frac{v^2}{c^2}}$
so $\omega´ =\omega \cdot \sqrt{1-\frac{v^2}{c^2}}$
so $T´=T \cdot \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$​

in force-based interpretation:

$T´=T \cdot \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$
$T´=2 \cdot \pi \cdot \sqrt\frac{F´}{m \cdot r}$
$T=2 \cdot \pi \cdot \sqrt\frac{F}{m \cdot r}$
so $F´=\frac{F}{1-\frac{v^2}{c^2}}$
so $F´=\frac{F \cdot c^2}{(c+v)\cdot(c-v)}$​

such condition is easy to find for any 2-body system.

 

[Dale]

such condition is easy to find for any 2-body system.

It isn't the condition that is in doubt. It is the behavior of forces. In your derivation of this condition you used relativity, not Newtons laws. Starting from Newtons laws you don't get time dilation.

Besides, your math really neglects the fact that forces are vectors.

 

[olgerm]

Starting from Newtons laws you don't get time dilation.

I know, I have not claimed ,that Newton physics are exactly in line with the general relativity. Time dilation is not explained by Newtonian physics. I have asked about interpretation to general relativity. Of course in that interpretation gravity would not be described be Newton´s law of universal gravitation.

In your derivation of this condition you used relativity, not Newtons laws.

Yes, I did.Can anybody prove that forming such force-based interpretation to theory of relativity is impossible?
Can anybody form such interpretation to theory of relativity?

 

[olgerm]

Besides, your math really neglects the fact that forces are vectors.

Just wanted to show basic idea of how forces may describe time dilation in another interpretation.

 

[Dale]

I know, I have not claimed ,that Newton physics are exactly in line with the general relativity.

Newtonian physics is the theory of physics that defines forces, so...

Can anybody prove that forming such force-based interpretation to theory of relativity is impossible?

This becomes almost meaningless. You cannot use Newtonian forces to replicate time dilation, so then what do you mean by "forces". You would need to make up a new definition of "force". In that definition maybe you could put all of relativity, but then you would just wind up with normal relativity except that you are labeling something as "force". This "force" in relativity would not be force as we know it from Newtonian physics, so in what way is this a force-based interpretation?

I don't think that it is impossible, just useless. It would be nothing more than arbitrarily sticking the label "force" somewhere in relativity.

Just wanted to show basic idea of how forces may describe time dilation in another interpretation.

By treating a vector as a scalar you didn't show anything.

 

[olgerm]

By treating a vector as a scalar you didn't show anything.

The pointmasses are orbiting circularly so they´r net force is always directed to they´r center of mass. F length of force-vector.
The pointmasses are orbiting circularly so they´r acceleration is always directed to they´r center of mass. a is length of acceleration-vector.
In the example situation the second frame of reference is moving in direction ,in which the pointmasses speeds are always 0, in respect of first frame of reference.

 

[olgerm]

I try to make more clear what I am asking.
Is it possible to use such interpretation to theory of relativity in which:

• physical-space is ordinary 3-dimensional Euclidean space.
  
• time is same for all observers.
  
• Newton´s second law remains in force.
  
• Since it is interpretation to theory of relativity ,it must be in accordance with the theory of relativity.

In that interpretation obviously:

• Newton´s law of universal gravity does not hold true.
• gravityfield is not 3-dimensional vector field.

nota bene: by saying that the interpretation is force-based I do not mean that gravity is only a force. I mean that gravityfield affects bodies by impressing force on them.

 

[PAllen]

What you seek is already impossible without any gravity because what you suggest violates SR for empty space with no charges and nothing except tiny test bodies. Your only possible way out is the LET interpretation of SR where there is an undetetectable preferred frame whose time you declare to be absolute, but it does not correspond to any possible measurements. However, this cannot, in principle, be generalized to match GR for reasons I have mentioned twice and you have comletely ignored. GR predicts non-trivial topology. There is no possible interpretation that can be layed on flat space + time that can modify its topology.

So that answer is that it is trivially provable that what you seek is impossible. You could accomplish it for weak field predictions of GR, but not for the whole of GR. (Weak field assumption allows you to ignore the cases where GR predicts non-trivial topology).

 

[Dale]

The pointmasses are orbiting circularly so they´r net force is always directed to they´r center of mass. F length of force-vector.
The pointmasses are orbiting circularly so they´r acceleration is always directed to they´r center of mass. a is length of acceleration-vector.
In the example situation the second frame of reference is moving in direction ,in which the pointmasses speeds are always 0, in respect of first frame of reference.

Exactly. Is there any linear transformation which has these properties? Specifically that the length of the vector changes as described and the direction of the vector remains pointing towards the center of mass in both frames.

By treating F as a scalar and trying to derive a necessary condition on its transformation you did not show anything because F is a vector and so you need to derive conditions on the transformation of a vector. This is why it is important to do your math carefully. As written your equations 2, 4, 9, and 10 are simply wrong.

 

[Dale]

• Newton´s second law remains in force.
  
• Since it is interpretation to theory of relativity ,it must be in accordance with the theory of relativity.

In addition to PAllen's point, these two desiderata are incompatible. Newton's second law is incompatible with relativity.

Whatever idea of "force" you might create in order to make a force-based interpretation would have to be a new ad-hoc concept of "force" created expressly and solely for the purpose of making the interpretation work. It cannot be the standard concept of force from Newton's laws.

 

[PeterDonis]

Is it possible to use such interpretation to theory of relativity in which:

• physical-space is ordinary 3-dimensional Euclidean space.
  
• time is same for all observers.
  
• Newton´s second law remains in force.
  
• Since it is interpretation to theory of relativity ,it must be in accordance with the theory of relativity.

Bullet #1: No, because "space" has no invariant definition in relativity, it depends on your choice of coordinates. The only invariant geometry is that of spacetime.

Bullet #2: No, because "time" has no invariant definition in relativity, it depends on your choice of coordinates. The only invariant is that of proper time along a particular worldline, and that depends on the worldline.

Bullet #3: No, because, as Dale already pointed out, Newton's second law is incompatible with relativity.

Bullet #4: No, because, as shown above, the other three points can't be made consistent with relativity.

 

[olgerm]

Okay, as I understand it is impossible even for special relativity. Probably I should have asked about SR first. I have been learning gravitoelectromagnetism and got an idea that maybe GR is similar - just more parameters for one point and interpreted differently, but I was wrong. Did I get it right: not-classical topology and geometry of relativity theory are physical reality, not part of interpretation?

However, this cannot, in principle, be generalized to match GR for reasons I have mentioned twice and you have comletely ignored. GR predicts non-trivial topology. There is no possible interpretation that can be layed on flat space + time that can modify its topology.

Can´t any interpretation with Euclidean space characteristic give same predictions as non-trivial topology interpretation gives?

 

[PAllen]

Okay, as I understand it is impossible even for special relativity. Probably I should have asked about SR first. I have been learning gravitoelectromagnetism and got an idea that maybe GR is similar - just more parameters for one point and interpreted differently, but I was wrong. Did I get it right: not-classical topology and geometry of relativity theory are physical reality, not part of interpretation?

Can´t any interpretation with Euclidean space characteristic give same predictions as non-trivial topology interpretation gives?

Yes, you were wrong. The non-trivial topology relates directly to physical predictions and cannot be replicated on a theory with trivial topology. For example, a wormhole cannot be represented in a simply connected topology. Nor can the interior of a Kerr BH. Nor can a spatially closed cosmology (this doesn't apply to our universe, but is a clear prediction of GR if energy density of the universe were different).

 

[PeterDonis]

For example, a wormhole cannot be represented in a simply connected topology. Nor can the interior of a Kerr BH. Nor can a spatially closed cosmology

I don't think "simply connected" is the right term here. A spatially closed cosmology has topology $S^3 \times R$, which is simply connected, but it is not a possible solution in the "gravity is a field on flat spacetime" model.

The correct statement, I think, is even more restrictive: the only topology that is a possible solution in the "gravity is a field on flat spacetime" model is the topology of Minkowski spacetime, $R^4$. This rules out almost all of the interesting solutions--for example, even the interior of a Schwarzschild black hole doesn't have this topology.

 

[Dale]

Did I get it right: not-classical topology and geometry of relativity theory are physical reality, not part of interpretation?

This is an interesting question, I think that you may be a little unfamiliar with the role of interpretations in modern science.

A scientific theory consists of a mathematical framework and a mapping between the framework and experiments. For example, your mathematical framework might have a quantity, t, that maps to the reading on a clock. The mathematical framework generally also establishes some specific relationships between the various elements of the framework, such that if you measure some values you will be able to predict other measurable values. Experiments which do so test the validity of the theory.

In addition to the framework and the mapping to experiment (sometimes called the minimal interpretation), it is often convenient to introduce additional philosophical interpretations of the framework. These philosophical interpretations do things like state that some quantity in the model represents something "real" while some other quantity is "not real". So the question of "physical reality, not interpretation" is a little backwards, since it is the only the interpretations that make claims about physical reality, not the theory itself.

That said, the geometry and topology are testable parts of the mathematical framework of relativity.

 

[haushofer]

Without having read all the posts, you can start from an interacting spin-2 theory in special relativity, called Fierz-Pauli theory. You can then introduce higher order interaction terms via an iteration-process, which results in GR. General covariance then shows up as a consisteny requirement to remove ghosts, as in QED. As such spacetime curvature can also be described as 'a coherent states of gravitons on a flat background', like in String Theory.

 

[PeterDonis]

you can start from an interacting spin-2 theory in special relativity

But, as has already been said, this theory assumes that spacetime has topology $R^4$, the topology of Minkowski spacetime. So it can't accommodate solutions that have a different topology.

 

[PAllen]

One other thought on this is that for most topologies, there is no globally flat Minkowskian realization. You can make a flat background in some finite area of interest, but not globally. For example, S³XR has no globally flat realizatiion.

 

[Ilja]

Are there any interpretation to general relativity that described gravity as field (which do not have to be vectorfield, but may have 10 components) and physical-space as classical euclidean space?
Can it be mathematically proven that such interpretation can not exist?

I am not talking about approximation ,but an interpretation ,which where exactly in line with the general relativity.

There exists such an alternative interpretation.

To be accurate, it is only an interpretation of the Einstein equations of GR. And it adds a coordinate condition, namely the harmonic condition, which is, in this interpretation, a physical equation. So, this interpretation can be, formally, considered as a different physical theory: There are solutions of GR which cannot be interpreted in this interpretation, thus, observing them would falsify the interpretation. But the equations are the same - the Einstein equations of GR, in harmonic coordinates. To introduce harmonic coordinates is locally always possible.

The interpretation is quite simple. It is an ether interpretation, the gravitational field defined density, velocity and stress tensor of the ether. The formulas are
g^{00}\sqrt{-g} = \rho\\<br /> g^{0i}\sqrt{-g} = \rho v^i\\<br /> g^{ij}\sqrt{-g} = \rho v^iv^j - sigma^{ij}
or, in other words, all one needs to transform the harmonic coordinate condition into the continuity and Euler equations of standard condensed matter theory.

This makes sense only if \rho&gt;0. This conditions is equivalent to the preferred harmonic time being time-like. And, of course, the harmonic coordinates have to cover the whole solution. (More accurate, if complete harmonic coordinates, defined for all values -\infty&lt;x^i&lt;\infty cover only a part of the complete GR solution, this part is interpreted as already defining a complete solution.)

There is nothing published about this interpretation, AFAIK, but it is the limit \Xi, \Upsilon\to 0 of an alternative theory of gravity published in a peer-reviewed journal, see http://arxiv.org/abs/gr-qc/0205035 The theory has further advantages, because it has a Lagrange formalism (by adding the harmonic condition to the Einstein equations the Lagrange formalism is destroyed) and local energy-momentum conservation laws.

Whatever, if one allows for such additional restrictions (excluding solutions with nontrivial topology, with causal loops, and other things not yet observed), a different interpretation of the Einstein equations themself is possible.

 

[Ilja]

Okay, as I understand it is impossible even for special relativity.

No, for special relativity another interpretation is possible, and well-known as the Lorentz ether. It is, essentially, the original interpretation of special relativity, which was originally named Lorentz-Einstein theory. The modern spacetime interpretation is what Minkowski has added.

While the remarks that all the solutions with nontrivial topology hardly allow other interpretations are correct, don't forget that we have not yet observed any solutions with any nontrivial topology. So, all the observational support for GR does not require that we use an interpretation which allows nontrivial topologies.

 

[Ilja]

Bullet #1: No, because "space" has no invariant definition in relativity, it depends on your choice of coordinates. The only invariant geometry is that of spacetime.

Correct, but an alternative interpretation can add such a definition of space, by distinguishing some special choice of coordinates as preferred. So, in the alternative interpretation there may exist an ordinary 3-dimensional space. In the interpretaion described in #27 it exists. The Euclidean distances are different from the distances measured with rulers, which is interpretated as the result of distortion of rulers by the gravitational field.

Bullet #2: No, because "time" has no invariant definition in relativity, it depends on your choice of coordinates. The only invariant is that of proper time along a particular worldline, and that depends on the worldline.

Correct, but, similarly, the alternative intepretation may propose some special choice of coordinates as preferred. Which would define a preferred absolute time. In the interpretaion described in #27 it exists. This time is different from the time measured by clocks, which is interpretated as the result of distortion of clocks by the gravitational field.

The interpretation described in #27 does not cover all solutions of GR, but excludes some of them, or parts of them. But, given that the local equations - the Einstein equations in harmonic coordinates - are identical to the GR equations (even if only in particular coordinates), and we have not yet observed any nontrivial topologies, this interpretation is compatible with relativity, at least if we interpret "relativity" not as a set of fundamental metaphysical beliefs, but as those equations of physics supported by empirical evidence.

 

[PAllen]

No, for special relativity another interpretation is possible, and well-known as the Lorentz ether. It is, essentially, the original interpretation of special relativity, which was originally named Lorentz-Einstein theory. The modern spacetime interpretation is what Minkowski has added.

While the remarks that all the solutions with nontrivial topology hardly allow other interpretations are correct, don't forget that we have not yet observed any solutions with any nontrivial topology. So, all the observational support for GR does not require that we use an interpretation which allows nontrivial topologies.

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

 

[Dale]

It is an ether interpretation

No, for special relativity another interpretation is possible, and well-known as the Lorentz ether. It is, essentially, the original interpretation of special relativity

In what way is that a "force based" interpretation? Newton's 2nd law is not compatible with LET any more than SR, and the aether is not typically considered a force field even with a modified definition of "force".

 

[haushofer]

But, as has already been said, this theory assumes that spacetime has topology $R^4$, the topology of Minkowski spacetime. So it can't accommodate solutions that have a different topology.

Yes, and I can remember that there was a topic about this issue before. But it still puzzles me, because you seem to get the full Einsteineqns after the iteration. So it seems like you're going from a background-dependent theory to a background-independent one.I have to think about this a bit more ;)

 

[PAllen]

Yes, and I can remember that there was a topic about this issue before. But it still puzzles me, because you seem to get the full Einsteineqns after the iteration. So it seems like you're going from a background-dependent theory to a background-independent one.I have to think about this a bit more ;)

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.

 

[PAllen]

I guess in additions to some topologies being inaccessible to the spin 2 field, what could happen is you get an incomplete solution that (treated as a manifold with metric) is geodesically incomplete but extendable to a become complete. Thus, for a high density cosmology, you might end up with a description where points are removed from each 3-sphere such the result is compatible with R⁴

 

[Ilja]

In what way is that a "force based" interpretation?

In no way. The base of the interpretation is the ether. It is a positive answer to the original question "Are there any alternative interpretations to GR?", but I see no connection to any "force-based" interpretation.

The ether does not have this anti-force attitude one knows from the geometric interpretation. A flat background exists, thus, in principle inertial motion can be defined, and once motion deviates from it, one can describe this as caused by some forces. But what would be the point of having forces? The advantage?

Forces derived from potentials, and, in general, equations of motion derived from minimum principles of a Lagrange formalism, are more attractive than simply forces. (That's why I prefer the modification of the GR equation which revives the Lagrange formalism).

 

[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.