Assuming only rest mass as source of gravity and real observations

[Quantum Immortal]

If we use only rest mass as the source of gravity, but use the rest of general relativity. What are the differences with observations. Excluding cosmology and dark energy.

I mean real observations, not just theory. I exclude from the question, cosmology, dark matter and dark energy. In my opinion 30% dark mater and 90% dark energy are a bit too suspicious.
Gravity is so weak, that rest mass alone should still be good enough for 99% of observations. What are the 1% that fail? Of observations, and not cosmology! Don't tell me about the gravitational attraction between photons... we can't measure that.

If you don't understand my exclusion criteria, just state all the differences you can think off, i'll just sort out what interest me.

Can we detect differences in the solar system?

Observed system of twin neutron stars close together? Just ignore the insides of the neutron stars, we can just assume there's more matter inside ( a difference i know of). Is there a difference in the way there orbit decays as they radiate gravity waves?

About stars, only neutron stars would be affected in an observable fashion?

We never observed a twin system of black holes right?

My guess, that in general only very energetic phenomena are problematic. Of witch, only a few are observable.

 

[PeterDonis]

If we use only rest mass as the source of gravity, but use the rest of general relativity.

You can't. Using only rest mass as the source of gravity is inconsistent with the rest of GR.

Perhaps the question you meant to ask is, what observations other than cosmological ones tell us that rest mass is not the only source of gravity? If so, here are a couple:

(1) All of our observations indicate that there is a maximum mass limit for white dwarfs and neutron stars. But if rest mass is the only source of gravity, there is no maximum mass limit: as more mass is piled on and the object gets denser, its pressure just goes up to balance the increased gravity. The maximum mass limit comes from the fact that pressure is also a source of gravity, so as the object gets denser and the pressure goes up, the gravity pulling it inward also goes up, leading to a situation where even infinite pressure cannot prevent the object from collapsing once enough mass is piled on.

(2) Binary pulsar systems emit gravitational waves; this is well confirmed by observations. As the gravitational waves are emitted, the total mass of the system goes down; this is also confirmed by observations (the orbital parameters change). However, the rest masses of the individual pulsars do not change; the change in the total mass of the system is due to the change in the relationship between the two pulsars, i.e., to the change in binding energy. If binding energy did not contribute to the source of gravity, these observations could not be explained.

 

[PAllen]

There is also evidence, outlined in the following paper that kinetic energy contributes to gravitation.

http://arxiv.org/abs/gr-qc/9909014

(Note, the re-analyzed experiments described establish that KE contributes to passive gravitational mass (more precisely, that passive gravitational mass = inertial mass only if KE/c^2 is included in both. However, a difference between passive and active gravitational mass would be un-precedented.)

 

[Quantum Immortal]

You can't. Using only rest mass as the source of gravity is inconsistent with the rest of GR.

Elaborate?
Its not a frequent approximation to just use the rest mass?
You can't just put zeros in the rest of the stress-energy tensor?

(1) All of our observations indicate that there is a maximum mass limit for white dwarfs and neutron stars. But if rest mass is the only source of gravity, there is no maximum mass limit: as more mass is piled on and the object gets denser, its pressure just goes up to balance the increased gravity. The maximum mass limit comes from the fact that pressure is also a source of gravity, so as the object gets denser and the pressure goes up, the gravity pulling it inward also goes up, leading to a situation where even infinite pressure cannot prevent the object from collapsing once enough mass is piled on.

Yea, i was suspecting something like that...
But the pressure keeps increasing, isn't the increased pressure change the nature of the object?
white dwarfs will not just become neutron stars?
and neutron stars just become black holes?

(2) Binary pulsar systems emit gravitational waves; this is well confirmed by observations. As the gravitational waves are emitted, the total mass of the system goes down; this is also confirmed by observations (the orbital parameters change). However, the rest masses of the individual pulsars do not change; the change in the total mass of the system is due to the change in the relationship between the two pulsars, i.e., to the change in binding energy. If binding energy did not contribute to the source of gravity, these observations could not be explained.

total mass of the system? How you measure that?
You mean, the rate of decay of the orbits, are consistent with a reduction in the gravitational self energy, that it self contributes as a source to the gravitational field... non linear stuff...
It doesn't behave just like two electric charges with constant charges...





The solar system is fine right?
Any others?
So, only very extreme situations need more then the rest mass...
right?

There is also evidence, outlined in the following paper that kinetic energy contributes to gravitation.

http://arxiv.org/abs/gr-qc/9909014

(Note, the re-analyzed experiments described establish that KE contributes to passive gravitational mass (more precisely, that passive gravitational mass = inertial mass only if KE/c^2 is included in both. However, a difference between passive and active gravitational mass would be un-precedented.)

that is observable? Or is equivalent of measuring the gravitation pull between photons?

 

[pervect]

I think I already mentioned this in some other thread, but theoretical calculations of the induced velocities from a relativistic flyby shows that the induced velocities are not proportional to either the rest mass of the flyby object, nor to the relativistic mass of the flyby object. In the ultrarelativistic limit, the relativistic induced velocities are similar to those induced by a Newtonian body flying by whose Newtonian mass is twice the relativistic mass of the flyby object.

See http://dx.doi.org/10.1119/1.14280

 

[Quantum Immortal]

I think I already mentioned this in some other thread, but theoretical calculations of the induced velocities from a relativistic flyby shows that the induced velocities are not proportional to either the rest mass of the flyby object, nor to the relativistic mass of the flyby object. In the ultrarelativistic limit, the relativistic induced velocities are similar to those induced by a Newtonian body flying by whose Newtonian mass is twice the relativistic mass of the flyby object.

See http://dx.doi.org/10.1119/1.14280

That's just theory right? Not experimentally measured...
I was asking about real observations.

 

[PeterDonis]

Elaborate?

"The rest of general relativity" requires that the source of gravity be the full stress-energy tensor, not just rest mass. That is, there is no freedom in GR to choose whether or not the source of gravity is just rest mass, or includes other stuff; the source of gravity is the stress-energy tensor, which includes the other stuff.

Sometimes the other stuff is negligible compared to the rest mass, but that doesn't mean you're only using rest mass as the source of gravity; it means the source just happens to be such that other stuff besides the rest mass is negligible in that particular case. If the latter is what you meant in the OP, then OK, but it didn't seem like it; it seemed like you were asking whether you could "use the rest of GR" but somehow modify the equations so only the rest mass could act as a source of gravity. You can't.

Its not a frequent approximation to just use the rest mass?

For some cases, yes. For others, no. But it didn't seem like you were talking about approximations. See above.

You can't just put zeros in the rest of the stress-energy tensor?

That amounts to assuming that the other stuff besides rest mass is negligible. It is not the same as modifying the field equation so only rest mass appears. See above.

But the pressure keeps increasing, isn't the increased pressure change the nature of the object?

white dwarfs will not just become neutron stars?
and neutron stars just become black holes?

Sort of. If you could somehow take a white dwarf and compress it and compress it, you could, in theory, cause it to change structure to a neutron star. But there would be a whole range of unstable states in between where you would not be able to control the process; it might collapse to a neutron star or it might end up collapsing straight to a black hole.

At any rate, the key point is that the pressure of white dwarfs and neutron stars has to be a source of gravity, not just their rest mass; otherwise there wouldn't be a maximum mass limit for these objects.

total mass of the system? How you measure that?

By the orbital parameters of the pulsars. Over time enough measurements of enough parameters have been collected to solve a system of simultaneous equations that gives, among other parameters, the total mass of the system.

You mean, the rate of decay of the orbits, are consistent with a reduction in the gravitational self energy, that it self contributes as a source to the gravitational field... non linear stuff...

This is one way of putting it, yes; but you have to be careful here, because if you think about gravitational self-energy as a "source" of gravity, you are using the term "source" in a different way than we were using it before. Gravitational self-energy does not contribute to the stress-energy tensor; it can't be localized, so it can't be captured in any tensor. The best you can do is to look at integral quantities over finite spacetime regions (the "total mass of the system" referred to above is one such quantity), and show that, for example, the total mass of a bound system is not just the sum of the rest masses of its constituents; there is a (negative) contribution from what is usually called "gravitational binding energy".

The solar system is fine right?

Not if you include binding energy. The Sun and planets all have masses (measured, for example, by looking at the orbital parameters of bodies orbiting them) that are less than the rest masses of their constituents. So gravitational binding energy makes a (negative) contribution to their gravity.

AFAIK pressure in all these objects is too small to make a significant contribution to their gravity. It is possible to have a star that is massive enough for pressure to be significant in determining its gravity, but the Sun is too small for that.

 

[Quantum Immortal]

Ok see it as an approximation. If you take the approximation that the rest mass is the source of gravity, what observational differences you get?

You mean, that you can't make a black hole, just by piling up matter, if the only source of gravity is rest mass?
Or a neutron star? There density would just reach some maximum?

 

[Bill_K]

Ok see it as an approximation. If you take the approximation that the rest mass is the source of gravity, what observational differences you get?

No light deflection, and an incorrect precession of Mercury. What you're talking about is a scalar theory of gravity.

 

[Quantum Immortal]

its not what i mean
Still use the left side of GR

 

[Dale]

You can't just put zeros in the rest of the stress-energy tensor?

The rest mass is not part of the stress energy tensor. There is no "rest of the stress-energy tensor".

the source of gravity is the stress-energy tensor, which includes the other stuff.

Sometimes the other stuff is negligible compared to the rest mass,

I think you mean that sometimes the other stuff is negligible compared to the energy. The rest mass is not part of the stress energy tensor at all.

 

[Bill_K]

The rest mass is not part of the stress energy tensor. There is no "rest of the stress-energy tensor".

My understanding is that the closest analog of the rest mass is the trace of the stress energy tensor. "Coupling to the rest mass" means coupling to the trace of the stress energy tensor, as in scalar gravitation theories.

 

[Bill_K]

its not what i mean
Still use the left side of GR

I don't think you know what you mean. "Putting zeroes in the rest of the stress energy tensor" is not a Lorentz invariant restriction. So what this amounts to is picking a particular rest frame and doing the slow motion approximation.

 

[PAllen]

that is observable? Or is equivalent of measuring the gravitation pull between photons?

Did you even look at the paper? There is a whole section on experimental verification. Yes, it is verified experimentally.

 

[Dale]

My understanding is that the closest analog of the rest mass is the trace of the stress energy tensor.

Agreed, but then even the trace isn't "part" of the SET, it is more like a summary of the entire SET.

While you could certainly talk about scalar gravitation theories which perhaps use the trace of the SET, those are different theories so I would hesitate to call them "GR with just the trace".

 

[WannabeNewton]

Ok see it as an approximation. If you take the approximation that the rest mass is the source of gravity, what observational differences you get?

No gravitomagnetism, which is a shame really because gravitomagnetism is probably the coolest thing in general relativity

 

[PeterDonis]

I think you mean that sometimes the other stuff is negligible compared to the energy. The rest mass is not part of the stress energy tensor at all.

I guess this is partly a matter of terminology. "Rest mass" could mean "energy density in the rest frame, in mass units", which would mean the 0-0 component of the SET in the rest frame. Or it could mean the trace of the SET, as Bill_K says. Or it could mean something like the invariant length of the 4-momentum that you get when you integrate the SET (contracted with some timelike unit vector) over a spacelike slice, which is not even a local concept, and only really makes sense if the spacetime is asymptotically flat (it's more like the "total mass of the system" I referred to in a previous post). I'm not sure which of these would be the "canonical" meaning in GR, though the last comes closest to matching up with the standard definition in SR. But I suspect the OP meant something more like the first.

 

[PeterDonis]

You mean, that you can't make a black hole, just by piling up matter, if the only source of gravity is rest mass?

Yes; if pressure is not a source of gravity, pressure would always be able to balance gravity and keep the object stable without collapsing. It's the fact that pressure is a source of gravity that forces collapse above some maximum mass (because as the pressure increases to balance increased gravity, it makes gravity increase more, which requires more pressure, which increases gravity more, etc., to the point where above some maximum mass there is no equilibrium possible at all).

Or a neutron star?

A neutron star is a stable equilibrium state, so it could still exist if pressure were not a source of gravity.

 

[Quantum Immortal]

Did you even look at the paper? There is a whole section on experimental verification. Yes, it is verified experimentally.

I just read the abstract, i couldn't believe you can measure that...

No gravitomagnetism, which is a shame really because gravitomagnetism is probably the coolest thing in general relativity

I'm pretty sure that statement is not correct. In GR its just frame dragging. Even in a relativistic scalar theory you'll get a "magnetism" like effect. Its exactly what is happening in electromagnetism. Magnetism, is just a relativistic effect of electrostatics...

Yes; if pressure is not a source of gravity, pressure would always be able to balance gravity and keep the object stable without collapsing. It's the fact that pressure is a source of gravity that forces collapse above some maximum mass (because as the pressure increases to balance increased gravity, it makes gravity increase more, which requires more pressure, which increases gravity more, etc., to the point where above some maximum mass there is no equilibrium possible at all).

A neutron star is a stable equilibrium state, so it could still exist if pressure were not a source of gravity.

hmmm
It seams to me, that the mass limits simply get increased.
You still get a collapse in the end

I think that you are just seeing the non relativistic models for these things. You forget that the materials will just hit there limits eventually. Electrons eventually combine with protons to form neutrons, then neutrons merge to form a quark plasma, then sub quark physics takes over. Eventually you get a BH...

I have hard believing, that there is some type of matter that can withstand arbitrary amounts of pressure.

 

[WannabeNewton]

I'm pretty sure that statement is not correct. In GR its just frame dragging. Even in a relativistic scalar theory you'll get a "magnetism" like effect. Its exactly what is happening in electromagnetism. Magnetism, is just a relativistic effect of electrostatics...

No you won't. The Ampere equation which governs the generation of magnetostatic fields, given canonically by $\vec{\nabla}\times \vec{B} = \vec{j}$, is directly sourced by an electric current $\vec{j}$. If we have a Lorentz covariant gravitational field equation that is only sourced by rest mass then you obviously can't have a gravitomagnetic field because you need a mass current term in the Lorentz covariant gravitational field equation in order to source a gravitomagnetic field just like you need an electric current term in the Ampere equation to source a magnetic field. So at the least you need a vector theory of gravity in order to get gravitomagnetic fields. You have a fundamental misunderstanding of the relationship between relativity and electromagnetism because you're trying to use a single electrostatic Maxwell equation (Gauss's law) when you need to be using all four Maxwell equations in order to achieve Lorentz covariance.

 

[pervect]

If we use only rest mass as the source of gravity, but use the rest of general relativity. What are the differences with observations. Excluding cosmology and dark energy.

You seem to imagine some theory that uses rest mass as a source of gravity, and you assume it exists, but it's not particularly obvious whether or not what you are imagining does exist.

Sorry if I'm putting words in your mouth, but my understanding of your question is that you want to modify GR in a manifestly non-covariant way, and ask "what happens". The answer, assuming I'm understanding you correctly, is that you don't even get a self-consistent theory that way, so there isn't any need (or any way!) to test it experimentally, because it doesn't make self-consistent predictions.

I'm not sure you even understand the word "covariance" (as I said, I don't know your background). Covariance is what allows you to work a problem from multiple viewpoints (for instance, multiple coordinate systems) and always get the same answer for any measurement.

In GR, the mathematics that guarantee that the physical theory is covariant is (guess what!) tensor theory. Something that I gather you're not familiar with, my impression is (again) that you don't understand why people use them, and don't feel much motivation to learn or understand them. I'd go so far to say as you seem almost hostile to them, though I can't imagine why.

Various candidate covariant theories which do make gravity dependent on a scalar (which may not be rest mass, but may be what you want anyway) have been suggested by other posters, but they are not not "general relativity", they are some other theory. Most such scalar theories have already been falsified, and have problems with correctly predicting light deflection, etc.

Mostly textbooks teach tensor theory, they don't explain in detail to the enthusiastic amateur why you need them. I'm afraid I don't know how to address that lack, it's generally assumed that the student wants to learn tensors, rather than needs motivation before they'll try to learn them.

Let's go on to experimental tests of gravity in the solar system. The recognized framework for such testing is the Paramaterized Post Newtonian formalism, or PPN theory.

http://en.wikipedia.org/w/index.php?title=Parameterized_post-Newtonian_formalism&oldid=570033531

What you seem to be asking is "do we have any solar system experimental tests of the PPN parameter $\beta4$ - which is assumed in GR to have the value unity.

There are several solar system tests of $\beta$, but I'm not aware of any tests that directly measure $\beta4$. I suspect that such direct tests solar system tests will be impossible for a long time, unless and until we have detailed knowledge of the density and pressure profiles of the sun (and perhaps some of the major planets), based on actual measurement rather than theoretical models. The contribution of $\beta4$ would be small in any event, so such measurements would probably have to be unreasonably precise to be able to detect the presence of $\beta4$ at all.

However, I'm not aware of any covariantly formulated theories that have $\beta=1$ and $\beta4=0$. It's not particularly obvious if any such theories exist.

I would say that the cosmological evidence is the best evidence that pressure causes gravity, though it's rather indirect. We have theoretical models of the cosmological radiation pressure and its gravitational effects (which are significant - see the Friedmann equations), and this affects things we can (and have) measured like the acoustics of the pressure wave in the big bang, which is a hot topic in modern cosmology (look up the WMAP observations).

I hope you don't want to exclude cosmology just because it doesn't support your argument, that's not really a good habit to get into when considering scientific questions.

 

[Quantum Immortal]

... misunderstandings >:P
in my mind, current = moving charge (charge in general, not just electric charge)
movement is not so strange as to consider that a current and a charge have nothing in common
moving thing --> special relativity
in my mind, if you have a current, you use special relativity
special relativity + coulombs law --> Ampers law
the explanations are a bit sketchy of course

 

[WannabeNewton]

special relativity + coulombs law --> Ampers law

Again, the misunderstanding is contained in your personal conclusion(s) arising from the quoted statement. When one combines special relativity with Coulomb's law one obtains the Maxwell equations; it can explicitly be shown that the Maxwell equations are Lorentz covariant. This process is equivalent to simply writing down the Maxwell equations in a manifestly Lorentz covariant form: $\nabla^{\mu}F_{\mu\nu} = -4\pi j_{\nu}$, $\nabla_{[\gamma}F_{\mu\nu]} = 0$. The former approach indirectly obtains the Maxwell equations by using special relativity and Coulomb's law whereas the latter directly writes down the Maxwell equations in geometric form using the flat derivative operator $\nabla_{\mu}$ associated with the Minkowski metric $\eta_{\mu\nu}$. But regardless of the approach, the end result will achieve Lorentz covariance in conjunction with the introduction of an electric current.

If you start with a Lorentz covariant scalar theory of gravity wherein the gravitational field equations have only a single scalar source term then you can't "further combine" it with special relativity to produce some kind of current because it's already fully relativized, it's already combined wit special relativity. In a very loose sense that's what Lorentz covariant means. So the single scalar source term is all you get, there will be no mass current and hence no gravitomagnetic field.

 

[Quantum Immortal]

@pervect

if you want to understand something
you ... brake it and see what happens
thats all. I just want to see the experimental limits of GR

I exclude cosmology, because then you assume 95% dark stuff
i don't know about you, but having an ad hoc constant that represents 95% of the result, renders any theory meaningless. Seriously 95%!
I'll wait and see until more data is gathered on the issue.
Until then, i treat cosmology like meteorology. >:P

No, this is not my crack pot theory.

PPN goes too far. If i understand correctly, it even exclude special relativity effects.
I'm interested in gravitational effects.

Its not a Ph.D. dissertation, if its a bit messy, it doesn't matter. Its just a post in a forum.
Just consider the case that there's only rest mass in the stress energy tensor.
It's an approximation.

The answers i liked
1. the pressure inside neutron stars and white dwarfs
2. 2gammaM
3. missing mass of planets
4. binary pulsar system
...other? surely there's more

I'll derail my own thread too much if i answer all your points in detail.

 

[Quantum Immortal]

Again, the misunderstanding is contained in your personal conclusion(s) arising from the quoted statement. When one combines special relativity with Coulomb's law one obtains the Maxwell equations; it can explicitly be shown that the Maxwell equations are Lorentz covariant. This process is equivalent to simply writing down the Maxwell equations in a manifestly Lorentz covariant form: $\nabla^{\mu}F_{\mu\nu} = -4\pi j_{\nu}$, $\nabla_{[\gamma}F_{\mu\nu]} = 0$. The former approach indirectly obtains the Maxwell equations by using special relativity and Coulomb's law whereas the latter directly writes down the Maxwell equations in geometric form using the flat derivative operator $\nabla_{\mu}$ associated with the Minkowski metric $\eta_{\mu\nu}$. But regardless of the approach, the end result will achieve Lorentz covariance in conjunction with the introduction of an electric current.

If you start with a Lorentz covariant scalar theory of gravity wherein the gravitational field equations have only a single scalar source term then you can't "further combine" it with special relativity to produce some kind of current because it's already fully relativized, it's already combined wit special relativity. In a very loose sense that's what Lorentz covariant means. So the single scalar source term is all you get, there will be no mass current and hence no gravitomagnetic field.

B[
I meant just this
http://en.wikipedia.org/wiki/Relativistic_electromagnetism
I think you just took me too literally...

 

[PAllen]

PPN goes too far. If i understand correctly, it even exclude special relativity effects.
I'm interested in gravitational effects.

This is not true. The post Newtonion limit assumes weak gravity and slow speeds. The 1PN formalism describes this limit. However, the PPN methodology has been extended well beyond 3.5PN order, allowing consideration of fairly strong gravity and relativistic orbital motions. See section 4.1 of the following for minimum PN levels required for different cases:

http://relativity.livingreviews.org/Articles/lrr-2006-3/

 

[Dale]

Hi Quantum Immortal,

This comment by WBN is particularly important in this context.

When one combines special relativity with Coulomb's law one obtains the Maxwell equations

Note, that when one does combine special relativity with Coulomb's law the source of the field goes from being a scalar (the charge density) in Coulomb's law to being a four-vector (the four current) in Maxwell's equation. In other words, the process of "relativising" increased the rank of the tensor of the source.

The same thing happens with gravity. The rank of the source tensor increases for the relativistic theory compared to the non-relativistic theory.

 

[Quantum Immortal]

Hi Quantum Immortal,

This comment by WBN is particularly important in this context.
Note, that when one does combine special relativity with Coulomb's law the source of the field goes from being a scalar (the charge density) in Coulomb's law to being a four-vector (the four current) in Maxwell's equation. In other words, the process of "relativising" increased the rank of the tensor of the source.

The same thing happens with gravity. The rank of the source tensor increases for the relativistic theory compared to the non-relativistic theory.

Yea, i have some understanding of electromagnetism.
i think this is playing with semantics. Or a philosophical difference if you want. A current is just a moving charge. It didn't transform to something completely different. It is still a charge. For movement, you don't need a new theory, special relativity should suffice.

Magnetism is just an imbalance in electrostatics due to relativistic length/time contractions of the currents ( =special relativity). We can experience the effects of these tiny contractions at slow speeds because electrostatics is extremely strong inside mater.

I don't want to go further in detail here about this.

Yes, i admit my OP is a bit messy
When i said that the source is just the rest mass i meant it to be like the electric charge in electrmagnetism. The rest mass can have a current/move, and special relativity still applies.
I'm not asking about special relativity effects though, only stuff that are purely GR.

 

[PeterDonis]

I think that you are just seeing the non relativistic models for these things.

Assuming that only rest mass is a source of gravity *is* non-relativistic; as has been said several times already in this thread, "rest mass" in the sense you appear to be using the term is not covariant, but the source of gravity has to be covariant in a relativistic theory.

You forget that the materials will just hit there limits eventually. Electrons eventually combine with protons to form neutrons, then neutrons merge to form a quark plasma, then sub quark physics takes over. Eventually you get a BH...

I have hard believing, that there is some type of matter that can withstand arbitrary amounts of pressure.

In relativity, it's true that it is currently believed that any physically reasonable type of matter must obey what are called "energy conditions", one of which is, more or less, that the pressure can't be greater than the energy density. This does place a limit on the kinds of equation of state that matter can have, which places a limit on how dense a piece of matter can be and still support itself against gravity with some kind of pressure.

However, it is perfectly possible to have a stress-energy tensor that violates the energy conditions and still gives a self-consistent solution in GR. In fact, our best current belief is that the effective stress-energy tensors generated by quantum fields do violate the energy conditions in some physically relevant scenarios. One of them is Hawking radiation by black holes; another is the Casimir effect, in which vacuum fluctuations between two uncharged conducting plates create a small attractive force between the plates--this has actually been measured.

Having said all that, let me go back to a comment a little earlier in your post:

It seams to me, that the mass limits simply get increased.

*If* you could somehow concoct a relativistic theory in which only rest mass, not pressure, was a source of gravity, but which still obeyed the energy conditions I referred to above, then yes, I think you would still get a mass limit for objects like white dwarfs and neutron stars, but it would be a significantly higher limit than the one we actually observe. However, I'm not sure you could concoct a consistent theory of this type, for reasons which I and others have already given multiple times in this thread.

 

[Dale]

i think this is playing with semantics. Or a philosophical difference if you want. A current is just a moving charge.

If that is playing with semantics then so is your worry about the stress energy tensor. A stress is just a moving momentum flux. The various components of the stress energy tensor have a similar relationship to each other as to charge and current density.

Furthermore, the point remains that current (regardless of its connection to charge) is not part of Coulombs law. To get a relativistically invariant theory form Coulombs law you have to increase the rank of your source tensor as well as add the other equations which describe those additional components.

With gravity it is the same.

I think that what you should be doing is trying to understand the stress energy tensor. You realize that charge and current are related, but don't seem to understand that the various components of the SET are also similarly related.