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Equations of Motion a consequence of the Field Equations?


elfmotat

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The Bianchi identities along with the EFE's with zero CC imply:

 

[math]\nabla_{\nu} G^{\mu \nu} = \nabla_{\nu} T^{\mu \nu}=0[/math]

 

This implies that, for arbitrary [math]\xi_\mu[/math], the following holds true:

 

[math]\xi_\mu \nabla_{\nu} T^{\mu \nu}=0[/math].

 

Therefore:

 

[math]\nabla_{\nu}(\xi_\mu T^{\mu \nu}) = T^{\mu \nu} \nabla_{\nu} \xi_\mu[/math]

 

Now let's say we're considering a free point particle traveling along some worldline with stress-energy given by:

 

[math]T^{\mu \nu} (s) = m \frac{dx^\mu}{ds} \frac{dx^\nu}{ds}[/math]

 

If we integrate both sides over some region containing the worldline and allow [math]\xi_\mu[/math] to vanish at the boundary, the left side goes to zero by Stokes. The right side must therefore be zero regardless of the size of the region, so we can simply take the integral over the path:

 

[math]\int \nabla_{\nu} \xi_\mu \frac{dx^\mu}{ds} \frac{dx^\nu}{ds} ds=0[/math]

 

Now we recognize that [math]\frac{dx^\nu}{ds}\nabla_{\nu} \xi_\mu = \frac{D \xi_\mu}{ds}[/math]. So if we integrate by parts, we get:

 

[math]\int \frac{D}{ds}\left [ \frac{dx^\mu}{ds} \right ]\xi_\mu ds=0[/math]

 

Since [math]\xi_\mu[/math] is just an arbitrary one-form, we have that the following must be true everywhere:

 

[math]\frac{D}{ds}\left [ \frac{dx^\mu}{ds} \right ]=0[/math].

 

This is just the geodesic equation. So it seems we can get the equations of motion for a single point particle in GR directly from the field equations. Is this a unique case? Will a similar argument lead to the equations of motion for, say, dust or a perfect fluid? Is there some way to generalize this for all matter distributions? And also, is the fact that the equations of motion come out of the theory's field equations something unique to GR?

Edited by elfmotat
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I'm not versed in tensor calculus, but it may be a consequence of the fact that GR is derived from the light equations directly, which are algebraic equations of motion themselves.

Also, I think it isn't unique to GR, but it must be inherent to the Minkowski metric, wherever applied, because it is derived from SR, and logically, from c=x/t and so forth. Minkowski simply imported the time dimension from ct into ds, so we are basically tracking infinitesimal space coordinates "over" infinitesimal times, right? If this is so, isn't the metric itself just an expanded equation of motion?

Edited by altergnostic
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What you have shown is that the geodesic equation can be derived from the field equations of general relativity. Basically, you model particles as singularities in the field and this then means that their motion is governed directly by the field equations. From another point of view, the fact that the field equations are highly non-linear means that one cannot really separate the motion of the sources from the field equations. This is well-known.

 

I am not aware of anyway of generalising this to arbitary distributions of matter, but stand ready to be corrected if there are very special cases.

 

This results seems rather special to general relativity, it is not obvious (to me anyway) how this would generalise to a charged particle in electromagnetism. The Lagrangian of the test particle is given independently of Maxwell's equations.

 

It maybe possible that other non-linear field equations could share this property with general relativity, but I am not aware of real concrete statments here.

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I'm not versed in tensor calculus, but it may be a consequence of the fact that GR is derived from the light equations directly, which are algebraic equations of motion themselves.

 

What are the "light equations?"

 

Also, I think it isn't unique to GR, but it must be inherent to the Minkowski metric, wherever applied, because it is derived from SR, and logically, from c=x/t and so forth. Minkowski simply imported the time dimension from ct into ds, so we are basically tracking infinitesimal space coordinates "over" infinitesimal times, right? If this is so, isn't the metric itself just an expanded equation of motion?

 

I'm not really sure what you mean here either.

 

What you have shown is that the geodesic equation can be derived from the field equations of general relativity. Basically, you model particles as singularities in the field and this then means that their motion is governed directly by the field equations. From another point of view, the fact that the field equations are highly non-linear means that one cannot really separate the motion of the sources from the field equations. This is well-known.

 

I am not aware of anyway of generalising this to arbitary distributions of matter, but stand ready to be corrected if there are very special cases.

 

This results seems rather special to general relativity, it is not obvious (to me anyway) how this would generalise to a charged particle in electromagnetism. The Lagrangian of the test particle is given independently of Maxwell's equations.

 

It maybe possible that other non-linear field equations could share this property with general relativity, but I am not aware of real concrete statments here.

 

Thanks for the response. I did some digging and found this in Wald as well:

 

 

 

 

The final remark concerns the equations of motion of matter . As we have
presented the theory, the equations of motion of particles, continuous matter, and
fields are postulated first, and then Einstein's equation relating the matter distribution
to the curvature of spacetime is given . However, Einstein's equation implies the
relation ∇aTab = 0, and this relation contains a great deal of information on the
behavior of matter . Indeed, for a perfect fluid, the relation ∇aTab = 0 is the entire
content of the equations of motion . Thus for a fluid we may economize our assumptions
by merely postulating the form of Tab = 0 ; the equations of motion of the fluid are
already contained in Einstein's equation . Notice that for a perfect fluid with P = 0,
i .e ., a fluid composed of grains of "dust" which exert no forces upon each other, the
fluid equation of motion (4 .3 .8) implied by ∇aTab = 0 tells us that the individual dust
particles move on geodesics . More generally, it can be shown (Fork 1939; Geroch
and Jang 1975) that the relation ∇aTab = 0 implies that any sufficiently "small" body
whose self-gravity is sufficiently "weak" must travel on a geodesic . Thus, Einstein's
equation alone actually implies the geodesic hypothesis that the world lines of test
bodies are geodesics of the spacetime metric . This demonstrates an important selfconsistency
of Einstein's equation with the basic framework of general relativity .
Note however, that bodies which are "large" enough to feel the tidal forces of the
gravitational field will deviate from geodesic motion . The equations of motion of
such bodies also can be found from the condition ∇aTab = 0 (Papapetrou 1951 ; Dixon
1974) .
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Wald's book is very good. The comments he makes sound reasonable to me. One point here is that the bodies are assumed to be acted upon by gravity only, which is fine. The equivalence principle tells us that the motion of structureless test particles in a gravitational background field is determined only by the space-time geometry- the worldlines are geodesics.

 

If the test particle is not structureless, let us say has electric charge or spin, then the motion is not determined by geodesics. There are "corrections" due to electromagnetic/spin orbital forces. The relation [math]\nabla^{a}T_{ab}=0[/math] will still hold, but I think one will need to either postulate the form of the equations of motion or equivalently the Lagrangian for the test particle.

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The light equations I was referring to are x=ct and x'=ct'.

And regarding Minkowski, he started off from the same set of equations, and brought time into the metric via ct:

ds2 = - dx2 - dy2 - dz2 + c2dt2

 

From this I believe we could always expect GR to describe any kinematic situation, due to the fact that the foundations of the theory rely on equations that deal with motion in the first place.

I would be amazed if it was not possible to reduce GR's equations to equations of motion, really.

Edited by altergnostic
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The light equations I was referring to are x=ct and x'=ct'.

And regarding Minkowski, he started off from the same set of equations, and brought time into the metric via ct:

ds2 = - dx2 - dy2 - dz2 + c2dt2

 

Again, I'm really not sure what any of this has to do with anything.

 

 

I would be amazed if it was not possible to reduce GR's equations to equations of motion, really.

 

I was certainly surprised. It seems like something unique to GR. There's no electromagnetic analogy, for example; the equations of motion are a separate postulate from its field equations.

Edited by elfmotat
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Well, think about it. The lorentz transforms translate coordinates between frames (positions and times and speeds). Without the transforms, we are describing simple velocities.

With the light equations, even though we are dealing with electromagnetism, the form of the equations also describe a simple velocity. What I mean is that, in SR, we focus on relative motion without acceleration.

If GR includes acceleration but also reduces to SR, which in turn reduces to simple speeds for low non relativistic speeds, it follows that GR must be able to reduce to the equations of motion. The postulates are not present in GR because they are implied in the fact that we deal with positions, times, velocities and accelerations, in agreement with Galileo and Newton for slow speeds.

And with Minkowski, ct is important because it comes from x=ct which is a simple velocity. Simply put, it is basically the description of the positions of a body in space and time using SR postulates, with the possibility of incorporating accelerations. The process of reducing the field equations like you did I had never seen before, and maybe it didn't surprise me because I am not too familiar with GR operations. I have always thought of GR as nothing more than SR+acceleration, so the equations of motion should be expected. Without the EM part, it is simply coordinates, velocities and accelerations, so why wouldn't you expect the equations of motion were fundamentally included?

 

PS: It is not unique to GR in the sense that it is a consequence of Minkowski's metric, which is in turn used by Einstein to build GR. It isn't a characteristic of GR, but of Minkowski's metric. It is the metric that includes the relation between space and time.

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... What I mean is that, in SR, we focus on relative motion without acceleration.

 

....I have always thought of GR as nothing more than SR+acceleration,...

You can deal with accelerations in special relativity; there is the notion of proper acceleration, which is fine on Minkowski space-time.

Without the EM part, it is simply coordinates, velocities and accelerations, so why wouldn't you expect the equations of motion were fundamentally included?

In the original formulation of general relativity the fact that test particles travel along geodesics was assumed in addition to the field equations. This seems very natural, but it is at this stage an assumption.

 

The point is that by taking the obvious energy-momentum tensor for a free massive point particle one gets the geodesic equation straight from the field equations. You don't have to do any calculus of variations here. Note that here we get the time-like geodesics.

 

Similarly one can propose a simple Lagrangian for a free massive point particle on a curved space--time and apply the Euler-Lagrange equations. If you do this you also get at the geodesic equation.

 

You can even understand geodesics as Hamiltonian flows.

 

 

 

 

 

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You can't deal with non-inertial frames in SR, that's what I mean when I say that GR is SR+acceleration.

Anyway, that the motion of test particles along geodesics was an assumption is not important, what matters is that this assumption is inherently part of the theory. The theory is built upon a curved spacetime and deals directly with the motions in this 4D metric. It should be expected that equations of motion underlie the field equations.

You can actually find such equations in Einstein's papers (and Minkowski's). This one is in the book Relativity, pg 146:

d^2xτ/ds^2 = Γτμν(dxμ/ds)(dxν/ds)

 

(i'm typing from my cellphone, sorry for the formatting)

 

And a quick search on "equations of motion general relativity" show some results concerning this topic, like this page:

http://www.grc.nasa.gov/WWW/k-12/Numbers/Math/Mathematical_Thinking/field_equations.htm

 

I didn't even read it, really, I'm just trying to point out that what you found out is not a novelty, and as I said, I would be surprised if it was not possible to reduce the field equations to equations of motion, since the field equations were developed to deal with relative motions in non-inertial frames of reference to start with. And of course GR must be consistent with SR which had to be consistent with Galileo, hence, GR should agree with Galileo and Newton under specific circumstances. Isn't this logical?

 

PS: the fact that time entered the metric as part of a spatial dimention ct which was subtracted from the euclidean metric (not added like x y and z) sugests that we should be able to reverse the process and separate the time-like geodesic for a particle moving in an euclidean space.

Edited by altergnostic
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Anyway, that the motion of test particles along geodesics was an assumption is not important, what matters is that this assumption is inherently part of the theory.

 

I didn't know that the geodesic equation fell out the field equations until recently, and Einstein didn't realize it himself when he first formulated his theory. That's essentially the topic of the this thread. It appears to be a feature unique to GR, because the equations of motion for a theory are usually postulated separately from its field equations.

 

For example, Maxwell's equations tell us how electromagnetic fields are generated and how they behave, but they don't tell us how matter placed in an electromagnetic field will behave. The Lorentz Force Law is a separate postulate from Maxwell's equations, and it is this equation (the equation of motion) which tells us how matter behaves in the presence of an EM field.

 

In (the field theory formulation of) Newtonian gravitation, the Poisson equation tells us how the gravitational field is generated and how it behaves, but it doesn't tell us how matter behaves in the presence of a gravitational field. Postulated separately is the equation of motion: [math]d^2x^i / dt^2=-\partial_i \phi[/math], where [math]\phi[/math] is the Newtonian potential.

 

Likewise, the Einstein Field Equations tell us how the gravitational field is generated and how it behaves. There's no immediately obvious reason to assume the field equations for this theory should contain the equations of motion, or Einstein wouldn't have bothered postulating geodesic motion separately. In fact, all experience tells us that the EFE's probably shouldn't contain the equations of motion.

 

 

 

The theory is built upon a curved spacetime and deals directly with the motions in this 4D metric.

 

What are "motions in the metric?"

 

 

 

It should be expected that equations of motion underlie the field equations.

 

I fail to see how this follows from anything you just said.

 

 

You can actually find such equations in Einstein's papers (and Minkowski's). This one is in the book Relativity, pg 146:

d^2xτ/ds^2 = Γτμν(dxμ/ds)(dxν/ds)

 

 

 

This is the geodesic equation. It's the same equation I derived in my original post, and it's the equation Einstein postulated separately from the Field Equations in original papers.

 

 

And a quick search on "equations of motion general relativity" show some results concerning this topic, like this page:

http://www.grc.nasa.gov/WWW/k-12/Numbers/Math/Mathematical_Thinking/field_equations.htm

 

I didn't even read it, really, I'm just trying to point out that what you found out is not a novelty,

 

I'm not sure what your link has to do with anything.

 

 

and as I said, I would be surprised if it was not possible to reduce the field equations to equations of motion, since the field equations were developed to deal with relative motions in non-inertial frames of reference to start with.

 

The Field Equations were developed to explain how the gravitational field is related to the energy-momentum distribution in spacetime, not to "deal with relative motions in non-inertial frames of reference." Of course GR can deal with such scenarios, but they really have minimal to do with the Field Equations.

 

 

And of course GR must be consistent with SR which had to be consistent with Galileo, hence, GR should agree with Galileo and Newton under specific circumstances. Isn't this logical?

 

SR and GR aren't consistent with Galileo! And I'm not really sure what point you're trying to make here anyway - what do low-energy/velocity limits have to do with the equations of motion in GR or its field equations.

 

PS: the fact that time entered the metric as part of a spatial dimention ct which was subtracted from the euclidean metric (not added like x y and z) sugests that we should be able to reverse the process and separate the time-like geodesic for a particle moving in an euclidean space.

 

I honestly have no idea what that means.

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I see your point.

First of all, of course SR and GR agrees with galileo (for small distances and slow speeds), that was one of the first goals of Einstein, to unite the classical model with maxwell.

Now, I think that postulating equations of motion separately in GR is redundant to start with. The equations of motion were already postulated separately in the derivation of SR, so SR already contains them. You found the geodesic equation from the field equations and this can ONLY mean that it was built-in to start with. I think (but I don't know it for a fact) that it entered the field equations in that form through Minkowski, who imported SR into a 4D metric. Are you saying that the equation of motion you derived exists by accident in the field equations?

 

Finally, the goal of GR as a whole was to generalise SR to non-inertial frames. In other words, to describe relative motion under the influence of gravity in relativistic scenarios. The field-equations were developed for that purpose. Isn't the energy-momentum distribution in spacetime fundamentally a motion in spacetime? If it is simply Minkowski's description of SR in four dimensions, and if SR already contains the equations of motion, you should be able to deduce them from the field equations, right?

Let me ask you, how did you come up with the idea of looking for equations of motion in the field equations in the first place?

 

And regarding my PS, Minkowski metric is SR in 4D, but the time dimension is subtracted from the 3 spatial dimensions and it comes in through x=ct. It isn't just another dimension like the others. The metric is reducible to an euclidean description of space if you graph time separately. My intention was to point out that not only the time-like geodesic equation must be derivable from GR, but also the classical linear equations of motion, tying GR all the way back to Galileo. I have no idea how to do it, though, it's just a thought.

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You can't deal with non-inertial frames in SR, that's what I mean when I say that GR is SR+acceleration.

This is not quite correct, you are free to use any coordinate system you like in special relativity, including non-inertial ones. What special relativity tell us is that out of all possible coordinate systems there are a kind of "preferred" class and this is the class of inertial coordinates.

 

When in non-inertial frames, within the context of special relativity, you can have frame specific effects and the speed of light is not usually c!

 

What is true is that when you use non-inertial frames the mathematics and ideas become closer to general relativity. In some sense, non-inertial frames in special relativity get you half-way to general relativity.

Anyway, that the motion of test particles along geodesics was an assumption is not important, what matters is that this assumption is inherently part of the theory.

Theories usually come with some kind of assumptions, your statement is quite consistent.

 

The theory is built upon a curved spacetime and deals directly with the motions in this 4D metric. It should be expected that equations of motion underlie the field equations.

I don't quite follow. In general relativity it is true that the equations of motion for massive (uncharged) test particles need not be postulated, but follows from a little geometry and the field equations. In hindsight, I agree that it should not be too surprising that such test particles follow geodesics as pure gravity manifests itself as the local geometry.

 

 

I didn't even read it, really, I'm just trying to point out that what you found out is not a novelty...

Yes, we all now acknowledge that this is well known.

 

 

...and as I said, I would be surprised if it was not possible to reduce the field equations to equations of motion...

That would be very interesting, but I am not confident that that is really possible. You can probe the causal structure of the space-time in that way and attempt to reconstruct the metric. However, reconstructing the metric from the geodesics is not one-to-one. If do not specify a particular parametrisation of the geodesic then you are in the world of projective connections. This means you do not have a unique connection associated with a particular (unparamaterised) geodesic.

 

Anyway, you want to argue that general relativity is the only theory that permits structureless massive test particles to move along time-like geodesics. I don't know if you can do that and I don't know what further assumptions you would need. For example, the equivalence principle may help. But we speculate here.

 

since the field equations were developed to deal with relative motions in non-inertial frames of reference to start with.

This is not quite true, but okay. See my earlier comments.

 

And of course GR must be consistent with SR which had to be consistent with Galileo, hence, GR should agree with Galileo and Newton under specific circumstances. Isn't this logical?

Yes, and we know how to take the Newtonian limit in general relativity.

 

Are you saying that the equation of motion you derived exists by accident in the field equations?

This made me smile.

 

What we can say is that it is not obvious straight from the start that the field equations directly include the geodesic equation as the equation of motion of a massive test particle. I don't see that it is obvious that general relativity is the only possible theory to have the feature of including the EQM of test particles in its field equations.

 

What we do know is that typically one needs to include equations of motion for the test particles in addition to field equations. Typically this may be in the form of a Lagrangian for both particles and fields. It is atypical for just the Lagrangian of the field to give you exactly the equations of motion for test particles "for free". This I suspect is why Einstein missed this in the early papers, there were no similar examples.

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Ok then, agreed.

Just a little remark, SR couldn't deal with non-inertial frames originally, it was only after Einstein started working with Minkowski metric that it was possible to do so. GR was the attempt to generalise SR to describe gravitation consistently, but prior to that, SR was simple algebra and the postulates only made sense in inertial frames.

 

But I agree, and I don't know of another theory that includes the equations if motion like GR does. Where do you believe they entered the field equations from?

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Just a little remark, SR couldn't deal with non-inertial frames originally, it was only after Einstein started working with Minkowski metric that it was possible to do so. GR was the attempt to generalise SR to describe gravitation consistently, but prior to that, SR was simple algebra and the postulates only made sense in inertial frames.

I don't know the exact history, but it is clear that one needs to think more geometrically than "standard" when dealing with special relativity in non-inertial frames. This is not really a thread about history, so we can use the knowledge of 100+ years of special relativity as we see fit.

Where do you believe they entered the field equations from?

It seems to be a consequence of the the Bianchi identities and "\nabla T =0". This is a mixture of geometry and physical ideas.

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I don't know the exact history, but it is clear that one needs to think more geometrically than "standard" when dealing with special relativity in non-inertial frames. This is not really a thread about history, so we can use the knowledge of 100+ years of special relativity as we see fit.

This is not a matter of history. Operating SR in non-inertial frames leads to the falsification of SR's postulates, especially on the constancy of the speed of light, and the contradiction is only resolved with GR. You can use non-inertial frames in SR only by neglecting Einstein's postulates and axioms.

 

It seems to be a consequence of the the Bianchi identities and "\nabla T =0". This is a mixture of geometry and physical ideas.

We could try the same logic on the Lorentz-invariant theory of gravitation and see if it also works (and if it is unique to GR). Anyone? My skills handling these equations are equivalent to the skills of a newborn handling a pen.

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Operating SR in non-inertial frames leads to the falsification of SR's postulates, especially on the constancy of the speed of light, and the contradiction is only resolved with GR. You can use non-inertial frames in SR only by neglecting Einstein's postulates and axioms.

 

 

The postulates of special relativity are statements specifically made with reference to inertial coordinate systems. Using more general coordinate systems does not invalidate the postulates as such, but it does make it harder to extract the meaningful physics. One has to be very careful about making statements in non-inertial frames and how they relate to out understanding in inertial frames. But still you are free to use any coordinate system on Minkowski space-time you like.

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We could try the same logic on the Lorentz-invariant theory of gravitation and see if it also works (and if it is unique to GR). Anyone? My skills handling these equations are equivalent to the skills of a newborn handling a pen.

 

I probably should have said that this property appears to be unique to metric theories of gravity. The equations of motion for Brans-Dicke theory, Nordstrom's (second) theory of gravity, etc. can all be derived from ∇aTab = 0, which holds in general for theories with diffeomorphism invariant actions.

 

LITG doesn't have the property that the equations of motion can be found "in" the field equations by analogy with Maxwell's equations, which also don't display this property. LITG is essentially the same theory as classical electromagnetism.

Edited by elfmotat
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If it is the spacetime metric that allows the field equations to spit out equations of motion, Minkowski must have something to do with it. If this is so, I would like to know specifically what step in his derivation is the source of this feature.

The first thing that comes to my mind is that time was brought into the metric as a fourth dimension through ct, but if x=ct, and x is one of the 3 spatial dimensions, it follows that ct is a second measurement of (euclidean) a spatial dimension. This is ok because time can be thought of as a secondary measurement of space. We can measure time by comparing two subsequent positions of a point moving in space and let the interval of change between states stand for time.

This implies that the metric automatically turns any position into a velocity, and any velocity into an acceleration. If a spacetime coordinate represents two subsequent spatial coordinates, we have the units of a velocity (even if v=0, but hardly if t=0). Likewise, a velocity would describe an acceleration. This would mean that any spacetime coordinate inherently represents motion in euclidean space, but that can only be so if the metric can describe motion mathematically, in which case we may assume that it does so via equations of motion, either implicitly or explicitly.

 

Does this make any sense?

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