Is gravitation faster than light?

[JesseM]

Waves are only created when an object accelerates.

Since gravitational waves depend on the quadrupole moment, constant acceleration won't produce them, only a changing acceleration will produce them. Electromagnetic waves depend on the dipole moment, so any change in velocity of a charge (acceleration) will produce electromagnetic waves.

All around our planet is the vacuum. Move far enough away from suns, and planets, and everything else, and it's pretty empty. There isn't much in it. What a field is, is some kind of warping or curving of that flat, empty vacuum. Warp it this way, you get gravity; warp it that way, you get an electric field.

Classical fields like electromagnetism are not traditionally understood in terms of changing the curvature of space, they're just force vectors attached to every point in ordinary flat space. Are you thinking of the Kaluza-Klein theory of electromagnetism?

Now, these warpings or curvatures are continuous; they aren't periodical in space, like a water wave is periodical. There aren't peaks and troughs; there are just levels of strength in this particular... direction (although it isn't any direction you can imagine) that aren't zero, or aren't as close to zero as they could be.

I'm not sure if there's actually a way to get curvature of spacetime to correspond to the idea of a force pulling in a particular direction--as I understand it, GR is really a fundamentally different picture of gravity, one that doesn't involve "forces" at all. Those schematic diagrams where they show gravity wells as depressions in a rubber sheet are a bit misleading, it's not like there's anything pulling "down" on objects as they move along the sheet, you could just as easily represent gravity wells as humps rather than depressions--the idea is that objects moving in the absence of other forces always follow a "geodesic" path, which on a curved 2D surface would mean the path with the shortest distance (like a segment of a 'great circle' on a globe), but in curved spacetime means the path with the greatest proper time (this is explained a little on this page).

 

[Nereid]

Per CarstenDierks' request (which I am fully in accord with), we're now in SR & GR.

To the regular contributors to this thread: I think you'll now find that most of your questions and issues will be at least grounded in the modern understanding of GR ...

 

[pervect]

I'm not sure if there's actually a way to get curvature of spacetime to correspond to the idea of a force pulling in a particular direction--as I understand it, GR is really a fundamentally different picture of gravity, one that doesn't involve "forces" at all.

Under the proper circumstances, the curvature of spacetime does reduce rigorously to the idea of a force. Very strong fields or very high velocities prevent this approximation from working rigorously, however.

To give the gory details, the geodesic deviation equation gives the equation of a path that a freely falling particle of negligible mass will take. This is

$$\frac{d^2 x^a}{d \tau^2} + \Gamma^a{}_{bc} \frac{dx^b}{d \tau}\frac{dx^c}{d \tau} = 0$$

When spatial curvature is negligible, on can apprxomiate this as

$$\frac{d^2 x^a}{d \tau^2} + c^2 \Gamma^a{}_{00} + 2 c \Gamma^a{}_{0b} \frac{d x^b}{d \tau} = 0$$

This throws out all the space-space curvature terms.

This means that the geodesic path of a particle acts just as if it experienced a pair of forces on it - a velocity independent force due to the $$\mbox{\Gamma^0{}_{00}}$$ term plus a velocity dependent force due to the second term. The first term, which is Newtonian gravity, can be considered to be analogous to the Columb force on a charge. The second term can be considered to be analogous to the magnetic force on a moving charge.

The anaology between weak-field gravitation an electromagnetism is sufficiently good that one can write a variant of Maxwell's equations for the two "force" components.

A good reference for this, if one can get a hold of it, is "Analogy between general relativity and electromagnetism for slowly moving particles in weak gravitational fields" by Harris. Thanks to Pete for pointing this paper out to me, even if he DID lose his copy :-).

To skip around a bit and address some of the questions that other posters made after skim-reading this thread

In standard general relativity gravity propagates at exactly 'c', the speed of light.

It's worthwhile checking out the sci.physics.faq How does gravity get out of a black hole

Note that neither light nor gravity waves can escape from a black hole, but both the electrostatic columb force of a charged black hole and the gravitational force of a massive black hole can escape. This has nothing whatsoever to do with the speed of gravity, though, as can be seen from the fact that the columb force also escapes a charged black hole. Note also that the only information that escapes from a black hole is it's charge (which generates the columb force of a charged black hole), and its mass (which generates the gravitational force).

 

[CarstenDierks]

Hi Nereid,

Thank you for moving. I hope we can get some additional interesting input in this group.


Schneibster and all others:

Thank you for the answers. I am still keeping up reading and looking up some papers on virtual particles, fields and the like.

However, before moving on, I have still a problem with some statements in prior post that seem contradictory to me.

So I would like to clarify some questions first:

(1) Gravitation curves spacetime.

(2) The curvature of spacetime (associated with gravitation) is only evoked by gravitational waves. A gravitational wave can constitute, aggravate, impair or erase the curvature.

(3) Every change in the extent of the curvature has to be evoked by a gravitational wave.

(4) If mass particles move, the curvature moves with them.

(5) Gravitational waves are only evoked if mass is in accelerated motion.
or:
(6) Gravitational waves are only evoked by changes in the acceleration of mass?

(7) Out of (2), (3), (4), (5) and (6): How does the curved spacetime of an object in linear motion move along with the object (since no gravitational waves are send out to "update" the "curvature information")?

I hope someone knows is a simple solution to this. Schneibster, do you?

Carsten

 

[JesseM]

(1) Gravitation curves spacetime.

You might also say that mass/energy curves spacetime, and that what we call "gravitation" is really just an effect of the fact that all objects follow geodesic paths through curved spacetime (a geodesic is the closest thing to a 'straight line' in curved space or curved spacetime, like how the shortest distance between two points on a sphere is determined by the great circle which passes through both points). As pervect explained above, in certain limits these geodesic paths will look just like the paths you'd get if you imagine gravity as a force pulling on objects in flat spacetime.

(2) The curvature of spacetime (associated with gravitation) is only evoked by gravitational waves. A gravitational wave can constitute, aggravate, impair or erase the curvature.

What do you mean by "only evoked by"? Spacetime can be curved by systems which are not generating any gravitational waves.

(3) Every change in the extent of the curvature has to be evoked by a gravitational wave.

No. It's easier to think about this in analogy with electromagnetism--any acceleration of a charge will produce electromagnetic waves, but a charge moving at constant velocity will not. For a charge moving at constant velocity, the electromagnetic field is changing--other charges will act as though they are always attracted to the current position of the moving charge. This does not imply the force is acting faster than light though, it's more as if the the electromagnetic field can "extrapolate" the position of a charge moving at constant velocity, so if the charge suddenly accelerates, other charges will continue to act as if they are attracted to the position the charge would have been if it had continued to move at constant velocity, until they receive an "update" in the form of an electromagnetic wave traveling at the speed of light, which was generated by the original acceleration.

In a similar way, the curvature of spacetime can apparently "extrapolate" constant acceleration (including 0 acceleration, or movement at constant velocity)--it is only when there is a change in the rate of acceleration that gravitational waves are generated.

(4) If mass particles move, the curvature moves with them.

Yes, at least locally.

(5) Gravitational waves are only evoked if mass is in accelerated motion.
or:
(6) Gravitational waves are only evoked by changes in the acceleration of mass?

#6.

(7) Out of (2), (3), (4), (5) and (6): How does the curved spacetime of an object in linear motion move along with the object (since no gravitational waves are send out to "update" the "curvature information")?

Again, you can think of it as a sort of "extrapolation". Presumably you could prove this by looking at solutions to the equations of GR involving masses moving at constant velocity (or at constant acceleration).

 

[pervect]

So I would like to clarify some questions first:

(1) Gravitation curves spacetime.

Yes. Or, alternatively, gravitation is curved space-time.

(2) The curvature of spacetime (associated with gravitation) is only evoked by gravitational waves. A gravitational wave can constitute, aggravate, impair or erase the curvature.

(3) Every change in the extent of the curvature has to be evoked by a gravitational wave.

No, to both of the above. Gravitational waves are important only under the most extreme conditions. Gravity can, and does exist as a curvature of space-time without the existence of gravitational waves.

Think of electrostatics. A pair of unlike charges attract each other. It is not necessary for an actual electromagnetic wave to exist for like charges to attract. Do not confuse actual electromagnetic waves with 'virtual" particles or waves.

 

[pervect]

Gravity & Electromagnetism - a comparison, and a perspective of "how gravity propagates".

Let's start with electromagnetism. There are a couple of ways to describe the electromagnetic field, but one of the simplest is in terms of the electric and magnetic fields. The electromagnetic field at any point in space-time has six components - 3 components of the electric field, and 3 components of the magnetic field. The fields arise ultimately from charges and the motion of charges. It takes 4 variables at any point in space-time to describe the charge and the motion of charge (current) there.

These six components of the electromagnetic field must satisfy a set of linear differential equations called Maxwell's equations that describe how the field variables interact with each other and with charges. The wave-like properties of light, the fact that light always travels at a speed less than or equal to c (exactly equal to c in empty space, lower in regions where there is matter), and the fact that any general changes in the electromagnetic field propagates at a speed less than 'c' are all determined by the nature of these linear differential equations.

Now, let's look at gravity. Instead of 6 variables to describe the electromagnetic field, one has 20 indpendent quantities which make up the Riemann curvature tensor, which describes the curvature of space-time at a individual point. Alternatley, one can describe space-time and it's local curvature by the 10 independent components of the metric tensor. The metric tensor is probably a little easier to grasp than the Riemann curvature tensor - it describes how one computes the distance between any two points. The 10 components of the metric tensor entirely determine all 20 components of the Riemann curvature tensor. This happens because the 20 components of the Riemann curvature tensor satisfy a set of differential equations known as the Bianchi iidentity.

There's more that can be said to make the Riemann tensor a bit more intuitive, but unfortunately this post is already getting to be a bit on the long side, so I'll leave this material out.

The source of the gravitational field is not charge, but energy - any sort of non-gravitational energy. Usually, though, the energy in the rest mass of matter dominates all other forms of energy. The description of the energy distribution in GR is given via the stress-energy tensor, which requires 10 variables at any point in space time - quite a bit more than the 4 that it took for electromagnetic theory.

Einsteins' equation, which describes gravity, is a NONLINEAR differential equation which relates the 20 components of the Riemann curvature tensor to the 10 components of the stress energy tensor at any point in space time. This is very similar to the way that Maxwell's equations relate the 6 electric and magnetic field comonents to the 4 components of charge and current at any poitn in space-time.

The non-linearity is an important difference between Einstein's equation and Maxwell's equation that makes gravity a lot harder. However, because these differential equations form a "quasi-linear, diagonal, second-order hyperbolic system" (whew!) it turns out that the solution of these differential equations locally exhibit much of the same "wave-like" properties that the solutions of Maxwell's equations do - specifically, changes in the field configuration always propagate at a speed slower than some value, 'c'.

[add]
http://www.physlink.com/Education/AskExperts/ae98.cfm
was a useful web source/refrerence for some (not all) of the points above.

 

[yogi]

Field vs Wave - some previous commentors have asserted authoritatively that waves, unlike static fields, must dissipate energy. Faraday first envisioned EM waves as fields that detached themselves - as such the wave retains its field energy as it propagates through a perfect medium - there is nothing inconsistent with the notion of a wave, wavelet, photon, or a deBroglie matter wave being able to communicate its intrinsic conservative field energy into some form of force.. e.g., in the case gravity - perhaps a pressure wave. If I recall, Caroline is a advocate and contributor to the inflow theory of gravity which depends upon a real aether.

Almost every force I can think of is somehow connected to a dynamic - motion of something, as Dirac once commented when faced with having to describe the vector potential in free space.

 

[pervect]

Field vs Wave - some previous commentors have asserted authoritatively that waves, unlike static fields, must dissipate energy.

I must have missed that part. Standard theory does not predict that electromagnetic waves in a vacuum lose energy, for instance - though they can and do "spread out", so their energy also spreads out over a larger volume. It sounds to me like it is being proposed that the energy in the waves is disappearing somehow, which doesn't sound like a very promising theory as it violates the conservation of energy in a major way.

Faraday first envisioned EM waves as fields that detached themselves - as such the wave retains its field energy as it propagates through a perfect medium - there is nothing inconsistent with the notion of a wave, wavelet, photon, or a deBroglie matter wave being able to communicate its intrinsic conservative field energy into some form of force.. e.g., in the case gravity - perhaps a pressure wave. If I recall, Caroline is a advocate and contributor to the inflow theory of gravity which depends upon a real aether.

If we have a positive charge and a negative charge sitting there in free space, attracting each other, but held back so that they are not accelerating, it is not standard to say that there are "waves" involved in the attraction. There are no actual light waves that can be detected in such a case (as with a camera, a radio receiver, or some other instrument that detects electromagnetic radiation).

There are some ways of looking at this attraction that involve "virtual particles", but it's not very much used for actual calculations. Hence my note about not confusing "real" waves with "virtual" ones.

The same applies to gravity. A pair of masses just sitting there attracting each other are not going to be emitting actual gravity waves, of the sort that could be detected with a gravity wave detector (LIGO, or one of its successors).

Onto the next point I want to talk about - differential equations.

The usual notion of physics relies heavily on differeintial equations. I hope this isn't scaring anybody, differential equations are sometimes as simple as

f = ma

This is a differential equation, because the accleration is the second derivative of the position. Usually the force is a function of position, and because of Newton's law above, the acceleration is proportional to the force. This means the second derivative of the position is a function of the position. Thus we have - a differential equation. Newton's laws are just differential equations.

Newton's laws are differential equations, and so are Maxwell's equations, Just about all of physics is differential equations. General relativity is not any different, in spite of the fact that it deals with some unusual notions like curved space-time. When you get behind curved space-time to the math that describes it, you see differential equations, just like the rest of physics. For instance, we say that mass in general relativity travels along a geodesic, which is a litle more general than saying that it experiences a force. How do we describe a geodesic? You guessed it (I hope!) - we describe a geodesic with a differential equations.

 

[yogi]

pervect - Good post re differential equations. I was referring to post #51 - I think I know what the author of it is trying to say - but as worded it implies that the EM wave is losing energy to the environment - that may happen with ocean waves where a close look reveals only an up and down motion of the particles in a friction medium. Of course if you look even closer, you will get your face wet.

 

[Garth]

Standard theory does not predict that electromagnetic waves in a vacuum lose energy, for instance - though they can and do "spread out", so their energy also spreads out over a larger volume. It sounds to me like it is being proposed that the energy in the waves is disappearing somehow, which doesn't sound like a very promising theory as it violates the conservation of energy in a major way.

But in the standard theory do not photons lose energy with cosmological/gravitational red shift?

That theory, GR, conserves energy-momentum, i.e. particle 'rest' masses rather than energy.

If we treat the electromagnetic waves as quanta, individual photons, then the photons from a distant galaxy or quasar are emitted at one frequency $$\nu$$ with energy $$E = h\nu$$ and are received much later in cosmological time with a smaller frequency and presumably a smaller energy. They have traveled across space-time, in zero proper time, with no forces acting on them and no work done on or by them, so where has their energy gone?



Garth

 

[pervect]

But in the standard theory do not photons lose energy with cosmological/gravitational red shift?

GR does, of course, have its own issues with energy conservation, which you and I have talked about quite a bit.

For those who came in late and are still with us, the sci.physics.faq

Is energy conserved in General Relativity

is a good reference to the issue of energy conservation in GR.

However, there are several extremely important concepts of energy conservation that GR does have that the theory being prposed seems to lack.

The first concept that GR has is the concept that the divergence of the stress energy tensor is zero. This is called by many the "local conservation of energy", though this terminology seems to confuse mathemeticans. (See Wald pg 286 for an example of this usage). This is the differential form of the energy conservation law as described by the sci.physics.faq reference.

The second concept of energy conservation in GR applies only in asymptotically flat space times, and on the cosmological scale space-time in the actual universe isn't asymptotically flat, so this notion doesn't apply.

A third notion of energy conservation requires static spacetiems, and also doesn't apply.

However, the proposed theory of light in free space losing energy "just because" doesn't seem to have any notion whatsoever of energy conservation in any sense whatsoever - a serious lack, IMO.

 

[Garth]

However, the proposed theory of light in free space losing energy "just because" doesn't seem to have any notion whatsoever of energy conservation in any sense whatsoever - a serious lack, IMO.

A serious lack IMHO as well!

Consider model GR universe filled with a photon gas - the CMB - with no matter at all, i.e. the completely radiation dominated universe. The energy of each photon decreases inversely proportionally to the scale factor of the universe:-
For each photon:-

As lambda_p = lambda_p0R(t)/R₀

E_p = h.nu_p = h.c/lambda_p = E_p0.R₀/R(t).

But if photon number is conserved then the total energy contained in the photon gas decreases inversely with R(t).

Where has the energy gone? A standard answer might: be into the energy absorbed by the expansion of the universe; but how?

Each photon is traveling along a null geodesic with no forces acting on it and no work done on or by it, so what is the mechanism by which it exchanges energy with the gravitational field? Does not the exchange of energy require a mediating force and would not such a force acting on the photon be a violation of the equivalence principle?

Just a thought or two.

Garth

 

[pervect]

A serious lack IMHO as well!

Consider model GR universe filled with a photon gas - the CMB - with no matter at all, i.e. the completely radiation dominated universe. The energy of each photon decreases inversely proportionally to the scale factor of the universe:-
For each photon:-

As lambda_p = lambda_p0R(t)/R₀

E_p = h.nu_p = h.c/lambda_p = E_p0.R₀/R(t).

But if photon number is conserved then the total energy contained in the photon gas decreases inversely with R(t).

Where has the energy gone? A standard answer might: be into the energy absorbed by the expansion of the universe; but how?

Each photon is traveling along a null geodesic with no forces acting on it and no work done on or by it, so what is the mechanism by which it exchanges energy with the gravitational field? Does not the exchange of energy require a mediating force and would not such a force acting on the photon be a violation of the equivalence principle?

Just a thought or two.

Garth

Yes, this is a rather disturbing notion. On the other hand, if you take a little tiny cube of flat space-time, the rate of change of energy stored in the cube is still equal to the net inflow or outflow of energy through the faces of the cube - the differential conservation law tells us this. And you can always find a coordinate system where the space-time is flat. (Or you can use the more general form of the theorem where one uses the covariant derivative and not even need the notion of flat space time to make the same point, i.e.)

$$\nabla^a T_{a0} = 0$$

rather than taking the divergence of the stress energy tensor in a locally flat coordinate system.

Note that one can replace the zero with an arbitrary index - the zero index makes statements about energy conservation, a non-zero index makes statements about momentum conservation.

So this form of the conservation law is saying that yes, the energy density at any point in space-time is going down, but it's going down because energy is flowing out of the cube as the universe expands. This is not too surprising, if we have a fixed volume cube, and the universe expands, and the energy distribution is isotropic, that the energy has to be flowing out of the cube to fill all of space as the universe expands, and the energy density in a cube of fixed volume is going down as time goes down.

I'm not sure exactly how to reconcile these two points of view at this moment. I'm a bit suspicious of the assumption that the photon number is constant, but I'm not totally convinced this is the real explanation yet.

[add]
I think there's a much better explanation. The energy is going into "the gravitational field". The gravitational self energy is higher when the universe is smaller than when it is larger. This gravitational self-energy doesn't show up in the differential conservation law, it only shows up in finite volumes as discussed in the physics FAQ. I don't think there is anyway to make this idea rigorous, though, unless I'm wrong about not finding any timelike Killing vectors in the flat FRW spacetime. Without asymptotic flatness or a timelike Killing vector, there's no way that I know of to rigorously define the energy in the gravitational field.

 

[yogi]

Maybe not the G field as such - but if the universe is under tension, expansion will increase the stress energy - the loss of energy in the photon gas may correspond to the energy gained by the spatial volume.

 

[pervect]

I've been reading up on this point a little more.

It looks like that during the matter dominated phase, when the pressure of the universe is nearly zero, (density of mass energy) * (volume) remains essentially constant. But it's definitely not a standard idea to interpret this number as the "mass of the universe". Basically the FRW universe just doesn't fit any of the standard conditions in GR where an energy can be defined over a finite volume (asymptotic flatness, or staticity). Note that the differential energy consevation law still works without a hitch, though.

When the pressure isn't zero (this happens when there is a gravitationally significant amount of radiation), the above function isn't constant.

If we let PV = (density of mass energy) * (volume)

then d(PV)/dt = -(pressure) d (volume)

MTW, pg 705 is the reference for the above.

So what we have in the matter dominated era is the universe expanding, and the energy density per unit volume going down, to maintain a constant product. However, we resist calling this constant product the total energy of the universe.

In the case where radiation is present, the universe does sort of "cool off" as it expands, a lot like any expanding gas.

 

[Chronos]

Only problem with that is the universe was not matter dominated in the early inflationary phase, it was radiation dominated.

 

[pervect]

Only problem with that is the universe was not matter dominated in the early inflationary phase, it was radiation dominated.

I'm not sure what problem you are referring to Chronos. If your problem is that PV is not constant during the early radiation phase, I agree. Further reading gives the result that (pressure)*(volume)*(4/3) is a constant during the early radiation dominated phase, when we make the plausible assumption that all pressure is due to radiation.

This, then, is a good reason not to regard PV as the "mass of the universe". (I already mentioned that this wasn't a good idea, but it probably isn't a bad idea to make this point more explicit).

I've gotten off track, and fumbled around a bit (it was educational) - but where I started was with the issue of how to reconcile cosmological energy loss with the differential conservation law. The fact that the energy lost is proportional to the volume is sufficient to allow the reconcilliation. As we take the limit to zero volume, the energy loss goes to zero, so there is no conflict with the differential conservation law.

This also means that if we take a small enough piece of space-time, the energy loss can be neglected. In the current epoch of the universe, because the radiation pressure is so low, a "small enough" piece of space-time is quite large even on a cosmological scale. On a human scale, the volume required for cosmological energy loss is enormous. This means that our everyday expeiences and experimental results that show energy being conserved are not impacted by the cosmological issues - the cosmological issues are real, but don't have any impact on experiments carried out on a human timescale.

 

[Garth]

This also means that if we take a small enough piece of space-time, the energy loss can be neglected

Are you happy with that? It seems that anything can be neglected if you make it small enough, however it is the whole universe that we are supposed to be dealing with here!

Adding pressure does not resolve the energy problem, it deepens it; a Friedmann dust universe does preserve mass-energy - because density ~ R^-3, however pressure makes the expansion, counter intuitively, harder and slows it down as it adds another form of energy to the system. The density now decreases faster and mass is absorbed by the cosmic expansion, density ~ R^{-(3 + a)}, and M = density x volume; but by what mechanism? Do atoms or dust particles simply disappear into the 'aether'?

Garth

 

[pervect]

Are you happy with that? It seems that anything can be neglected if you make it small enough, however it is the whole universe that we are supposed to be dealing with here!

Adding pressure does not resolve the energy problem, it deepens it; a Friedmann dust universe does preserve mass-energy - because density ~ R^-3, however pressure makes the expansion, counter intuitively, harder and slows it down as it adds another form of energy to the system. The density now decreases faster and mass is absorbed by the cosmic expansion, density ~ R^{-(3 + a)}, and M = density x volume; but by what mechanism? Do atoms or dust particles simply disappear into the 'aether'?

Garth

Well, I'm not exactly "happy" with that, but the universe isn't constrained to operate in a manner that pleases me :-). There are certain aspects of quantum mechanics that disturb me more than the energy problem, actually. Like particles passing through both slits in a two-slit experiment and interfering with themselves, for example.

It's possible that there is some sort of missing "scalar field", or as in your SCC theory, or some other sort of non-scalar field that accounts for the "missing" energy - but we don't have any evidence for such a thing, yet. It's certainly worth looking for.

It's also possible that energy conservation is only approximate. We've seen a lot of other symmetries in physics that broke down under extreme enough conditions. The universe did have to get created somehow, after all - and if energy is conserved, we either have to believe that the energy of the universe is zero, or that energy conservation can be violated somehow or other.

It is somewhat useful to be able to put some sort of bounds on how much energy is being "lost", though, though of course the answer is coordinate system dependent,

 

[yogi]

Some interesting relationships are arrived it by considering inflation to be an on going phenomena - the universe then is in a state of increasing negative pressure - energy is continually added in the form of spatial stress - the total energy is proportional to the surface area of the Hubble Sphere, the density is proportional to 1/R, there is no singularity at the beginning, the total energy (negative potential plus stress) is always zero ...