What Determines the Speed of Gravitational Waves?

[WannabeNewton]

Hey guys. So in the einstein gauge the linear field equations can reduce to the wave equations:
\partial^{2} _{t}h_{ab} - \nabla^{2}h_{ab} = 0 where here h_{ab} is the symmetric tensor containing the deviation of the gravitational field from minkowski space for the weak field. How can one tell from this that gravitational waves propagate at light speed? I guessed and thought it was because the laplacian is usually followed by a c^{2} for the speed of the wave and in this case c^{2} is 1 and since this is in geometric units I assumed that this was the indicator that gravitational waves travel at light speed. Of course I am probably completely off so if you guys could tell me why they do travel at said speed I would really appreciate it. Thanks.

 

[bcrowell]

What you said sounds right to me.

 

[GTOM]

Hi. Well I have to say I'm not a physicist but I don't really understand, why should gravity waves only travel with the speed of light.
Gravity can bend the light, even stop it beyond the event horizont of a black hole. Than why should it obey to the laws of electromagnetic waves?
How do you define the speed of light, when gravity makes a four dimensional curvature, and even meter and second isn't the same in that coordinate system, as in ours?
What proofs do we have, that tells us, that gravity waves can only propagate with a speed of c?

 

[Phrak]

Forget about the velocity of light. In all of general relativity there are only two speeds: zero and c, and I'm not so sure about zero. If you come up with anything else--then you are not taking the weak field limit.

 

[GTOM]

As i said, I'm not a physicist, just a curios guy.
Can i ask you to explain that weak field thing to me?
What does GR tells us about black holes, where gravity really stops the light?
Yet there are things, that coming out from them. (Hawking radiation.) And the effects of gravity are also "comes out" from them.

 

[Forestman]

Hey GTOM

Basically gravity causes time dilation too. Anyway at the event horizon of a black hole gravity becomes so strong and time dilation so extreme that time stops relative to an outside observer, but if you were to go into the black hole time would pass normally for you.

About Hawking radiation. Virtual particles are always being created everywhere, and then disappearing. Virtual particles always come in pairs. A particle, and an anti particle. When virtual particles are created near a black hole sometimes one gets pulled in, while the other gets left behind. Since the virtual particles can no longer disappear, since they cannot come back together, they must become real particles. As a result mass is stolen from the black hole in a process that I don't quite understand that turns the virtual particles into real particles. When this happens we see the virtual particle that did not get pulled in seeming to come out of the black hole, but it did not, it was just created right outside of the black hole. And as a result of its creation mass was stolen from the black hole.

 

[Phrak]

As i said, I'm not a physicist, just a curios guy.
Can i ask you to explain that weak field thing to me?
What does GR tells us about black holes, where gravity really stops the light?
Yet there are things, that coming out from them. (Hawking radiation.) And the effects of gravity are also "comes out" from them.

Don't ask me. In my opinion you are asking how many angles can dance on the head of a pin.

 

[GTOM]

"As a result mass is stolen from the black hole in a process that I don't quite understand that turns the virtual particles into real particles."

I don't understand that, neither. :(


Anyway, what proofs do we have, that tells us, gravity also propagates with c? Not theories, proofs?

 

[tom.stoer]

In curved spacetime it is no longer possible to define "velocity" for distant objects; so velocity or "speed" can only be defined locally. But locally one can always chose a frame of reference which looks (locally!) like flat Minkwoski space. So one can "transform away" all gravitational effects.

But in flat Minkowsk space all symmetry principles known from special relativity available are valid (again locally). That means that local Lorentz invariance holds, i.e. the "generalized length" of a 4-vector (t,x) is invariant w.r.t. Lorentz transformations. That means that s²=(ct)²-x² is a Lorentz scalar.

So from this space-time symmetry one can derive that there is a unique constant c - which is not only the speed of light but the speed of all wave-fronts (classically; in quantum theory one would have to talk about quantum particles like photons). That means that all wave equations will look like vacuum wave equations with one preferred "speed of propagation". If there would be a wave equation with a different constant c' it would immediately violate Lorentz invariance defined via the original speed c.

This wave equation is uniquely defined via the differential operator mentioned in the first post which is nothing else but a "transformation
of s²=(ct)²-x² into momentum space".

 

[GTOM]

Thanks for the answer, but i still have questions.

"In curved spacetime it is no longer possible to define "velocity" for distant objects; so velocity or "speed" can only be defined locally."

So in this case, c won't remain ~3*10^8 m/s? Most people associates this value with the speed of light.

"So from this space-time symmetry one can derive that there is a unique constant c - which is not only the speed of light but the speed of all wave-fronts (classically; in quantum theory one would have to talk about quantum particles like photons)."

I read about an experiment, where they split a quantum in two halves, and when they changed one half, the other one changed instantly. Of course, they only realized it, when they compared data.

Ok, Lorentz invariance is really true for electromagnetic waves, but it seems to me, that gravity, and things between quantums are a different table.

 

[tom.stoer]

"In curved spacetime it is no longer possible to define "velocity" for distant objects; so velocity or "speed" can only be defined locally."

So in this case, c won't remain ~3*10^8 m/s? Most people associates this value with the speed of light.

Measuring the speed of distant objects can indeed result in different measured values; even the speed of light may appear as if light itself slows down; see Shapiro delay http://en.wikipedia.org/wiki/Shapiro_delay. But that does not mean that local measurements will result in different measured values; it only means that velocity or speed loses its global meaning.

"So from this space-time symmetry one can derive that there is a unique constant c - which is not only the speed of light but the speed of all wave-fronts (classically; in quantum theory one would have to talk about quantum particles like photons)."

I read about an experiment, where they split a quantum in two halves, and when they changed one half, the other one changed instantly. Of course, they only realized it, when they compared data.

True, but this quantum effect does neither transport energy nor information and does not violate Lorentz invariance.

Ok, Lorentz invariance is really true for electromagnetic waves, but it seems to me, that gravity, and things between quantums are a different table.

In some sense yes, but not with respect to Lorentz invariance. Relativistic quantum field theory respects Lorentz invariance for all known non-gravitational interaction. For gravity global Lorentz invariance does no longer hold due to spacetime curvature, but locally gravity / GR respects Lorentz invariance as well. For quantum gravity we still do not now - we neither have a fully established theory nor reliable data.

 

[GTOM]

"But that does not mean that local measurements will result in different measured values; it only means that velocity or speed loses its global meaning."

Well, it seems to me, that its kind of philosopy, how we see things. To me, velocity and distance has to have a global meaning, it is our instruments, that are not flawless, as they use EM waves, and what effects EM waves, also effects the instruments.
But i understand and accept your viewpoint.

"True, but this quantum effect does neither transport energy nor information and does not violate Lorentz invariance."

Is it quite sure, it cant' deliver information? For example, we can only detect neutrino when it hits something. Maybe we could use quantum dislocality to alter a quantum half, so it will hit something (what it shouldn't hit, unless altered), and then we transferred a bit.

Thanks for the rest. :)

 

[WannabeNewton]

"But that does not mean that local measurements will result in different measured values; it only means that velocity or speed loses its global meaning."

Well, it seems to me, that its kind of philosopy, how we see things. To me, velocity and distance has to have a global meaning, it is our instruments, that are not flawless, as they use EM waves, and what effects EM waves, also effects the instruments.
But i understand and accept your viewpoint.

"True, but this quantum effect does neither transport energy nor information and does not violate Lorentz invariance."

Is it quite sure, it cant' deliver information? For example, we can only detect neutrino when it hits something. Maybe we could use quantum dislocality to alter a quantum half, so it will hit something (what it shouldn't hit, unless altered), and then we transferred a bit.

Thanks for the rest. :)

Frame dependent quantities like velocity don't have a global value. Local co - moving reference frames on a manifold can have these quantities defined but if you look at the manifold on in its "Entirety" velocity doesn't have global meaning.

 

[tom.stoer]

@WannabeNewton: thanks for this clarification; but originally it was your thread / your question; are there remaining questions?

 

[Dale]

Gravity can bend the light, even stop it beyond the event horizont of a black hole. Than why should it obey to the laws of electromagnetic waves?

It doesn't. It follows the Einstein field equations, not Maxwell's equations.

 

[tom.stoer]

Gravity can bend the light, even stop it beyond the event horizont of a black hole.

I saw this sentence the first time. Gravity does NOT stop light. Even inside the event horizon light travels always with constant speed c - measured locally in a physical reference frame.

 

[ApplePion]

<<So in this case [curved space], c won't remain ~3*10^8 m/s? Most people associates this value with the speed of light. >>


No, the wave equation you wrote had coefficients attached to the second derivatives that actually are due to the values of the metric, values assumed to be Lorentzian. But, for example, in a region where the (0,0) metric component was 1.2 rather than 1, the coefficient for the second derivative with respect to time would now be 1.2 rather than 1.

So, approximately, the speed of a gravitational wave moving in the x direction would be the square root of (g00/g11).

There are also non-linear terms that are quadratic in the affine connection, so the situation involving an exact characterization is actually more complicated.

 

[Sam Gralla]

I think that already in my first week on these forums this is the second time this issue has come up. Maybe there should be a faq? In any case the point is that c really represents the maximum speed of information propagation, although for historical reasons it is referred to as the "speed of light". You can make this precise for hyperbolic wave equations, such as Maxwell's equation on a curved spacetime or the linearized Einstein equation (in Lorenz gauge), by proving that changing the initial conditions in one spot cannot change the field at spacetime locations "too far away to be reached in time". Locally near the area where the initial conditions were changed, the region that cannot be affected is that outside the ordinary light cone (speed c) of special relativity. Hence one says that fields cannot propagate "faster than light". Note that they can (and do) propagate slower than light, in that changes in initial data also lead to changes in the field within the light cone, not just on it. (Only a few special cases, such as Maxwell's equations on a flat spacetime, have the special propertiy that only the surface of the light cone is affected. This property is known as Huygen's prinicple and it does not hold in curved spacetime.)

By the way, a common misconception (even among professional theoretical physicists) is that the Tachyon equation (Klein-Gordon with negative mass squared) has faster-than-light propagation. In fact it still obeys the restrictions discussed above; it's just if you look at some "pattern speed" like phase or even group velocity, the pattern may move faster than c. So there is nothing wrong with the classical Tachyon equation and causality. (The quantized version, on the other hand, has some problems, but I'm not an expert there.)

Keeping information moving below some maximum speed is very important for preserving causality, which is about the most sacred principle in physics. So, we're all very happy that there is a maximum speed limit, c.

 

[ApplePion]

<< In any case the point is that c really represents the maximum speed of information propagation, although for historical reasons it is referred to as the "speed of light".>>

In a non-Lorentzian spacetime, speeds faster than 3 x 10^8 m/s are indeed possible.

Near the Earth the g00 value of the metric is approximately c^2 [1 + 2 phi/(c^2)] and g11 is approx 1 - 2phi/(c^2), where phi is the Newtonian gravitational potential.

For a radially moving beam of light the operative equation is: g00 times the second dervative of the electromagnetic 4-vector with respect to time + g11 times the second derivative of the electromagnetic 4-vector with respect to the radial coordinate = 0

Substituting the values for g00 and g11 into the wave equation given in the previous paragraph can yield a plane wave solution of the form f (r- vt) = constant, where v -= square root of c([1 + 2 phi/(c^2)]/ ([1 - 2 phi/(c^2)]) which is approximately c [1 + 2 phi/(c^2)].

So the speed would be a bit faster than c.

 

[Sam Gralla]

There are at least three things wrong with your argument, ApplePion. 1) You're computing a pattern speed, not an information speed 2) your "velocity" is a coordinate velocity dr/dt, 3) the gravitational potential is negative, so your velocity is less than c, anyway.

Since you're finding that the "speed" is modified by the redshift factor g_{00}, probably number 2) is the main issue here. This is the same type of confusion that could lead you to conclude that things don't fall into black holes, since your velocity would go to zero at the horizon of a black hole.

 

[GTOM]

"Gravity does NOT stop light. Even inside the event horizon light travels always with constant speed c - measured locally in a physical reference frame."

So how can it not come out from the black hole? We know gravity can bend light. But we couldn't notice the change if we measure locally neither.
Ok, let's say it doesn't stop light, just bends to itself. In other words, it has been stopped, or didnt?

"In a non-Lorentzian spacetime, speeds faster than 3 x 10^8 m/s are indeed possible."

Well most people can only imagine space in an ortogonal coordinate system. If we draw our solar system with ortogonal coordinates, and assume Sun is the origo, we can speak about absolute speeds in that frame.

"Keeping information moving below some maximum speed is very important for preserving causality, which is about the most sacred principle in physics. So, we're all very happy that there is a maximum speed limit, c."

So what about quantum dislocality? I mentioned an idea, that we couldn't send EXACT information through quantum halves faster than light, but maybe there is a chance, that we could use this phenonema to transfer binary information FTL.

 

[atyy]

http://nedwww.ipac.caltech.edu/level5/March01/Carroll3/Carroll6.html Eq 6.32

Also http://en.wikipedia.org/wiki/Pp-wave_spacetime

More on the interpretation of pp-waves are found in chapter 17 of http://books.google.com/books?id=Sb...&resnum=4&ved=0CDAQ6AEwAw#v=onepage&q&f=false . Griffiths and Podolský state that the covariantly constant null vector field means that there is a family of nulllines which are expansion-free, shear-free and twist free. So these nulllines are orthogonal to a family of 2 surfaces which we can consider wavefronts. Because the null vector field is covariantly constant, the wave fronts are planar, and the rays orthogonal to them are parallel.

The relationship between the perturbative and exact wave solutions are discussed on p210 of http://www.blau.itp.unibe.ch/lecturesGR.pdf

 

[ApplePion]

<<There are at least three things wrong with your argument, ApplePion. 1) You're computing a pattern speed, not an information speed>>

I have not heard the term "pattern speed" before. The speed I computed was both the group velocity and the phase velocity for linearized gravity waves in a non-Lorentzian metric.

<< 2) your "velocity" is a coordinate velocity dr/dt,>>

Yes, that is what it indeed is. Why do you consider that to be an error?

<< 3) the gravitational potential is negative, so your velocity is less than c, anyway.>>

I made a careless algebraic mistake. I got it as c times square root of (g00/g11). If one does the algebra more carefully one gets c times the sqyare root of (g11/g00)

<<Since you're finding that the "speed" is modified by the redshift factor , probably number 2) is the main issue here. This is the same type of confusion that could lead you to conclude that things don't fall into black holes, since your velocity would go to zero at the horizon of a black hole.>>

Indeed for a distant observer the speed does go to zero for an object approaching the event horizon. So the "problem" you claim is not something that was incorrect.

 

[ApplePion]

Me <<I made a careless algebraic mistake. I got it as c times square root of (g00/g11). If one does the algebra more carefully one gets c times the sqyare root of (g11/g00)>>

Just to make things clear--I am NOT agreeing with sqralla. I am saying that the speed does indeed exceed c, as I had originally claimed.

 

[Sam Gralla]

"I have not heard the term "pattern speed" before. The speed I computed was both the group velocity and the phase velocity for linearized gravity waves in a non-Lorentzian metric."

The "group velocity" (to the extent it is defined) refers to how fast some shape in the wave propagates, so some people call it a pattern speed. Anyway it is not a reliable indicator of the speed at which information propagates--it tells you the speed at which some pattern (shape) propagates. Wikipedia actually has a decent discussion of this. Physics classes sometimes teach that phase velocity is unreliable but group velocity is, but actually neither is reliable. The only sensible measure of "propagation speed" for a wave equation that I've ever heard of is the mathematical setup I described above, where you change the initial data and see that the solution changes only within a "light cone".

"<< 2) your "velocity" is a coordinate velocity dr/dt,>>

Yes, that is what it indeed is. Why do you consider that to be an error?"

If you choose some other coordinates, and compute dr/dt for those, you'll get a different answer. Which one represents the "speed of light propagation" in this spacetime? If you want to calculate something about light, you need to ask a question that doesn't depend on the choice of coordinates.

 

[ApplePion]

sgralla: << 2) your "velocity" is a coordinate velocity dr/dt,>>

Me <<Yes, that is what it indeed is. Why do you consider that to be an error?>>

sgralla <<If you choose some other coordinates, and compute dr/dt for those, you'll get a different answer. Which one represents the "speed of light propagation" in this spacetime? If you want to calculate something about light, you need to ask a question that doesn't depend on the choice of coordinates.>>

You are acting as if the speed of something should not depend on the coordinate system, when in fact it should.

A different coordinate system can indeed have a different speed for the wave. The metric components g00 and g11 will of course be different too, and thus the square root of g11/g00 formula still applies.

 

[Sam Gralla]

Okay, well at least we are in agreement that you are computing the speed of a shape in a particular coordinate system. The reader can now decide for himself whether this computation has anything to do with the "speed of light".

 

[tom.stoer]

"Gravity does NOT stop light. Even inside the event horizon light travels always with constant speed c - measured locally in a physical reference frame."

So how can it not come out from the black hole? We know gravity can bend light. But we couldn't notice the change if we measure locally neither.
Ok, let's say it doesn't stop light, just bends to itself. In other words, it has been stopped, or didnt?

If you travel into the black hole (e.g. free falling) you have the chance to make experiments and measure speed of light locally (in your free-fall reference frame). You will measure exactly the same speed of light as outside and even as in flat space.

That's what I am saying: "speed" loses its global meaning in curved spacetime; only local measurements (in a small region of space where curvature can be neglected) make sense.

 

[ApplePion]

<<Okay, well at least we are in agreement that you are computing the speed of a shape in a particular coordinate system. >>

Physical quantities often have different values in different coordinate systems. It is not as if only scalars are real physical quantities. It has meaning, for example, to say "I am driving my car at 75 miles per hour relative to the Earth".

Linear gravitational waves (just like light) have the same speeds in different INERTIAL coordinate systems, but not the same speeds more generally in different coordinate systems.

(BTW, I made quite a few high school algebra type errors earlier--it is not worth going thru them unless someone asks.)

 

[ApplePion]

<<That's what I am saying: "speed" loses its global meaning in curved spacetime>>

I don't see what you are getting at. "Electric field" has only a local meaning, too. Local quantities are perfectly legitimate.

 

[jtbell]

Local quantities are perfectly legitimate.

Sure. But you have to measure them locally. That is, you have to be at the location in question. If you measure them "remotely", from some other location, you have to take into account the spacetime curvature between the two locations.

 

[tom.stoer]

I am saying the following: Two observers A and B in curved spacetime at rest w.r.t each other but located at different spacetime points P_A and P_B will not agree on the velocity of a test body C w.r.t. to e.g. observer A:

v_C(w.r.t. A, measured by A = observed from P_A) != v_C(w.r.t. A, measured by B = observed from P_B)

That means the velocity is no longer globally valid. It does not only depend on the velocity of the two reference frames for A and B (wich is well-known from SR), but also on the location of these reference frames (= tangent-spaces) in spacetime.