Gravitational wave propagation in GR

[cianfa72]

TL;DR Summary

About Gravitational Waves propagation as spacetime metric change

Hi, I'd like to discuss in this thread the propagation of Gravitational Waves (GW) in the context of GR.

Just to fix ideas, let's consider a FW spacetime. It is not stationary (even less static), however the timelike congruence of "comoving observers" is hypersurface orthogonal.

Suppose at a given event in spacetime some kind of "explosion" occurs then, from my understanding, the spacetime metric begins to change.

This metric change represents the GWs. In a sense such GW propagation does not take place through spacetime since it is the spacetime itself that undergoes a change of metric.

Any comment ? Thank you.

 

[Ibix]

This metric change represents the GWs.

The metric doesn't change - the metric is defined everywhere through 4d spacetime. It may have a time or spatial coordinate dependence, but that's already true of the FLRW metric. I think what you want to say is that the metric is not (or not necessarily) FLRW in the future lightcone of the explosion event. Note that this is also true in the past lightcone of the explosion event, since an FLRW metric can't have an explosion at a point because everywhere is the same. You can choose to neglect that, but it's worth bearing in mind.

In a sense such GW propagation does not take place through spacetime since it is the spacetime itself that undergoes a change of metric.

In strong gravitational radiation then you have no choice but to solve the Einstein field equations and get a metric depending strongly on your time and space coordinates. And you are formally correct that gravitational waves are always "ripples" in the metric.

However, in the case where the waves are weak you can write the metric as an undisturbed background metric plus a "small" perturbation. Then you can see the perturbation as a tensor field propagating in the undisturbed spacetime. The textbook treatment I have seen assumes the background metric is Minkowski. I don't see why it wouldn't work in other spacetimes, although you'd need a bit more care about what "small" means.

 

[cianfa72]

The metric doesn't change - the metric is defined everywhere through 4d spacetime. It may have a time or spatial coordinate dependence, but that's already true of the FLRW metric. I think what you want to say is that the metric is not (or not necessarily) FLRW in the future lightcone of the explosion event.

Ah ok, as time and/or spatial dependence you mean the dependence of metric components $g_{\mu \nu}(.)$ in some coordinate chart in the future lightcone of that event.

However, in the case where the waves are weak you can write the metric as an undisturbed background metric plus a "small" perturbation. Then you can see the perturbation as a tensor field propagating in the undisturbed spacetime.

As above, a propagating metric tensor field does mean metric components "added" to the underlying undisturbed background (Minkowski) metric components in the future lightcone of the explosion event.

 

[Mordred]

An easy way to think of it is to set your global metric $g_{\mu\nu}$ the GW wave would be described using the permutation tensor $h_{\mu\nu}$

 

[weirdoguy]

permutation tensor

Isn't that a perturbation tensor?

 

[cianfa72]

An easy way to think of it is to set your global metric $g_{\mu\nu}$ the GW wave would be described using the permutation tensor $h_{\mu\nu}$

Ok, you simply mean add to underlying spacetime tensor metric $g_{\mu \nu}$ the perturbation tensor $h_{\mu \nu}$ in the same coordinate chart.

 

[Mordred]

Yeah it's common practice to separate local perturbations from the global metric then combine them
Something along the statement
$$ g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$$

Gravity waves are quadrupolar with a transverse traceless gauge on (if I recall the longitudinal polarizations are traceless) so you will have two polarizations + and ×. These would be shown in the permutation tensor with each polarization being it's own permutation tensor.

This article will provide the related mathematics

https://arxiv.org/pdf/physics/9908041

 

[Ibix]

permutation

Do you mean to keep writing "permutation tensor"? I think you mean perturbation. If you do mean "permutation" I'd appreciate a reference or short description.

 

[Mordred]

Sure perturbation works for some reason I keep thinking permutation but you are correct

 

[cianfa72]

What are the effects of GWs on interferometers like LIGO ?

 

[Mordred]

How do you mean ? Are you asking why LIGO requires the L shape to pick up the quadrupole wave characteristics as opposed to say an antennae for dipolar waves as per EM field ?
A useful analogy is squeeze a rubber ball on the x axis. The Y axis expands. Then switch, same if you squeeze the y axis.
That describes the quadrupole wave so that analogy also helps understand the L shape design. Now LIGO uses if I recall three Michelson like interferometers . The GW waves induce strain and this affects the distance travelled by the beams according to the above.

Once you clarify what effect you are specifically asking about We can go into greater detail but the above describes the basics.
One detail to take from this is that the characteristics are different from dipolar waves and LIGO by design can distinguish between the two. (This includes mechanicsl vibration which is dipolar.

 

[cianfa72]

How do you mean ? Are you asking why LIGO requires the L shape to pick up the quadrupole wave characteristics as opposed to say an antennae for dipolar waves as per EM field ?
A useful analogy is squeeze a rubber ball on the x axis. The Y axis expands. Then switch, same if you squeeze the y axis.
That describes the quadrupole wave so that analogy also helps understand the L shape design. Now LIGO uses if I recall three Michelson like interferometers . The GW waves induce strain and this affects the distance travelled by the beams according to the above.

From my understanding, in the future light cone's spacetime region of an "explosion" event, there is a perturbation metric tensor $h_{\mu \nu}$ that "adds up" to the underlying $g_{\mu \nu}$ in the same underlying coordinate chart.

Both laser beams of LIGO interferometers basically propagate "inside" each worldtube's spacetime region representing the associated LIGO's arm. One can calculate in any coordinate chart the (invariant/frame-independent) round-trip travel time for each laser beam respectively (at the common origin of the two arms).

In presence of GWs these two round-trip travel times should no longer be the same, hence the interferometer establishes the presence of GWs "hitting" LIGO's arms.

 

[ergospherical]

Look at a simple GW metric for a wave in the $z$-direction:$$g = -dt^2 + (1+h_{+})dx^2 + (1-h_{+})dy^2 + 2h_{\times} dx dy + dz^2$$Imagine one with no cross polarization, $h_{\times} = 0$. The linearised Einstein equations admit plane wave solutions so $h_{+}$ oscillates. Two particles at $\mathbf{x}_1 = (-\epsilon, 0, 0)$ and $\mathbf{x}_2 = (\epsilon, 0,0)$ have separation $ds^2 = (1+h_{+})4\epsilon^2$. Two particles at $\mathbf{x}_3 = (0,-\epsilon,0)$ and $\mathbf{x}_4 = (0,\epsilon, 0)$ have separation $ds^2 = (1-h_{+})4\epsilon^2$. If you draw it out, the four particles form a diamond shape which is squished and stretched in the $x$ and $y$ directions.

Real-life diagnostics involve more jargon. With both plus and cross polarisation modes, you can define a complex strain $H = -h_{+} + ih_{\times}$. It's conventional to project this onto spherical harmonics, $H_{lm} := \langle Y_{lm}^{-2}, H \rangle$. LIGO can measure these multipoles of the GW strain.

 

[cianfa72]

Look at a simple GW metric for a wave in the $z$-direction:$$g = -dt^2 + (1+h_{+})dx^2 + (1-h_{+})dy^2 + 2h_{\times} dx dy + dz^2$$Imagine one with no cross polarization, $h_{\times} = 0$. The linearised Einstein equations admit plane wave solutions so $h_{+}$ oscillates. Two particles at $\mathbf{x}_1 = (-\epsilon, 0, 0)$ and $\mathbf{x}_2 = (\epsilon, 0,0)$ have separation $ds^2 = (1+h_{+})4\epsilon^2$. Two particles at $\mathbf{x}_3 = (0,-\epsilon,0)$ and $\mathbf{x}_4 = (0,\epsilon, 0)$ have separation $ds^2 = (1-h_{+})4\epsilon^2$. If you draw it out, the four particles form a diamond shape which is squished and stretched in the $x$ and $y$ directions.

Your simple GW metric is given in the same inertial coordinate chart of flat spacetime. Particles $\mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3, \mathbf{x}_4$ of course follow timelike paths through spacetime and are actually "at rest" in that chart (i.e. in that chart they are described by fixed spacelike coordinates and varying $t$).

To get the separation for example between $\mathbf{x}_1$ and $\mathbf{x}_2$ you set $dt=0$ and evaluate $ds^2$ on the spacelike hypersurface on constant $t$.

Is actually $h_{+}$ a function of coordinate time $t$ (hence the metric $g$ is not stationary) ?

 

[ergospherical]

In TT gauge, the linearized vacuum Einstein equation is $\square h_{\mu \nu} = 0$, so $h_{\mu \nu} = A_{\mu \nu} e^{ik_{\rho}x^{\rho}}$. For a wave traveling in the $z$-direction, $k_{\mu} = \omega(-1,0,0,1)$, so $h_{\mu \nu} = A_{\mu \nu} e^{-i\omega(t-z)}$. With our choice of Lorenz + TT gauge, there are only two independent components $h_{xx} = -h_{yy} := h_{+}$ and $h_{xy} = h_{yx} := h_{\times}$.

For an arrangement of test particles which are initially at rest in the background inertial frame, such as the diamond in the previous post, then $u^{\mu} = (1,0,0,0)$ and the geodesic equation reduces to $\dot{u}^{\mu} = - \Gamma^{\mu}_{00} = 0$ in Lorenz gauge.

So, in Lorenz gauge, the particles remain at fixed coordinate positions. But the proper separations do change, c.f. the previous post.

 

[cianfa72]

Sorry, what is TT gauge ?

As far as I can understand, in Lorentz gauge, the four diamond particles stay "at rest" in the background inertial frame (fixed spatial coordinate positions), however their spacelike separations evaluated in each spacelike hypersurface of constant coordinate time $t$ do change (in coordinate time).

 

[renormalize]

Sorry, what is TT gauge ?

TT = "transverse traceless"

 

[cianfa72]

In TT gauge, the linearized vacuum Einstein equation is $\square h_{\mu \nu} = 0$, so $h_{\mu \nu} = A_{\mu \nu} e^{ik_{\rho}x^{\rho}}$. For a wave traveling in the $z$-direction, $k_{\mu} = \omega(-1,0,0,1)$, so $h_{\mu \nu} = A_{\mu \nu} e^{-i\omega(t-z)}$. With our choice of Lorenz + TT gauge, there are only two independent components $h_{xx} = -h_{yy} := h_{+}$ and $h_{xy} = h_{yx} := h_{\times}$.

Ok, therefore the non-zero component of the perturbation tensor $h_{+}$ depends on $e^{-i\omega(t - z)}$ where $A_{\mu \nu}$ are the components of the polarization tensor in the background inertial chart (to take it simple assume $h_{\times} = 0$).

The timelike paths of the four diamond particles are actually geodesics since their 4-velocities $u^{\mu}$ satisfy the geodesic equation in Lorenz gauge.

Whether the above is correct, I'd like to be sure that the coordinates in which one is working are actually the global inertial coordinates in the background flat spacetime.

 

[ergospherical]

Sounds alright to me. Everything I wrote so far is in inertial coordinates and Lorenz gauge (chosen such that $\partial^{\mu} \bar{h}_{\mu \nu} = 0$).

 

[cianfa72]

Everything I wrote so far is in inertial coordinates and Lorenz gauge (chosen such that $\partial^{\mu} \bar{h}_{\mu \nu} = 0$).

Since the TT gauge is a specific Lorenz gauge such that $$\eta^{\mu \nu}\bar {h}_{\mu \nu} =\bar{h}_{\mu}{}^{\mu} = \bar{h} = 0 \rightarrow h=0, \bar{h}_{\mu \nu} = h_{\mu \nu}$$ we get the following Linearized EFE in vacuum $$\square \bar{h}_{\mu \nu} = \square h_{\mu \nu} = 0$$
From this video at minute 8:45 it is clear that the "spatial distance" between timelike geodesics "at rest" in the chart one gets from Lorenz + TT gauge (geodesics w.r.t. the perturbated metric $\eta_{\mu \nu} + h_{\mu \nu}$), has to be understood as the "shortest distance" between the geodesics evaluated in each spacelike hypersurface of constant chart coordinate time $t$.

My point is: what is the "real" significance of such "proper distance" between those geodesics since in GR coordinates are essentially arbitrary ?

 

[ergospherical]

You said it yourself: the coordinates are arbitrary, which is why Lorenz gauge (where particles at rest initially remain at fixed coordinates) are useful. But the proper distances between the particles is the measurable distance, and these do change with time.

 

[cianfa72]

But the proper distances between the particles is the measurable distance, and these do change with time.

Sorry, since the proper distance between those particles is evaluated on each spacelike hypersurface of constant coordinate time (and it changes in time), implicitly we are picking a synchronization convention given by the underlying coordinate chart chosen.

 

[ergospherical]

I'm don't think I understand your objection. The varying proper separation is real and measurable (c.f. LIGO!). You can look at it another way: in the local frame field (tetrad) of a nearby observer. If you have a family of particles, then the separation vector $S$ between two geodesics satisfies the geodesic deviation equation $\nabla_{T} \nabla_{T} S = R(T,S)T$, where $T$ is the geodesic tangent vector. Expand $R(S,T)T$ in component form, project onto the tetrad and consider the resulting equation to first order. You find that the components of $S$ in the local frame field vary in the same way as before.

Refer to pages 79 - 82:
https://www.damtp.cam.ac.uk/user/hsr1000/part3_gr_lectures.pdf

*one subtlety is that, equation (7.54) in the above is written w.r.t. the local tetrad coordinates. But then, he uses the expression for $R_{\mu 0 0 \nu}$ derived in TT gauge. Why is this valid? It turns out that the components of the Riemann tensor are gauge invariant to first order (to show it, use the linearized expression for $R_{\mu \nu \rho \sigma}$ and make the transformation $h_{\mu \nu} \mapsto h_{\mu \nu} + \partial_{\mu} \xi_{\nu} + \partial_{\nu} \xi_{\mu}$)

 

[PeterDonis]

since the proper distance between those particles is evaluated on each spacelike hypersurface of constant coordinate time (and it changes in time), implicitly we are picking a synchronization convention given by the underlying coordinate chart chosen.

Not if you measure "proper distance" by round-trip light signals. That's how LIGO does it: it uses round-trip light signals along each arm.

 

[cianfa72]

Not if you measure "proper distance" by round-trip light signals. That's how LIGO does it: it uses round-trip light signals along each arm.

In a sense one employs null paths to "indirectly" measure a spacelike separation.

The linearized metric is not stationary, so does make sense to use round-trip light signals to measure "proper distance" on a given spacelike hypersurface between the two timelike paths representing the two ends of a LIGO arm ?

 

[PeterDonis]

In a sense one employs null paths to "indirectly" measure a spacelike separation.

But which spacelike separation? There is no invariant answer.

a given spacelike hypersurface

Which spacelike hypersurface? There is no invariant answer.

Note that there doesn't need to be an invariant answer to the above questions for LIGO to work. LIGO is not measuring a change in "distance" in any invariant sense. It is measuring a change in the round-light travel time--more precisely a change in the difference in round-trip light travel time along the two arms.

 

[cianfa72]

But which spacelike separation? There is no invariant answer.

Which spacelike hypersurface? There is no invariant answer.

Exactly, that was my point.

Note that there doesn't need to be an invariant answer to the above questions for LIGO to work. LIGO is not measuring a change in "distance" in any invariant sense. It is measuring a change in the round-light travel time--more precisely a change in the difference in round-trip light travel time along the two arms.

Ok, and the above does mean there was a GW "hitting" the LIGO's arms ?

 

[PeterDonis]

the above does mean there was a GW "hitting" the LIGO's arms ?

A variation in the difference between light travel times can mean that. But it can also mean other things: the apparatus could have been jiggled by a heavy truck driving by the lab, or by an earthquake tremor, or any number of other things. That's why LIGO checks to make sure the same signal is received at multiple detectors which are at widely separated locations on Earth--because that rules out causes that would only be there at one location, like the ones I listed above.

 

[PeterDonis]

that was my point

That wasn't at all clear from your posts.

 

[cianfa72]

A variation in the difference between light travel times can mean that.

As far as I can tell, each laser beam travels "inside" the spacetime region representing the worldtube of each LIGO arm. In Lorenz TT gauge coordinates, we said the endpoints of each arm are "at rest" (and the endpoint worldlines are actually timelike geodesics). Then one can write down the null geodesic equation that describes the round-trip laser path along an arm and from the proper time elapsed along the timelike path representing the "corner" endpoint of "L shape" between the sending and receiving back event of the laser beam, evaluate whether the "arm length" has changed or not.

 

[Ibix]

Then one can write down the null geodesic equation that describes the round-trip laser path along an arm and from the proper time elapsed along the timelike path representing the "corner" endpoint of "L shape" between the sending and receiving back event of the laser beam, evaluate whether the "arm length" has changed or not.

Yeah, but Peter's point is that the arm length also changes when a mouse sneezes somewhere near LIGO. In an ideal world, flight time change implies a GW. In the real world it just implies something caused a flight time change, and there is a lot of signal processing that goes on to pull GW signals out of the noise.

 

[cianfa72]

Then one can write down the null geodesic equation that describes the round-trip laser path along an arm and from the proper time elapsed along the timelike path representing the "corner" endpoint of "L shape" between the sending and receiving back event of the laser beam, evaluate whether the "arm length" has changed or not.

Or, to put it the other way around, whether there are not GWs hitting LIGO arms, then one can assume flat metric and arms "at rest" in a inertial coordinate chart (the timelike congruence given from each arm worldtube's worldlines is geodesic). Then the laser beam round-trip travel time actually "measures" the "proper distance" between each arm's endpoints (in this case there is a natural definition of "proper distance" for them).

 

[PeterDonis]

whether there are not GWs hitting LIGO arms, then one can assume the metric is flat

I think you mean when there are not GWs hitting LIGO arms.

Then the laser beam round-trip travel time actually "measures" the "proper distance" between each arm's endpoints (in this case there is a natural definition of "proper distance").

You are confusing yourself.

The issue that there is no invariant way to define the "proper distance" between the arms (or, to put it another way, to define the "distance the light travels" on its round trip) is an issue whether there is a GW passing or not. But we can ignore that issue because LIGO does not measure the proper distance. It measures the difference in round-trip light travel time between the two arms. That is an invariant independent of any issues about how "proper distance" is defined.

The question is, supposing that a single LIGO apparatus observes a difference in the round-trip light travel time between the two arms, what caused that difference? A GW is not the only possible cause. A GW does it by changing the spacetime geometry in the arms, while leaving the reflectors at the ends of the arms in free fall (actually they're not because they are being suspended in Earth's gravitational field, but we can ignore that and only look at horizontal motion and consider them to be in free fall horizontally). Other causes do it by moving the reflectors at the ends of the arms, i.e., by pushing them so they are not in free fall.

But a single LIGO apparatus has no way of telling which of those two things happened to cause a difference in the round-trip light travel time between the two arms. The only thing LIGO can do is have multiple detectors at widely different locations and look to see if the same signal appears in both of them. That is what would be expected to be the case only if the signal is due to a GW: other causes pushing on the reflectors at the ends of the arms would not be expected to cause the same signal in both detectors at widely different locations.

 

[cianfa72]

A GW does it by changing the spacetime geometry in the arms, while leaving the reflectors at the ends of the arms in free fall (actually they're not because they are being suspended in Earth's gravitational field, but we can ignore that and only look at horizontal motion and consider them to be in free fall horizontally).

Ok, so a GW acts by changing the spacetime geometry in the future lightcone of some "explosion" in spacetime. As you pointed out it leaves the reflectors at the ends of the arms in free-fall (their timelike geodesic worldlines are "at rest" in Lorenz TT gauge coordinates).

From the above it follows that the "frame-invariant" laser beam round-trip travel time (between reflectors at the ends of each arm) would indeed change, right?

 

[PeterDonis]

a GW acts by changing the spacetime geometry in the future lightcone of some "explosion" in spacetime.

"Changing" it from what it would have been if the explosion had not taken place, yes.

it leaves the reflectors at the ends of the arms in free-fall

Yes.

From the above it follows that the "frame-invariant" laser beam round-trip travel time (between reflectors at the ends of each arm) would indeed change, right?

Isn't that what I've already said, more than once?