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Traditionnal definition of speed in space-time

If I understood it correctly a speed is defined relatively to space-time, i.e. we can talk of the speed of anything moving or propagating “inside” space-time. This speed is measured in m / s. The space-time has a structural “speed limit” which is the upper limit at which any information can travel, this “speed limit” is usually named the “speed of light”: c.

Space-time is distorted and move

GR explains gravity is distorting space-time. So a moving mass will cause change “of” (as opposed to “inside”) the space-time which we could measure if we were able to witness it from “outside” of space-time. But there is no evidence this “outside” or “underlying” dimension exists. And there is less evidence we can make any kind of measure of our space-time in this dimension. Nonetheless this “change” of space-time is often described as having a “speed”.

What is the correct definition of speed of the refering system relative to itself

Is the term of “speed” appropriate to speak of the change of the medium in which we define and measure “speed”: the space-time? Aren't we here facing a recursive definition: we measure something (the space-time change) which is changing our basis of measure (space-time itself "change": (x,y,z,t) = f(x,y,z,t)).

Then what would be the correct definition of this “speed”?

In which unit should we measure it?

For example we can't speak of the space-time expansion measured in m/s because it isn't a speed of something happening "inside" the space-time. And this "pseudo-speed" doesn't have to respect the speed limit of the "space-time": c. See: https://physics.stackexchange.com/a/13390/12282

How might we measure it? With the meter before or beside the distortion? Or with the meter at the "now" and "there" of the distortion?

In an ideal space-time the expected answer to this question would be a simple definition of this "speed of space-time distortion" or "speed of gravity". Something not much more complex than the traditionnal concept of speed in GR.

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Related questions:

  • Speed of gravity

  • Speed of gravitational waves

  • Speed of the universe expansion

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    $\begingroup$ Are you familiar with the concept of a spacetime metric? $\endgroup$ – Jacopo Tissino Jun 2 at 9:23
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    $\begingroup$ @JacopoTissino : I am familiar with the Minkowski one. I would also appreciate any other reading helping me to sort out this apparent conflict of measuring the containing geometry from “inside”. $\endgroup$ – dan Jun 2 at 10:24
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    $\begingroup$ I've deleted a number of off-topic comments and/or responses to them. $\endgroup$ – David Z Jun 3 at 3:05
  • $\begingroup$ @dan I have tried to address the new points you have raised in my answer. For a reference explaining precisely what a general metric is and how it is used, I'd recommend Carroll still, but any textbook covering the basics of GR will discuss this topic in-depth really. The keyword would be "intrinsic curvature", which is precisely that which can be measured "from inside the manifold". $\endgroup$ – Jacopo Tissino Jun 3 at 7:24
  • $\begingroup$ @JacopoTissino: my new points weren't introduced to try to complexify my OQ, but to try to make it clearer. Thank you for your attempt to help, but my conception of topological fundamental difference between the content and the container seems to be rather hard to be understood. $\endgroup$ – dan Jun 3 at 11:01
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And there is less evidence we can make any kind of measure of our space-time in this dimension. Nonetheless this “change” of space-time is often described as having a “speed”.

This is where, I think, you have a misconception. GR is a geometric theory, and its most fundamental object is the metric, which allows us to compute the distance along any path in spacetime.

Einstein's Field Equations describe how this metric changes in the presence of energy and momentum. When we say that spacetime is "distorted" we mean that the metric has changed in a specific way. This is not some abstract effect which can only be measured from "outside" of spacetime: it means that distances, angles and times physically change! For instance, you can have triangles whose edges are locally "straight" but whose angles do not add to $180 ^{\circ}$.

Now, about the speed of gravity: this is actual, regular speed, measured in m/s. The simplest context in which one usually discusses this is that of linearized gravity: we can assume that spacetime is almost flat and that there is a small perturbation in it, and we only consider the first order in this perturbation.

If we do this, we find that the equation describing how this perturbation moves is a wave equation, the same one that the electric and magnetic fields obey in a vacuum; and where the propagation velocity is precisely $c$. By "in a vacuum", here, what is meant is "in the absence of electric charge or currents", in which case Maxwell's equations can be cast into a wave equation, $\square A^\mu = 0$ with an appropriate gauge choice.

When we detected gravitational waves, we did so by measuring with an interferometer by how much the distance between two points changed. The variation was tiny, to be sure, but it was measurable. Also, we have seen gravitational wave events with electromagnetic counterparts, which arrived within a couple of seconds of each other --- and they were coming from millions of light-years away! So, we are quite confident that gravitational waves do indeed travel at the speed of light.

To answer your question somewhat more formally: the spacetime metric is a tensor usually written as $g_{\mu\nu} (x)$; at each point in spacetime $x$ it assigns 16 numbers indexed by $\mu$ and $\nu$, which are indices which can both take values from 0 to 3, corresponding to a direction of time and three direction in space.

The speed of gravity is the speed of propagation of a perturbation in this metric.

To use an analogy, if you have waves in a pond you can describe them with a function like $h(\vec{x}, t)$, which assigns the height of the water $h$ to each point $x$ in the pond at each time $t$. You can have waves propagating in the pond: they can be described by a differential equation for $h$, like $\partial_{tt} h = c^2 \nabla^2 h$, where $c$ is the speed of propagation for the waves.

The water is not moving sideways, but its height is changing in a coherent way, such that if you look at it from outside you can see the wave rippling out. Crucially, this wave propagates: there is a causal relation between the wave being here now and it being there later.

The discussion of a wave in GR is similar, albeit with a few more mathematical complications.

Edit: A couple points which might clarify remaining doubts:

1) GR describes spacetime as a manifold, and even though we usually imagine a manifold as immersed in some larger space this is not something which is required by the definition --- instead, a manifold is simply a structure which can be identified locally with $\mathbb{R}^n$.

2) The metric tensor $g_{\mu\nu}$ fully describes the geometry of the spacetime, since if we have it we can measure any distance or angle we want. All the complicated tensors which describe the curvature of spacetime ultimately can be derived by a certain combination of derivatives of the metric.

Edit 2: addressing the comment by dan,

There stands for me the difference between a “speed” inside space-time and “another kind of speed” the speed of the geometry itself. What does this speed mean? Speed isn’t a concept built on vacuum but on meters and seconds… and they distort when space-time does, don’t they?

You are absolutely correct in that one must be careful when defining speeds in GR, but in fact there is a way to define the speed without ambiguities. In order to see this one must look at the anatomy of a gravitational wave. I'm not going to go in too much detail, section 6 in Carrol's GR notes is a good free reference for the basics of GW theory.

I'll just outline the assumptions: we take a perturbed flat metric, $g_{\mu\nu} (x) = \eta_{\mu\nu} + h_{\mu\nu}(x)$, where $\eta_{\mu\nu}$ is the Minkowski metric, while $h_{\mu\nu}(x)$ is a symmetric tensor whose components are small compared to 1.

We consider the vacuum case, meaning there is no energy nor momentum: the Einstein equations read $R_{\mu\nu} = 0$.

Then, it can be shown that the simplest plane-wave solution for a wave propagating can be written as: $$ h_{\mu\nu} (x) = \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & h_{+} & h_{\times} & 0 \\ 0 & h_\times & - h_+ & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] e^{ikz - i \omega t}, $$ where $\omega / k = c$: a wave propagating at the speed of light.

But, you may say, the issue still stands! How can we define the speed if spacetime is being distorted? The solution now appears: space is not being distorted in the direction of propagation! This GW is propagating along the $z$ axis, but it stretches space along the $x$ and $y$ directions, while the $z$ direction is unaffected. Further, it does not affect time either.

So, the definition is simple: speed simply means $$ \frac{\mathrm{d}z}{\mathrm{d}t}, $$ and there are no ambiguities since the GW does not affect $t$ nor $z$. The shape of the distortion of space is shown on page 154 by Carroll. So, this speed of gravity is very much the speed of a physical "thing": if you have GW detectors in a line, one will see the oscillatory signal from the deformation shifted by a certain time with respect to the other (and this is experimentally confirmed!). In this sense, a gravitational wave is a thing travelling through space with a certain speed as much as a photon is.

Now, this all holds in the linear regime, where we assume that the perturbation to the metric is small. This allows us to work analytically with Einstein's equations, which are nonlinear and therefore very difficult to deal with. In the general nonlinear case, the definition will have to be adapted; however, when people refer to the speed of GW they are usually talking about the linear regime.

Edit 3: addressing a point which was added to the question,

Aren't we here facing a recursive definition: we measure something (the space-time change) which is changing our basis of measure (space-time itself "change": (x,y,z,t) = f(x,y,z,t)).

I think you might be conflating two different things: changes of coordinates and the curvature of spacetime due to the presence of a nonvanishing stress-energy tensor ("stuff").

GR is invariant under changes of coordinates: if I use the coordinates $x^\mu$ and you use the coordinates $x^{\prime \mu}$, as long as they're well-behaved (there is a differentiable map between them and so on) we will make the exact same physical predictions. This is something which is deeply embedded in the mathematical framework of the theory.

The curvature of spacetime is a completely different thing: here, the metric changes in a way that has physically measurable effects (about which observers using different bases will agree).

In the end, the point is this: in GR predictions are not ambiguous. Sure, because of relativity different observers measure different things, but if the reference frame is agreed upon the predictions will be the same. Also, the speed of light is peculiar in that is an invariant, a trajectory which is lightlike for an observer is so for any other observer.

Edit 3a: regarding

Speed of the universe expansion

In cosmology, things are slightly different. Let us forget about perturbations and consider a very simple case, which is called flat FLRW geometry. This means that spacetime is spatially flat, the metric is similar to the Minkowski one, but we insert a scale factor for the spatial coordinates: $$ \mathrm{d} s^2 = - \mathrm{d} t^2 + a^2(t) \delta_{ij} \mathrm{d}x^i\mathrm{d}x^j\,. $$

This is the simplest mathematical context in which one may discuss the expansion of the universe. Here, we do not usually talk about "speed"; instead, what is used is the rate of expansion of the universe, which is measured in 1/s, and is defined as $H = \dot{a} / a$. This is the parameter which appears in Hubble's law.

So, in this context people may imprecisely say "speed" but if they are referring to the Hubble parameter $H$ they should say "rate".

Note that this is perfectly well-defined, with no ambiguities formally (although experimental determinations of it are difficult). In an ideal world, you can measure it by taking two points, setting them in free fall (which is really hard practically) and measuring their distance over time. It will change in a way that is proportional to $a$. The variation of this distance over time has the units of a speed and it is called the recession velocity, it however is not the speed of an object moving through space. This is why there is no issue with it exceeding $c$.

For more details on this aspect, check out the excellent paper by Davis and Lineweaver. It is not meant to be introductory, but it has a proper mathematical explanation for everything you are asking about.

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  • $\begingroup$ “the same one that the electric and magnetic fields obey in a vacuum; and where the propagation velocity is precisely 𝑐” Electro-magnetic fields obey to the geometry of space-time, they follow its distorsions. I feel that “in a vacuum” is a language simplification. There stands for me the difference between a “speed” ínide space-time and “another kind of speed” the speed of the geometry itself. What does this speed mean? Speed isn’t a concept built on vacuum but on meters and seconds… and they distort when space-time does, don’t they? $\endgroup$ – dan Jun 2 at 10:59
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    $\begingroup$ @dan I see your point, it's a reasonable one, the answer is kind of long so I will add it as an edit to my answer. $\endgroup$ – Jacopo Tissino Jun 2 at 13:06
  • $\begingroup$ I also added a clarification about the meaning of "vacuum" in the electromagnetic case. I was thinking about standard special-relativity electromagnetism, although you can certainly write Maxwell's equations in curved spacetime if you want. $\endgroup$ – Jacopo Tissino Jun 2 at 13:58
  • $\begingroup$ I like your analogy with waves on the surface of a pond. But would we be able to speak of "waves propagation speed" if we were living within this 3 dimensionnal "pond surface" and not measuring it within in our 4 dimensionnal space-time? $\endgroup$ – dan Jun 2 at 17:22
  • $\begingroup$ Sure! We're stretching the analogy here, but you can "detect" the water wave if you live on the surface since, by there being crests and throughs, distances change between nearby points. So, if you measure the wave in a place and later you measure it in another you can compute the speed as distance/time with respect to the "flat pond" metric. Here the analogy breaks since water waves do stretch "space" along the direction of propagation; however, we can recover the argument if we consider the perturbation to be small. $\endgroup$ – Jacopo Tissino Jun 2 at 17:49
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The distortion of spacetime due to a gravitational wave is entirely within the four dimensions of spacetime (as far as we know). There is no need for any additional dimensions "outside" of spacetime to explain gravitational waves. Although the distortion of spacetime by mass/energy is often illustrated by the analogy of heavy objects distorting a rubber sheet, this analogy is misleading in several ways. A better analogy would be think of a rubber sheet that stays flat but is pulled and stretched by different amounts in different directions.

In general relativity, gravitational waves travel at the same speed as massless particles such as the photon i.e. at the speed of light. This has been confirmed to an accuracy of better than $1\%$ by observing the orbital decay rate of binary pulsars - see this Wikipedia article for details.

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  • $\begingroup$ Fully agreed with your better analogy. This is my way of understanding distortion of space-time itself. The rubber sheet is fully wrong since it is the result of Earth’s gravitation perpendicular to the rubber sheet. This is very bad analogy everyone should try to remove. BTW I also fully understand your proof there is no need of extra-dimension for a ”distortion” to exist. But on purpose I didn’t use the word “move” here. $\endgroup$ – dan Jun 2 at 11:06
  • $\begingroup$ The problem with the rubber sheet analogy is that the stretch has to extend into another dimension which is additional to the kind of universe being described. It's easier to stop pretending that it's the fabric of space being stretched, and that it's just the things in space which are being modified. In the same way that when we use a magnet to attract a nail on a worktop, it's much easier to see the sense that the magnet is moving the nail by the presence of a magnetic field, not that space is being stretched or condensed so as to bring the nail closer without it actually moving. $\endgroup$ – Steve Jun 2 at 17:38
  • $\begingroup$ @Steve the analogy is not exact, but GR works very well and does not need extra dimensions. Comments here are not the appropriate place to insinuate that GR is wrong. $\endgroup$ – fqq Jun 2 at 22:56
  • $\begingroup$ @fqq, I'm not insinuating GR is wrong. I'm saying the analogy is wrong, as it simply raises more questions than it answers. It's the sort of analogy that is beloved of those who possess no critical faculties, no inclination to ask the relevant question of the analogy, which is that if the 3rd dimension is what the 2d sheet stretches into, then into what does space-time stretch? It is well-known that "distortion of space-time" is merely an interpretation of GR (along with many other philosophical oddballs in physics), and it can be jettisoned entirely without impugning the correctness of GR. $\endgroup$ – Steve Jun 2 at 23:14
  • $\begingroup$ I should add as well, gandalf61's reformulation of the analogy as being (effectively) a redistribution of the rubber sheet's density across its surface (rather than a stretching of its surface into an orthogonal dimension) is not much better, because it still begs the question of what it means for space itself to have a density, when density is a measure of how much of something is found "within a locality of space". Jettison the "distortion of space entirely", and just admit you're employing the concept of a Lorentzian aether but refusing to speak its name. $\endgroup$ – Steve Jun 2 at 23:27
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I would add one more clarification to great Jacopo Tissino answer and gandalf61 nice rubber analogy

If I understood it correctly a speed is defined relatively to space-time

This is not so. The speed is defined only relative to some observer. It so happens in relativity, that speed of light is (by design) same for all the observers, but it is still concept defined only w.r.t to some observer.

The "absolute" measure of motion in spacetime is actually 4-velocity. But this 4-velocity is always the same length $c$, so the information about the motion of real objects is always encoded only in its direction. The fact that gravitational waves propagate with speed of light is then saying, that they propagate only along special directions called null.

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  • $\begingroup$ What I meant when writing "If I understood it correctly a speed is defined relatively to space-time" was that there is no meaning of speed "outside of space-time". I never wrote or thought anything that would contradict the fact that space-time is relative to some observer: its speed (RR) and its gravitationnal environnement (GR). $\endgroup$ – dan 9 hours ago

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