Do Objects in Motion Create Gravitational Waves?

[PeterDonis]

The subject was referred to as classical dynamics, electrodynamics, geometrodynamics ect

The physicists who use the term "geometrodynamics" are not, as far as I can tell, the same ones who say that spacetime is not "dynamical". But neither are the physicists who use the term "geometrodynamics" all (or even a majority, as far as I can tell) of the physicists working in general relativity or some closely related field.

 

[Paul Colby]

The physicists who use the term "geometrodynamics" are not, as far as I can tell, the same ones who say that spacetime is not "dynamical". But neither are the physicists who use the term "geometrodynamics" all (or even a majority, as far as I can tell) of the physicists working in general relativity or some closely related field.

It's a term coined by Wheeler back in the day. There are whole sections of MTW that treat the local time evolution of 3-space so this is a concept which has technical meaning. I certainly could get behind there not being a globally defined universal time parameter but I would stop short of redefining the meaning of "dynamical" (which has been a part of physics for hundred of years) just so one could say that space-time isn't "dynamical". Sounds like a small group of researchers may have adopted this terminology but unless it has a precise meaning I see little benefit gained in adopting it.

 

[pervect]

There is no such thing as "the space component". The separation into space-like slices is quite arbitrary and can be done in many different ways.

It is rather ambiguous. I've got my own preference for how to make a split, via a mechanism that passes for "an observer". In simple terms for the OP, I'd describe this mechanism is just the set of worldlines of observers who are considered "at rest" and form an extended frame. (Some writers call this a frame, but sometimes other writers mean something else by a frame :( ). The technique works best in unchanging, i.e. static or stationary, space-times, where it's fairly obvious what these observers are, but in a pinch any set of observers will do, as long as they are agreed-upon. The wordline of this infinite set of observers forms a timelike congruence which fills space-time. Then using this congruence, one can use the Bel decomposition https://en.wikipedia.org/wiki/Bel_decomposition to decompose the RIemann (there's another technical term, which, for the benefit of the OP, I'll explain as representing the fundamental entity that describes the curvature of space-time). The Riemann decomposes , in the context of general relativity (GR), into three parts. (See wiki for when you might need four, but it's off-topic, it's not GR). One of these parts, calls the "topogravitic" part, can be interpreted as representing the purely spatial part of the curvature, as defined by the set of observers.

A basic issue I have though, is that while I think of things this way, it's unclear how many other people do - though it is in the literature, though not as prominently as it's explanatory power merits (in my opinion). So the techincal reader needs a lengthly explanation, and the non-technical reader has their own concepts, which may or may not be fundamentally compatible with GR, and which as a writer, we have no idea of what they are.. (Unfortrunately, it's rather likely that the non-technical readers concept of space and space-time is not even compatible with special relativity, which means automatically that it's not compatible with GR as GR is built on SR, special relativity. This makes writing about GR rather challenging.).

Anyway, with all this background out of the way, we can provide a answer to the original question, if one accepts the basic approach. Ideally we'd agree on the set of observers, but basically the electrogravitic part of the Riemann is equal to the topogravitic part in a vacuum. So with this definition, we'd say that there is a "space curvature" component of the GW. Otherwise we'd need both the electrogravitic and topogravitic parts to be zero, leaving the only source of curvature as the magnetogravitic part, and I don't believe that's possible.

One other thing I want to mention. I think that many people expect the set of observers regarded as being "at rest" to at least approximate a rigid body, as I think that's probably the most common approach to understanding space. This follows from the old, pre-relativistic idea of distance being measured with meter sticks, which are presumed to be rigid bodies. One will find a lot of explanations of GW's that basically do not take this approach, because it's mathematically more convenient not to. I feel that this causes a lot of conceptual mis-understandings and mis-communications, but it's unclear what to do about it. In a very common approach, observers "at rest" are considered to be those observers who follow geodesics, i.e. observers who have no proper acceleration. These observers do not maintain even approximately a constant separation :(.

I should mention that my second-favorite approach to the Bel decomposition for conceptual understanding is the usage of Fermi-normal coordinates, with the idea that these represent what happens if we choose coordinates which allow one to apply a lot of one's Newtonian intuitions. But the issue isn't fundamentally coordinate dependent, it can be talked about in non-coordinate (tensor) terms, which is why I prefer the Bel decomposition approach.

 

[PeterDonis]

It's a term coined by Wheeler back in the day. There are whole sections of MTW

Yes, I'm familiar with them and with the "superspace" formalism that Wheeler helped to develop. I'm just pointing out that that's not the only way to approach GR, and that other GR textbooks don't have the same emphasis on this particular aspect. (I don't remember Wald using the term "geometrodynamics" at all, for example.)

I certainly could get behind there not being a globally defined universal time parameter

It's more than that. It's that, although it is certainly possible to approach GR by viewing it as describing the "time evolution" of 3-geometries (as the superspace approach does), it is not necessary to do so. One can also take the viewpoint that a solution of the EFE describes a single 4-geometry. I personally think both viewpoints can be useful, and I don't think it's either useful or necessary to insist on adopting just one of them and not using the other at all.

I would stop short of redefining the meaning of "dynamical" (which has been a part of physics for hundred of years) just so one could say that space-time isn't "dynamical". Sounds like a small group of researchers may have adopted this terminology

I don't think it's a small group, at least not in the GR community. But I have not tried to come up with any actual numbers.

 

[PeterDonis]

redefining the meaning of "dynamical" (which has been a part of physics for hundred of years)

I'm also not sure this is true in the precise sense of "dynamical" you are using (i.e., that "dynamical" means "described by hyperbolic differential equations"). The term "dynamical" has been around for quite a while, but I'm not sure it has been understood as having that precise meaning for that same amount of time. But that's a question of history, not physics.

 

[laymanB]

Spacetime, as a model, does not have "time" as an external parameter, and does not describe a succession of "states" as a function of "time". It just describes a single 4-dimensional geometry.

This discussion is beginning to shed some light on my questions from the thread I started on infinite versus finite space. Very interesting and illuminating.

Does this spacetime model (viewing spacetime as a single 4-dimensional, non-evolving geometry) apply only to the universe as a whole, or can it be used for a subset of the whole universe (ie. the observable universe)?

 

[laymanB]

I don't claim to understand all of this but let me ask a couple questions.

If there are multiple solutions to the Einstein Field Equations that give you a model of spacetime that works for each, what is the justification for using the FRW metric as opposed to some other metric?

Is this idea of splitting up space and time into arbitrary components similar to the idea of no absolute reference frame in relativity? The mathematical models still account for the observational data and make accurate predictions by choosing an arbitrary way to slice spacetime, but you are left with artifacts that only correspond uniquely to your choice of how to slice it?

 

[PeterDonis]

Does this spacetime model (viewing spacetime as a single 4-dimensional, non-evolving geometry) apply only to the universe as a whole, or can it be used for a subset of the whole universe (ie. the observable universe)?

You can view any open region of spacetime (such as the observable universe) in the way I described.

 

[PeterDonis]

Is this idea of splitting up space and time into arbitrary components similar to the idea of no absolute reference frame in relativity?

It's the same idea.

The mathematical models still account for the observational data and make accurate predictions by choosing an arbitrary way to slice spacetime, but you are left with artifacts that only correspond uniquely to your choice of how to slice it?

In principle you don't have to choose an arbitrary way to slice spacetime in order to make predictions; you can formulate everything in terms of coordinate-independent quantities. In practice choosing coordinates (i.e., choosing an arbitrary way to slice up spacetime into space and time) often makes it much easier to make predictions, which is why it's so often done.

 

[pervect]

I don't claim to understand all of this but let me ask a couple questions.

If there are multiple solutions to the Einstein Field Equations that give you a model of spacetime that works for each, what is the justification for using the FRW metric as opposed to some other metric?

Is this idea of splitting up space and time into arbitrary components similar to the idea of no absolute reference frame in relativity? The mathematical models still account for the observational data and make accurate predictions by choosing an arbitrary way to slice spacetime, but you are left with artifacts that only correspond uniquely to your choice of how to slice it?

A lot depends on what you mean by "different solution". If you have some solution to EInstein's field equations, and you derive a new solution from the old solution by a mathematical transformation (in this case, the appropriate technical name for the appropriate transformation would be a diffeomorphism), do you regard the solution as "different"?

I suspect from your question that you do, but I regard the solutions as equivalent, and I'd describe changing the coordinates via a transformation to a new set of coordinates gives a different representation of the same solution, not a different solution.

To borrow an analogy, suppose you have a map of some section of land. And you rotate the map by some angle. Is it a "different map" after you rotate it, or is it "the same map, rotated"?

Some thought about what the "observations are" is helpful. On the map analogy, "observables" might be the length of trips (curves) that we draw on the map. Then the mathematical point is that rotating the map doesn't change the length of any curve, of any trip.

I say "borrow an analogy" because the original inspiration for this is a section called "The Parable of the Surveyor" from Taylor & Wheeler's "Space-time physics". Note that a "change in reference frame" in special relativity is called a Lorentz boost (it's also a diffeomorphism, like the others, a specific example that applies to SR), and it's mathematically quite similar to the mathematics that describe rotating a map, which is the original point of the analogy.

 

[laymanB]

A lot depends on what you mean by "different solution". If you have some solution to EInstein's field equations, and you derive a new solution from the old solution by a mathematical transformation (in this case, the appropriate technical name for the appropriate transformation would be a diffeomorphism), do you regard the solution as "different"?

I suspect from your question that you do, but I regard the solutions as equivalent, and I'd describe changing the coordinates via a transformation to a new set of coordinates gives a different representation of the same solution, not a different solution.

To borrow an analogy, suppose you have a map of some section of land. And you rotate the map by some angle. Is it a "different map" after you rotate it, or is it "the same map, rotated"?

Some thought about what the "observations are" is helpful. On the map analogy, "observables" might be the length of trips (curves) that we draw on the map. Then the mathematical point is that rotating the map doesn't change the length of any curve, of any trip.

I say "borrow an analogy" because the original inspiration for this is a section called "The Parable of the Surveyor" from Taylor & Wheeler's "Space-time physics". Note that a "change in reference frame" in special relativity is called a Lorentz boost (it's also a diffeomorphism, like the others, a specific example that applies to SR), and it's mathematically quite similar to the mathematics that describe rotating a map, which is the original point of the analogy.

Thanks, most of this is above my current knowledge but I think I understand what you are saying about derivations of prior equations not being "new" ones. And I think I can understand qualitatively about changing coordinate systems through transformations not altering invariant quantities.

Let me start with some more basic questions.

Are there solutions to the EFEs that are logically and mathematically consistent, yet no one uses them?

How do metrics like FRW correspond to the EFEs?

Thanks for your time.

 

[PeterDonis]

Are there solutions to the EFEs that are logically and mathematically consistent, yet no one uses them?

If by "uses them" you mean uses for practical modeling of something we expect to actually observe, sure: the Godel metric is the first example that comes to my mind. I'm sure there are many others that nobody has bothered to discover--after all, there are an infinite number of possible solutions, mathematically speaking.

How do metrics like FRW correspond to the EFEs?

They are solutions of them.

 

[Ibix]

How do metrics like FRW correspond to the EFEs?

Einstein's field equations relate the geometry of spacetime to the matter and energy distribution. You specify how the matter behaves and where it is at some point. You feed that into the EFEs and get sixteen simultaneous non-linear differential equations. If you've chosen a friendly setup many of those equations turn out to be 0=0 and you may be able to solve the rest. If not, you need a powerful computer.

Either way you end up with a metric tensor, which describes the geometry of spacetime given that distribution of matter. Schwarzschild started with a universe empty except for a spherically symmetric mass and derived the Schwarzschild metric, which works well for things like the solar system where 99%-ish of the mass is the Sun. Friedmann, Lemaitre, Robertson and Walker started with a universe completely filled with a same-everywhere mass distribution and came up with the FRW or FLRW metric, which works well on the scale where galaxies are dust grains.

There are a fair few known solutions, but an awful lot of interesting stuff (e.g. the binary mergers LIGO detects) can only be done numerically.

 

[laymanB]

Einstein's field equations relate the geometry of spacetime to the matter and energy distribution. You specify how the matter behaves and where it is at some point. You feed that into the EFEs and get sixteen simultaneous non-linear differential equations. If you've chosen a friendly setup many of those equations turn out to be 0=0 and you may be able to solve the rest. If not, you need a powerful computer.

Either way you end up with a metric tensor, which describes the geometry of spacetime given that distribution of matter. Schwarzschild started with a universe empty except for a spherically symmetric mass and derived the Schwarzschild metric, which works well for things like the solar system where 99%-ish of the mass is the Sun. Friedmann, Lemaitre, Robertson and Walker started with a universe completely filled with a same-everywhere mass distribution and came up with the FRW or FLRW metric, which works well on the scale where galaxies are dust grains.

There are a fair few known solutions, but an awful lot of interesting stuff (e.g. the binary mergers LIGO detects) can only be done numerically.

Thanks. That helps my understanding.

 

[StandardsGuy]

I'm not sure the question has been answered with all the pushback. I thought it took two massive objects moving near each other to produce gravitational waves. Is that correct? If you say it only takes one, can you prove that with observed data?

 

[Ibix]

I'm not sure the question has been answered with all the pushback. I thought it took two massive objects moving near each other to produce gravitational waves. Is that correct? If you say it only takes one, can you prove that with observed data?

So far we have three data points, all of which are for two bodies. However - join the two bodies by a string. Then you've got one body that isn't massively different from a system we've spotted emitting gravitational waves.

More seriously, the source term for gravitational waves is a time varying quadrupole moment - so in principle a dumbell or a rod spinning will produce them. In practice, to produce detectable quantities of gravitational waves you're going to need stellar masses, and we don't have any materials strong enough to avoid simply collapsing into a near-sphere under their own weight at that scale. So I think all detectable sources are likely to be two-body sources for some time to come.