I Physical meaning of a spacelike geodesic

[accdd]

I understand what is the physical meaning of a timelike geodesic, but what is the physical meaning of a spacelike geodesic?

 

[Dale]

It is a path that goes straight from one place to another.

 

[martinbn]

It is a path that goes straight from one place to another.

He is asking about a physical meaning. This seems more like a geometric meaning.

 

[Dale]

Geometry is an important part of physics. I would even say it is central

 

[martinbn]

Geometry is an important part of physics. I would even say it is central

Well, technically geometry is not part of physics. Of course it is important for physics, but it is not part of it. That aside, I think that when people ask about physical meaning, they don't mean geometry.

 

[Ibix]

A geodesic passing through two events is a path with an extremum of distance (or interval, in this case) between the events.

If one exists, a spacelike geodesic passing through two events is the longest (usually) distance between those two events. In curved spacetime there can be multiple geodesics through two events and each one is an extremum locally, meaning that all nearby paths are longer or all nearby paths are shorter.

 

[accdd]

Thank you all.
Dale answered my question.
For me, the geometric point of view counts as physical meaning.

 

[martinbn]

I'd argue that spacelike geodesics do not have physical meaning.

 

[PeterDonis]

I'd argue that spacelike geodesics do not have physical meaning.

I would argue that they do. If you don't want to accept @Dale's statement as it stands, consider:

Arc length along a timelike geodesic represents the proper time of an object in free fall.

Arc length along a spacelike geodesic represents the proper length of an object in free fall.

 

[Dale]

Well, technically geometry is not part of physics. Of course it is important for physics, but it is not part of it. That aside, I think that when people ask about physical meaning, they don't mean geometry.

I completely disagree with this. Any time you have an equation with an $x$ or a $t$ or something similar then geometry is part of that equation. In physics there are many equations that have geometry built into the equation, so geometry is indeed part of physics. I haven't gone through to count, but I would suspect that in physics there are more equations with geometry than without. So geometry is most definitely part of physics, as much as mass or energy or any other quantity that we put into our equations.

I'd argue that spacelike geodesics do not have physical meaning.

There are substantially fewer equations that use spacelike geodesics, so I wouldn't argue strongly against this, but I don't share that opinion. The phrase "physical meaning" is the slippery part here.

 

[martinbn]

I would argue that they do. If you don't want to accept @Dale's statement as it stands, consider:

Arc length along a timelike geodesic represents the proper time of an object in free fall.

That I understand and it makes perfect sense to me. It has physical meanin (whatever that may mean) a clock with that worldline would be measuring the arc length.

Arc length along a spacelike geodesic represents the proper length of an object in free fall.

This I don't understand this withoud some further specification. It seems that it has implicit underlining assumptions and conventions. My guess would be the following. An extended object is represented by its worldsheet in spacetime. I am guessing that you need a spacelike foliation and the intersection with each leaf and the worldsheet would give the length. There is a lot of freedom in the choice of a folitation without the necessity of it having any physical meaning. I suppose that you have in mind a special choice that reflects the being at rest for the extended object and hence giving proper length, but I am not sure what that might be.

But I still cannot see what the possible phycial meaning could be. Any atempt to measure any sort of length would fail (it seems to me) unless you have a static or stationary spacetime and/or have some convention in mind. It is very similar to claiming that there is physical meaning to some implicit choice of simultaneity convention in special relativity.

 

[martinbn]

I completely disagree with this. Any time you have an equation with an $x$ or a $t$ or something similar then geometry is part of that equation. In physics there are many equations that have geometry built into the equation, so geometry is indeed part of physics. I haven't gone through to count, but I would suspect that in physics there are more equations with geometry than without. So geometry is most definitely part of physics, as much as mass or energy or any other quantity that we put into our equations.

May be I misunderstood what you meant by geometry is part of physics. I thiught you meant it in the same way as in electrodynamics is part of physics, that the subject geometry is a subfield of physics. In any case I agree that a lot og geometry is used in physics. But I still disagree that geometric meaning answers the question about physical meaning. Well, the OP clarified that it was fine, so that's that.

There are substantially fewer equations that use spacelike geodesics, so I wouldn't argue strongly against this, but I don't share that opinion. The phrase "physical meaning" is the slippery part here.

I agree that physical meaning is unclear and can lead us in any of a number of directions.

 

[Dale]

May be I misunderstood what you meant by geometry is part of physics. I thiught you meant it in the same way as in electrodynamics is part of physics, that the subject geometry is a subfield of physics.

Oh, I can see how my statement would have made you think I intended to say that. My apologies. I don’t mean as an academic discipline that geometry is a sub discipline within physics. Unfortunately I don’t know a better way to say what I intended.

I mean that there is a geometrical relationship between physical objects. Geometry is therefore part of the description of how physical objects interact. This is reflected in the equations we use to describe the objects and their interactions which almost universally include geometrical variables.

 

[cianfa72]

An extended object is represented by its worldsheet in spacetime. I am guessing that you need a spacelike foliation and the intersection with each leaf and the worldsheet would give the length. There is a lot of freedom in the choice of a folitation without the necessity of it having any physical meaning. I suppose that you have in mind a special choice that reflects the being at rest for the extended object and hence giving proper length, but I am not sure what that might be.

But I still cannot see what the possible physical meaning could be. Any attempt to measure any sort of length would fail (it seems to me) unless you have a static or stationary spacetime and/or have some convention in mind. It is very similar to claiming that there is physical meaning to some implicit choice of simultaneity convention in special relativity.

That's my viewpoint too. Unlike the timelike geodesic case, there is no way to attach any "intrinsic/independent" physical meaning to a spacelike geodesic until you pick some convention for a spacelike foliation of the extended object worldtube (as you said it is alike the simultaneity convention).

We had a thread on this topic some time ago...

 

[PeterDonis]

There is a lot of freedom in the choice of a folitation

In general, yes, but if you have a specific worldline that is a geodesic, then there is a unique foliation defined on at least some "world tube" surrounding that worldline: just take the spacelike 3-surface that is orthogonal to the worldline at each event. That "stack" of 3-surfaces, within the "world tube" where none of them cross (eventually some will in a curved spacetime, but there will be some finite "world tube" surrounding the worldline where they do not), is the unique foliation. It is unique because of the orthogonality condition.

Technically you can do this for a non-geodesic worldline as well, but because of the path curvature of the worldline, the surfaces of the foliation might cross sooner.

 

[cianfa72]

just take the spacelike 3-surface that is orthogonal to the worldline at each event. That "stack" of 3-surfaces, within the "world tube" where none of them cross (eventually some will in a curved spacetime, but there will be some finite "world tube" surrounding the worldline where they do not), is the unique foliation. It is unique because of the orthogonality condition.

From what we said in other threads on this topic, if the worldtube geodesic congruence has not zero vorticity then the spacelike hypersurfaces othogonal to the congruence's worldlines at each event will not 'join' together (i.e. Frobenius's theorem condition will not be met hence the worldlines' orthogonal spacelike 3D distribution at each event will not be integrable).

 

[Orodruin]

there can be multiple geodesics through two events and each one is an extremum locally, meaning that all nearby paths are longer or all nearby paths are shorter.

This is not true for saddle points (such as the long part of the great circle passing through two points on a sphere).

 

[PeterDonis]

if the worldtube geodesic congruence has not zero vorticity then the spacelike hypersurfaces othogonal to the congruence's worldlines at each event will not 'join' together

If extended out far enough, yes. But even in this case, there will still be a finite "world tube" around the worldline in which the foliation I described can be used without the issue you describe arising. As the vorticity increases, the finite size of this "world tube" will in general decrease.

 

[cianfa72]

But even in this case, there will still be a finite "world tube" around the worldline in which the foliation I described can be used without the issue you describe arising. As the vorticity increases, the finite size of this "world tube" will in general decrease.

Ah ok, basically you are saying that if we pick a worldline and consider the neighboring worldlines in the congruence in a small region of spacetime around it, then the orthogonal 3D spacelike distribution at each event along them meets locally the Frobenius's conditions.

 

[PeterDonis]

basically you are saying that if we pick a worldline and consider the neighboring worldlines in the congruence in a small region of spacetime around it, then the orthogonal 3D spacelike distribution at each event along them meets locally the Frobenius's conditions.

No. The congruence still has nonzero vorticity and still does not meet the Frobenius conditions. But the non-integrability due to those conditions not being met is not a local property. (In general, anything involving integrals is not a local property; the whole point of integrals is to evaluate properties that involve extension over some region, not just properties at a point.)

 

[cianfa72]

But the non-integrability due to those conditions not being met is not a local property. (In general, anything involving integrals is not a local property; the whole point of integrals is to evaluate properties that involve extension over some region, not just properties at a point.)

So basically what you are saying amounts to the following: take the set of orthogonal 3D spacelike subspaces in tangent spaces at each event along the given worldline and for each of them evaluate the exponential map. Such maps will define in a finite region of spacetime around the given worldline the spacelike foliation you were talking about.

 

[Ibix]

This is not true for saddle points (such as the long part of the great circle passing through two points on a sphere).

Fair enough. Is a great circle arc really a saddle point? I'd have thought that by symmetry nearby paths on one side of the arc had the same lengths as ones on the other side of it. Or is there some sense of 'variation' that I'm missing?

 

[Orodruin]

Fair enough. Is a great circle arc really a saddle point? I'd have thought that by symmetry nearby paths on one side of the arc had the same lengths as ones on the other side of it. Or is there some sense of 'variation' that I'm missing?

They do. But a functional space is infinite dimensional. You cannot restrict yourself to a single shape of the perturbation.

Easy way to see that long way around is a saddle point: Use coordinates such that great circle is the equator. Expand perturbation in Fourier series. If there is less than $\pi R$ in distance between the points, all modes will lead to a longer path. If there is more, then the first mode will lead to a shorter path and the others to longer. Hence, a saddle point.

 

[watcher1]

I believe that the physical meaning of a spacelike geodesic between two points is simply the path a beam of light will follow between those points. Any other definition is, as many of the above responders have said, only a geometric meaning.

 

[DrGreg]

I believe that the physical meaning of a spacelike geodesic between two points is simply the path a beam of light will follow between those points. Any other definition is, as many of the above responders have said, only a geometric meaning.

We are discussing geodesics between events in 4D spacetime, not geodesics between points in 3D space. In spacetime light follows a null geodesic.

Whether or not light follows a geodesic in space depends on how you choose to extract a "space" from spacetime.

 

[PeterDonis]

take the set of orthogonal 3D spacelike subspaces in tangent spaces at each event along the given worldline and for each of them evaluate the exponential map. Such maps will define in a finite region of spacetime around the given worldline the spacelike foliation you were talking about.

Yes, that's it.

 

[Orodruin]

They do. But a functional space is infinite dimensional. You cannot restrict yourself to a single shape of the perturbation.

Easy way to see that long way around is a saddle point: Use coordinates such that great circle is the equator. Expand perturbation in Fourier series. If there is less than $\pi R$ in distance between the points, all modes will lead to a longer path. If there is more, then the first mode will lead to a shorter path and the others to longer. Hence, a saddle point.

To make it a bit more hands-on, I managed to dig out this post from my first months on PF:
Post in thread 'Understand the major arc connecting two points on a sphere'
https://www.physicsforums.com/threa...ng-two-points-on-a-sphere.767025/post-4829841

 

[cianfa72]

Easy way to see that long way around is a saddle point: Use coordinates such that great circle is the equator. Expand perturbation in Fourier series. If there is less than $\pi R$ in distance between the points, all modes will lead to a longer path.

You mean take two points on the great circle on the sphere (e.g. on the equator) and evaluate the distance between them along the great circle via the short way and the long way around. In the general case (i.e. not antipodal points) the two will be different.

Now for the long (geodesic) path the evaluated distance will be greater than $\pi R$. For this case your following argument applies (i.e. the long path around is actually a saddle point).

If there is more, then the first mode will lead to a shorter path and the others to longer. Hence, a saddle point.

 

[Demystifier]

It is a path that goes straight from one place to another.

A straight answer.

 

[Orodruin]

You mean take two points on the great circle on the sphere (e.g. on the equator) and evaluate the distance between them along the great circle via the short way and the long way around. In the general case (i.e. not antipodal points) the two will be different.

Now for the long (geodesic) path the evaluated distance will be greater than $\pi R$. For this case your following argument applies (i.e. the long path around is actually a saddle point).

Yes, the long way around is a saddle point. The short way is the global minimum.

 

[Orodruin]

A straight answer.

A geodesic answer.

 

[Demystifier]

I'd argue that spacelike geodesics do not have physical meaning.

How would you build the optimal road from one city to another? I would build a straight road, that is a spacelike geodesic.

 

[Ibix]

How would you build the optimal road from one city to another? I would build a straight road, that is a spacelike geodesic.

Well, it's a worldsheet with several spacelike geodesics lying in it. The intersection of that sheet with a member of some foliation of spacetime (i.e. given some definition of "now") may be a spacelike geodesic.

 

[cianfa72]

Well, it's a worldsheet with several spacelike geodesics lying in it. The intersection of that sheet with a member of some foliation of spacetime {i.e. given some definition of "now") may be a spacelike geodesic.

I believe the only physical requirement for a spacelike foliation to be effective is that the events on it cannot be connected by a light rays or by the paths of massive objects.

 

[Ibix]

I believe the only physical requirement for a spacelike foliation to be effective is that the events on it cannot be connected by a light rays or by the paths of massive objects.

Yes. That's why I said its intersection with the road's worldsheet may be a spacelike geodesic. It doesn't have to be.

 

[DrGreg]

It's perhaps worth mentioning, in passing, that there are some circumstances when you can extract "space" from spacetime as a quotient manifold (instead of by foliation). In this special case, "space" is not a submanifold of spacetime, and I suspect that geodesics in the quotient space do not correspond to geodesics in spacetime. Arguably the quotient space represents "physical space" in these cases. (I say "arguably" because I'm not sure everyone agrees.)

(For those not familiar with this, it becomes relevant if you are using a timelike congruence with vorticity, e.g. representing a rotating cylinder in flat spacetime, or more generally the timelike KVF of any stationary-but-not-static spacetime.)

 

[Orodruin]

It's perhaps worth mentioning, in passing, that there are some circumstances when you can extract "space" from spacetime as a quotient manifold (instead of by foliation). In this special case, "space" is not a submanifold of spacetime, and I suspect that geodesics in the quotient space do not correspond to geodesics in spacetime. Arguably the quotient space represents "physical space" in these cases. (I say "arguably" because I'm not sure everyone agrees.)

(For those not familiar with this, it becomes relevant if you are using a timelike congruence with vorticity, e.g. representing a rotating cylinder in flat spacetime, or more generally the timelike KVF of any stationary-but-not-static spacetime.)

Can you give a specific example of this construction?

 

[DrGreg]

Can you give a specific example of this construction?

For Born coordinates, which represent a rotating cylinder in flat spacetime, there's an example at Born coordinates § Radar distance in the small on Wikipedia.

The essential idea is that a "point in space" corresponds to a member of the congruence of timelike wordlines (the flow of the KVF in a stationary spacetime). "Lying on the same worldline" defines an equivalence relation on spacetime which allows you construct a quotient space. The metric on the quotient space defines the distance between two "infinitesimally close" points to be the orthogonal distance between two worldlines; in a stationary spacetime that distance is constant.

In practice you take a "stationary" coordinate system $(t, x^1, x^2, x^3)$ in which all components of the metric are independent of $t$, and "complete the square" on the metric quadratic form to get it in the form$$
\text{d}s^2 = A(\text{d}t - \omega_i \, \text{d}x^i)^2 - h_{ij} \, \text{d}x^i \, \text{d}x^j
$$and then $h$ is the metric on the quotient space with coordinates $(x^1, x^2, x^3)$. (In a static spacetime you can put $\omega_i = 0$.)

See also Stationary spacetime on Wikipedia.

 

[Orodruin]

For Born coordinates, which represent a rotating cylinder in flat spacetime, there's an example at Born coordinates § Radar distance in the small on Wikipedia.

The essential idea is that a "point in space" corresponds to a member of the congruence of timelike wordlines (the flow of the KVF in a stationary spacetime). "Lying on the same worldline" defines an equivalence relation on spacetime which allows you construct a quotient space. The metric on the quotient space defines the distance between two "infinitesimally close" points to be the orthogonal distance between two worldlines; in a stationary spacetime that distance is constant.

In practice you take a "stationary" coordinate system $(t, x^1, x^2, x^3)$ in which all components of the metric are independent of $t$, and "complete the square" on the metric quadratic form to get it in the form$$
\text{d}s^2 = A(\text{d}t - \omega_i \, \text{d}x^i)^2 - h_{ij} \, \text{d}x^i \, \text{d}x^j
$$and then $h$ is the metric on the quotient space with coordinates $(x^1, x^2, x^3)$. (In a static spacetime you can put $\omega_i = 0$.)

See also Stationary spacetime on Wikipedia.

I do not see how this is fundamentally different from providing a foliation of spacetime. It is a particular foliation for a stationary spacetime based on the existence of a timelike KVF.

 

[PeterDonis]

It is a particular foliation for a stationary spacetime based on the existence of a timelike KVF.

No, it isn't. The quotient space is not a "surface of constant time" corresponding to the timelike KVF and does not correspond to any foliation of the spacetime. We have had plenty of previous PF threads on this topic.

You can of course construct a foliation of Minkowski spacetime using the Langevin congruence (which is the congruence that is "at rest" in Born coordinates), but that foliation's leaves will be identical to surfaces of constant time in the inertial chart in which the center of rotation is at rest, and of course the geometry of those leaves will be flat. The geometry of the quotient space, however, is not flat, and there is no way to foliate Minkowski spacetime with a "stack" of copies of it.

 

[DrGreg]

I do not see how this is fundamentally different from providing a foliation of spacetime. It is a particular foliation for a stationary spacetime based on the existence of a timelike KVF.

But the metric on the quotient space is not the metric that would be induced on a 3-surface within the 4-manifold. In the example I gave, if you foliate into surfaces of constant $t$ you get the surface metric by putting $\text{d}t = 0$ into the 4-manifold metric i.e.$$
\text{d}s^2 = -A(\omega_i \, \text{d}x^i)^2 + h_{ij} \, \text{d}x^i \, \text{d}x^j
$$But that's not $h$.

The geodesics of $h$ need not correspond to geodesics in the 4-manifold, or geodesics in a surface of constant $t$. You can't even map a curve in the quotient space to a curve in 4-space, as one space isn't a subspace of the other.

 

[Orodruin]

But the metric on the quotient space is not the metric that would be induced on a 3-surface within the 4-manifold. In the example I gave, if you foliate into surfaces of constant t you get the surface metric by putting dt=0 into the 4-manifold metric i.e.ds2=−A(ωidxi)2+hijdxidxjBut that's not h.

Sure, but the general question becomes: Why is h of interest in and of itself? What physics can you extract from h and its geodesics?

 

[cianfa72]

but that foliation's leaves will be identical to surfaces of constant time in the inertial chart in which the center of rotation is at rest, and of course the geometry of those leaves will be flat.

Just to check my understanding: the inertial chart in which the center of rotation is at rest is actually unique up to Lorentz transformations either of timelike coordinate or any of spacelike coordinate.

 

[vanhees71]

Geometry is an important part of physics. I would even say it is central

From a mathematical point of view it looks as if physics is just a subset of geometry (nearly ;-)).

 

[PeterDonis]

the inertial chart in which the center of rotation is at rest is actually unique up to Lorentz transformations either of timelike coordinate or any of spacelike coordinate.

I'm not sure what you mean by "Lorentz transformations". It's unique up to time translations along the worldline of the center of rotation, and spatial rotations about that axis. But you can't apply a Lorentz boost since that would change which timelike geodesic is "at rest".

 

[DrGreg]

Sure, but the general question becomes: Why is h of interest in and of itself? What physics can you extract from h and its geodesics?

The claim was made earlier in the thread that a spacelike geodesic (in spacetime of course) somehow represented the "geometry of physical space" (loosely speaking, whatever that means). I just mentioned this as a counterexample to the converse. I would argue that $h$ represents the "geometry of physical space" for a Langevin observer (e.g.) but the geodesics of $h$ don't have any obvious correspondence (as far as I can tell) with spacelike geodesics of spacetime.

E.g. as a practical experiment, if you were sitting on a rotating disk and took your time to measure its geometry with a ruler or a tape measure (no clocks or simultaneity conventions required because the disk is Born rigid and its geometry is independent of time), the answer you would get would be $h$.

 

[PeterDonis]

The claim was made earlier in the thread that a spacelike geodesic (in spacetime of course) somehow represented the "geometry of physical space"

The claim I made in post #9 was more limited than that; it was about a spacelike geodesic representing the proper length of an object, not the "geometry of physical space". The former is more of a local concept; the latter is more of a global concept. In the case of the rotating disk, local measurements of the proper lengths of objects rotating with the disk would correspond to short segments of spacelike geodesics orthogonal to the worldlines of those objects, but the way those local measurements fit together globally into a "geometry of space" would correspond to the geometry of the quotient space you described.

 

[cianfa72]

The metric on the quotient space defines the distance between two "infinitesimally close" points to be the orthogonal distance between two worldlines; in a stationary spacetime that distance is constant.

In a stationary-but-not-static spacetime in the "stationary" coordinates by definition the 3D hypersurfaces of constant $t$ all have the same (spatial) geometry. So the distance between neighboring worldlines of the KVF timelike congruence evaluated on each of such hypersurfaces does not change. However it is not the same as the orthogonal distance between them (since the KVF timelike congruence is not hypersurface orthogonal).

 

[vanhees71]

I'm not sure what you mean by "Lorentz transformations". It's unique up to time translations along the worldline of the center of rotation, and spatial rotations about that axis. But you can't apply a Lorentz boost since that would change which timelike geodesic is "at rest".

Lorentz transformations in GR are local transformations, i.e., they refer to transformations in the tangent space at one spacetime point. A local inertial frame is defined in the tangent space at one spacetime point and a tetrad as basis. Choosing another tetrad of course leads to another inertial frame, and the corresponding transformation is a Lorentz transformation.

 

[martinbn]

How would you build the optimal road from one city to another? I would build a straight road, that is a spacelike geodesic.

One city and another city (considering them without extend so they are points) do not specify events in spacetime. You have some implicit assumptions. May be you are thinking of the two cities at the same time, which leads to simultaneity conventions. Or you are assuming a stationary spacetime. Or something else.