The spacetime length of finite spacelike intervals

[cianfa72]

TL;DR Summary

In the context of SR ad GR how interpret and measure the length of finite spacelike intervals

Hello,

I'm aware of the following topic has already been discussed here on PF, nevertheless I would like to go deep into the concept of "finite spacelike interval" in the context of SR and GR.

All us know the physical meaning of timelike paths: basically they are paths followed through spacetime from massive objects. Furthermore we also know how to measure this kind of path length: just use a wristwatch attached to it and then measure the time elapsed from the initial to the final event through the path itself.

What about a "finite spacelike path" ? We know it is not the path followed by any physical object. However, from a local point of view, we can interpret it as the spacelike direction of a spatial axis carried by an observer (basically the spatial axis of a tetrad field)

Then if we consider a finite spacelike interval how should we actually interpret and physically measure it ? Thanks.

 

[mitochan]

Hello. Wikipedia Metre says:
The metre is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second.

It means length is measured by light speed and second.
You send light and measure time light to comeback by reflection, say t, then space like distance to the reflection point is ct/2.

 

[martinbn]

Then if we consider a finite spacelike interval how should we actually interpret and physically measure it ? Thanks.

This depends on the choice of a spacelike hypersurface so it doesn't have that much physical meaning. It is also impossible to measure. If you are in a special spacetime, for example static, you could do what you would expect from non-relativistic intuition. Or if you have the spacetime metric and two events that belong to a chosen spacelike hypersurface you can compute the spatial distance between them. But as I said this has little significance.

 

[vanhees71]

Fortunately we can obviously measure distances in non-static spacetimes. E.g., in cosmology we can measure the luminosity distance to, e.g., Type 1a supernovae and relate it to the Hubble-Lemaitre red shift and determine the Hubble constant $H_0$, though that's far from being trivial.

 

[cianfa72]

This depends on the choice of a spacelike hypersurface so it doesn't have that much physical meaning. It is also impossible to measure. If you are in a special spacetime, for example static, you could do what you would expect from non-relativistic intuition. Or if you have the spacetime metric and two events that belong to a chosen spacelike hypersurface you can compute the spatial distance between them. But as I said this has little significance.

I hope the following definition of spacelike path makes sense: a path through spacetime having a spacelike vector as tangent vector to each point (event) on it.

As I take it is actually possible measure its length only when choosing a spacelike hypersurface on which the spacelike path lies on, otherwise it actually makes no sense.

 

[martinbn]

I hope the following definition of spacelike path makes sense: a path through spacetime having a spacelike vector as tangent vector to each point (event) on it.

As I take it is actually possible measure its length only when choosing a spacelike hypersurface on which the spacelike path lies on, otherwise it actually makes no sense.

This is the definition of a spacelike curve, and its length is well defined, you don't need hypersurfaces for this. But i don't think that this has any physical implications.

 

[cianfa72]

This is the definition of a spacelike curve, and its length is well defined, you don't need hypersurfaces for this. But i don't think that this has any physical implications.

Therefore for a spacelike curve the length is well defined however it has no physical meaning nor it can be physically measured alike timelike curves

 

[cianfa72]

This is the definition of a spacelike curve, and its length is well defined, you don't need hypersurfaces for this. But i don't think that this has any physical implications.

Thinking again about it: in the context of SR (flat spacetime) we actually 'attach' a physical meaning to spacelike intervals and we are also able to measure their lengths. Take for instance a physical latticework made of rigid rods and clocks in inertial coordinate system and consider the spacelike path made of simultaneous events (w.r.t the inertial coordinate system chosen) belonging to a rod world-tube.

 

[PeterDonis]

in the context of SR (flat spacetime) we actually 'attach' a physical meaning to spacelike intervals

Yes.

we are also able to measure their lengths

Only indirectly. See below.

Take for instance a physical latticework made of rigid rods and clocks in inertial coordinate system and consider the spacelike path made of simultaneous events (w.r.t the inertial coordinate system chosen) belonging to a rod world-tube.

This construction does not directly measure the spacelike path you describe, since no single measuring device can possibly be at both of a pair of spacelike separated events. The measurement relies on the physical latticework already having been set up, clocks already having been synchronized, etc., and all of those processes occur over timelike or null intervals. Only after all that has been done can we infer the length of the particular spacelike path you describe.

 

[cianfa72]

This construction does not directly measure the spacelike path you describe, since no single measuring device can possibly be at both of a pair of spacelike separated events. The measurement relies on the physical latticework already having been set up, clocks already having been synchronized, etc., and all of those processes occur over timelike or null intervals. Only after all that has been done can we infer the length of the particular spacelike path you describe.

Thanks, I believe I got your point. By the way I found the following link about the process can be used to measure the spacelike path length in case of stationary spacetime (i.e. spacetime admitting a timelike Killing symmetry)

https://physics.stackexchange.com/q...-and-lightlike-spacetime-interval-really-mean

 

[PeterDonis]

the process can be used to measure the spacelike path length in case of stationary spacetime

Where is such a process described on the page you linked to?

 

[pervect]

Summary:: In the context of SR ad GR how interpret and measure the length of finite spacelike intervals

Hello,

I'm aware of the following topic has already been discussed here on PF, nevertheless I would like to go deep into the concept of "finite spacelike interval" in the context of SR and GR.

All us know the physical meaning of timelike paths: basically they are paths followed through spacetime from massive objects. Furthermore we also know how to measure this kind of path length: just use a wristwatch attached to it and then measure the time elapsed from the initial to the final event through the path itself.

What about a "finite spacelike path" ? We know it is not the path followed by any physical object. However, from a local point of view, we can interpret it as the spacelike direction of a spatial axis carried by an observer (basically the spatial axis of a tetrad field)

Then if we consider a finite spacelike interval how should we actually interpret and physically measure it ? Thanks.

It's just the proper length of the curve, which is a spatial length. Mathematically, it has the opposite sign as the proper time that you've already discussed, but it's only a sign difference, it's mathematically a similar invariant interval.

The physical interpretation is a bit trickier, you might ask "the length according to what observer(s)"? The theory of timelike congruences comes in handy here, one can regard a timelike congruence of worldlines as an sort of generalized observer. I don't know if you are familiar with time-like congruences, I'm not sure how much to explain - I will assume for now that you are familiar enough with them that I can make my point, rather than digress into an explanation and lose the point I'm trying to make. If you are not familiar the point may get lost, but then I suppose we can try and discuss the background if you find the idea interesting to see how it applies to your specific quesiton.

The point is that if at least one timelike congruence of worldines exists that is everywhere orthogonal to the space-like curve in question, the invariant length of the curve is naturally the spatial length of the curve associated with said time-like congruence.

Mathematically, the time-like congruence of worldlines allows us to separate space and time by a projection operator. A projection operator that maps every event on a worldline in the congruence to the same point "projects" a space-time event into a spatial location. This then allows us to discuss "spatial length", and "spatial geometry", via the induced metric, as we've projected our 4d space-time into a 3d space. It's best for interpreation if the space-time is static.

 

[cianfa72]

Where is such a process described on the page you linked to?

Sorry my fault

https://physics.stackexchange.com/q...eodesics-or-what-is-the-physical-interpretati

 

[cianfa72]

Mathematically, the time-like congruence of worldlines allows us to separate space and time by a projection operator. A projection operator that maps every event on a worldline in the congruence to the same point "projects" a space-time event into a spatial location. This then allows us to discuss "spatial length", and "spatial geometry", via the induced metric, as we've projected our 4d space-time into a 3d space. It's best for interpreation if the space-time is static.

Thanks, I believe that was the point in the link attached so far (sorry I attached the wrong one)

 

[PeterDonis]

Sorry my fault

https://physics.stackexchange.com/q...eodesics-or-what-is-the-physical-interpretati

I don't see any process for directly measuring spacelike path lengths, either in a stationary spacetime or not, described here. Please quote specifically the part that you think does that.

I do see a "theoretical" process described, but the person who posted it immediately admits that it cannot be realized in practice because doing so would require a single measuring device to instantaneously make measurements at spatially separated locations, which is not possible.

The same post describes how having a stationary spacetime does help somewhat, because it at least allows you to measure, not instantaneously, a spacelike path length, by making measurements at different positions at different times and then time translating the results. "Stationary" means, roughly, "things don't change when you time translate", so if, for example, I measure the positions of two ends of a rod at different times, I can assume that the positions don't change with time, and nothing else that would affect the measurements (like the spacetime geometry) does either, so I can "time translate" one of the measurements to the same time as the other and thereby infer the path length along a spacelike curve that describes the rod "at one instant of time". This all works fine, but it still does not count as a direct measurement of a spacelike path length.

 

[PeterDonis]

Mathematically, the time-like congruence of worldlines allows us to separate space and time by a projection operator. A projection operator that maps every event on a worldline in the congruence to the same point "projects" a space-time event into a spatial location. This then allows us to discuss "spatial length", and "spatial geometry", via the induced metric, as we've projected our 4d space-time into a 3d space. It's best for interpreation if the space-time is static.

Unfortunately, even having a static spacetime is not really a help, because the important thing is not whether the spacetime is static but whether the congruence is. There are many timelike congruences in any static spacetime which are not static--because they are not formed from integral curves of the timelike Killing vector field.

What's more, if we want to find geodesic congruences that are static, in general, there will not be any even in a static spacetime. For example, Schwarzschild spacetime is static, but the static congruence is not geodesic; it is the congruence of "hovering" observers, who each stay motionless at the same altitude above the central gravitating mass, and must have nonzero proper acceleration to do so. They can indirectly measure spacelike path lengths between them, using the method I described in my previous post, and the spacelike curves whose lengths they are will be geodesics of the spacelike 3-surfaces in which they lie, but the observers' worldlines themselves will not be geodesics. I'm not aware of any static [Edit: curved] spacetime in which the Killing congruence is a geodesic congruence. (I do know of one example of a stationary, but not static, spacetime whose Killing congruence is geodesic: Godel spacetime. But Godel spacetime is widely viewed as physically unreasonable.)

 

[pervect]

For the purposes of physical insight, my opinion is that the most useful congruences are the ones where the tangents to the worldlines making up the congruence are Killing vectors, i.e. the worldlines of the congruence are integral curves of the Killing vectors. But sometimes one might want other choices. It doesn't matter that much to the point I wanted to make , which was that the congruence is one of the best tools for gaining physical insight, breaking apart space-time into space and time.

Fermi-Normal coordinates are a strong runner up for gaining physical insight, but they're not quite as general and useful (all my opinion) as congruences. But both are very important to get physical insight, unfortunately they're introduced at a rather late date (if at all) in most textbooks.

 

[PeterDonis]

For the purposes of physical insight, my opinion is that the most useful congruences are the ones where the tangents to the worldlines making up the congruence are Killing vectors, i.e. the worldlines of the congruence are integral curves of the Killing vectors.

I agree that such congruences are easy to interpret physically. The problem is that some very important congruences in commonly used spacetimes are not such congruences. For example, the congruence of "comoving" worldlines in FRW spacetime is not a Killing congruence. Also, as I noted before, almost all Killing congruences in known spacetimes are not geodesic congruences, which means that some important intuitions from inertial frames in flat spacetime do not carry over.

(Another issue is that important stationary spacetimes are not static, which means their Killing congruences are not hypersurface orthogonal. That also invalidates some important intuitions and properties.)

 

[pervect]

Congruences with vorticity, such that there is no spacelike hypersurface orthogonal to the congruence, can indeed be non-intuitive. This shows up in the "Erhenfest paradox" problem. One of the strong features of the congruence treatment though is that it is capable of handling Erhenfest's rotating disk within the formalism. It may not be intuitive, but the formalism works in this case. So it's useful for understanding the issue, such that appllying ones knowledge of congruences can give some insight into the rotating disk. In fact, that's one of the reasons I like congruences, there are some other systems which can work under more limited circumstances but are not capable of handling the rotating disk gracefully.

The other point I wanted to make is that the formalism of the latticework of clocks and rods that is often used for special relativity generates a congruence. Every point on the lattice has a worldline. A "point in space", is just one particular worldline of the congruence, the worldline contains all events that occur at any time at that "spatial location", and the projection operator projects this worldine in 4d space to a point in 3d space, giving each event a "location", a mapping from the 4d space-time to a 3d space.

When one pick a specific frame of reference with some particular velocity in SR, one picks a particular lattice structure (of which there are an infinite number of possible lattice structures). Thus the congruence is a natural extension of the idea of a "frame of reference", one that is more general than the lattice structure. And the lattice is just a special case of the more general congruence.

The important case of the FRLW "expanding universe" is another example of where the congruence tool works, and lattices do not, at least not without modifications. The fact that the congruence can handle the FRLW expanding space-time is another example of it's generality.

For a reference that goes into congruences in more detail, I'd suget "A relativists toolkit", by Eric Poisson. I believe it only handles geodesic congruences, which I find unfortunate, many useful congruences are not geodesic congruences , however, as you point out.

 

[PeterDonis]

the formalism of the latticework of clocks and rods that is often used for special relativity generates a congruence

Yes, but it's worth noting that since it's both a geodesic congruence (all of the worldlines are freely falling) and a Killing congruence, it has a set of properties that do not all generalize, and which ones do generalize depends on the particular scenario.

 

[cianfa72]

I don't see any process for directly measuring spacelike path lengths, either in a stationary spacetime or not, described here. Please quote specifically the part that you think does that.

I do see a "theoretical" process described, but the person who posted it immediately admits that it cannot be realized in practice because doing so would require a single measuring device to instantaneously make measurements at spatially separated locations, which is not possible.

Bold in the quote is mine. I believe the point you was referring to is the following: "Very unfortunately all that I said above is theoretical, in the sense that it cannot be realized in practice. This is because "tempus fugit"

The same post describes how having a stationary spacetime does help somewhat, because it at least allows you to measure, not instantaneously, a spacelike path length, by making measurements at different positions at different times and then time translating the results. "Stationary" means, roughly, "things don't change when you time translate", so if, for example, I measure the positions of two ends of a rod at different times, I can assume that the positions don't change with time, and nothing else that would affect the measurements (like the spacetime geometry) does either, so I can "time translate" one of the measurements to the same time as the other and thereby infer the path length along a spacelike curve that describes the rod "at one instant of time". This all works fine, but it still does not count as a direct measurement of a spacelike path length.

From my understanding, the measurement proposed actually measure --indirectly as you pointed out -- the minimum of the lengths in the set of spacelike curves joining the two given (spacelike separated) events constrained to lie down (belonging) to a chosen spacelike hypersurface.

 

[cianfa72]

The point is that if at least one timelike congruence of worldines exists that is everywhere orthogonal to the space-like curve in question, the invariant length of the curve is naturally the spatial length of the curve associated with said time-like congruence.

Not sure to fully understand...Does 'the existence of one timelike congruence of worldlines everywhere orthogonal to the given spacelike curve' actually imply there exists a spacelike hypersurface the given spacelike curve belongs to ?

I think that was actually the assumption in my previously referenced link (basically the $\sum_ {t}$ hypersurface he was talking about)

 

[vanhees71]

The point is we can only do local measurements, which are, as Einstein realized after some years of struggle with the interpretation of diffeomorphism invariance (involving the infamous hole argument), coincidences of events. Spatial distances have to be defined by such measurements. E.g., concerning cosmology most things we measure is electromagnetic radiation (and more recently also gravitational waves), and to define the distance of the source of this radiation you start with distances of not too far objects using the parallax method and then use "standard candles" like Cepheids or Type 1a Supernovae with a known brightness at the source and then you can measure the distance to very far objects by assuming that all the physical laws are valid everywhere and at any time of the universe, i.e., that the brightness at the source of some identified standard candle is the same everywhere and at everytime and then infer the distance from the luminosity at your place. In this way, e.g., the redshift-distance relation and with that the Hubble constant is measured.

It's far from being trivial to take into account all kinds of perturbations like "dust" (where there is a big debate between experts, how to properly take this into account) etc. At the moment there's also some tension between measurements of the Hubble constant using different methods, and the big question is, whether this is a problem with the measurements or whether our "cosmological standard model" ("$\Lambda\text{CDM}$") is really right.

 

[mitochan]

@chianfa
In SR as for time like intervals in rest IFR and other IFR the interval is common, i.e.,
$$c^2\triangle \tau^2=c^2\triangle t^2- \triangle x^2$$
so
$$\triangle x^2= c^2\triangle \tau^2-c^2\triangle t^2$$
Why don't you apply this formula for length in that IFR?

 

[vanhees71]

If you make your $\Delta$ a $\mathrm{d}$ then that makes sense.

 

[PeterDonis]

I believe the point you was referring to is the following: "Very unfortunately all that I said above is theoretical, in the sense that it cannot be realized in practice. This is because "tempus fugit"

Yes.

From my understanding, the measurement proposed actually measure --indirectly as you pointed out -- the minimum of the lengths in the set of spacelike curves joining the two given (spacelike separated) events constrained to lie down (belonging) to a chosen spacelike hypersurface.

Yes. The constraint that the curves have to lie in the chosen spacelike hypersurface is the key thing that allows the particular spacelike curve in question (the spacelike geodesic connecting the two events) to be the curve of minimum length.

Does 'the existence of one timelike congruence of worldlines everywhere orthogonal to the given spacelike curve' actually imply there exists a spacelike hypersurface the given spacelike curve belongs to ?

There will be an infinite number of possible spacelike hypersurfaces that the given curve belongs to. But only one of them will be orthogonal to the timelike congruence the way the curve itself is. That is the spacelike hypersurface we are interested in.

 

[cianfa72]

Yes. The constraint that the curves have to lie in the chosen spacelike hypersurface is the key thing that allows the particular spacelike curve in question (the spacelike geodesic connecting the two events) to be the curve of minimum length.

Just to be clear: consider two spacelike separated events and a given spacelike curve joining them that lies on the chosen spacelike hypersurface: that curve may not actually be the curve of minimum length in the set of all spacelike curves joining the two given events and belonging to the chosen spacelike hypersurface, right ?

It is our choice to search for the minimum length curve (basically a geodesic of the induced metric on the chosen hypersurface) I believe..

 

[PeterDonis]

consider two spacelike separated events and a given spacelike curve joining them that lies on the chosen spacelike hypersurface: that curve may not actually be the curve of minimum length in the set of all spacelike curves joining the two given events and belonging to the chosen spacelike hypersurface, right ?

Not if you're allowed to choose any spacelike curve you like, no.

It is our choice to search for the minimum length curve (basically a geodesic of the induced metric on the chosen hypersurface) I believe..

The "distance" between two spacelike separated events is defined as the length of the spacelike geodesic connecting them. That's how our intuitive concept of "distance" is made precise in relativity.

 

[cianfa72]

Not if you're allowed to choose any spacelike curve you like, no.

Sorry...I might have some misunderstanding with my poor English

I take it if we are allowed to choose any spacelike curve we like joining the two events --with the constrain that it must belong to the set of spacelike curves lying on the chosen spacelike hypersurface-- it need not necessarily be a geodesic (of the induced metric on the hypersurface itself)

 

[PeterDonis]

I take it if we are allowed to choose any spacelike curve we like joining the two events --with the constrain that it must belong to the set of spacelike curves lying on the chosen spacelike hypersurface-- it need not necessarily be a geodesic (of the induced metric on the hypersurface itself)

If we are allowed to choose any such curve (not just the one of minimal length), then yes, it need not be a geodesic. But what we are "allowed" to choose depends on what you are trying to do. If you are trying to define the "distance" between two events, the geodesic is the only curve it makes sense to choose.

 

[pervect]

Not sure to fully understand...Does 'the existence of one timelike congruence of worldlines everywhere orthogonal to the given spacelike curve' actually imply there exists a spacelike hypersurface the given spacelike curve belongs to ?

I think that was actually the assumption in my previously referenced link (basically the $\sum_ {t}$ hypersurface he was talking about)

I am pretty sure the answer is no, though I am am relying on an example, rather than a reference. The example is the rotating disk, which generates a set of worldlines. There is no hypersurface orthogonal to the worldlines comprising the disk, but one can find a non-closed spacelike curve representing the circumference of the disk that's orthogonal to all the worldlines comprising the disk.

The spacelike curve starts and stops on one particular shared wordline, but the two ends of said spacelike curve are timelike separated, there isn't any spacelike surface holding the whole curve.

So this spacelike curve representing the circumference has a length, and the construct of worldlines represents an array of rulers that fit around the circumference of the disk, each ruler being represented by a pair of worldlines a constant distance apart. So, we can find the length of this spacelike curve, and argue that it represents the circumference of the disk, but we do not have a spatial hypersurface in which the entire curve lies. This is because the ends are timelike separated even though the curve is everywhere spacelike.

 

[PeterDonis]

Does 'the existence of one timelike congruence of worldlines everywhere orthogonal to the given spacelike curve' actually imply there exists a spacelike hypersurface the given spacelike curve belongs to ?

I think that was actually the assumption in my previously referenced link (basically the $\sum_ {t}$ hypersurface he was talking about)

The $\Sigma_t$ spacelike hypersurface in that link is not "assumed", it is constructed. But it is constructed only from a single observer's 4-velocity at a given event, not from a congruence of worldlines. The construction being described in that link basically amounts to: treat the 4-velocity at the given event as the timelike basis vector of an inertial frame; then the spacelike hypersurface is just the "surface of constant time" in the same inertial frame that contains the given event.

Note, btw, that the spacelike curve @pervect described in his post #31 cannot be constructed the way I just described; it does not lie in any spacelike surface, certainly not a surface of "constant time" in an inertial frame.

 

[Dale]

What about a "finite spacelike path" ? We know it is not the path followed by any physical object. However, from a local point of view, we can interpret it as the spacelike direction of a spatial axis carried by an observer (basically the spatial axis of a tetrad field)

Then if we consider a finite spacelike interval how should we actually interpret and physically measure it ?

So a spacelike spacetime interval is just interpreted as a length. The interpretation is not problematic.

What is a challenge is to decide which spacelike path you want to determine the length of.

If you pick two arbitrary events then there are a few different paths that you can calculate. One is the spacelike path from one event to another in some reference frame where they are simultaneous. This has the advantage of not just being a general length, but specifically the distance between the two events in the chosen coordinate system. It has the disadvantage of being defined with respect to a reference frame. Another option is to pick the geodesic from one event to the other. This has the advantage of being completely frame independent. However, it may not be unique.

 

[Dale]

I'm not aware of any static spacetime in which the Killing congruence is a geodesic congruence.

Surely Minkowski spacetime fits that description, no?

 

[PeterDonis]

Surely Minkowski spacetime fits that description, no?

Yes, I meant to say static curved spacetime. I'll edit the post.