Is the proper distance the shortest distance in FLRW universe?

[andrewkirk]

In Appendix A of Davis & Lineweaver (2003) proper distance to a faraway galaxy is defined as the distance along a curve of constant time in the RW metric.

I was wondering whether that line of constant time is a geodesic of spacetime. If not then there will be a shorter-distance path from here to the faraway galaxy, that starts and finishes in cosmic 'now' but in between deviates into other cosmic times.

The RW coordinate system is, I think, orthogonal. But that is not sufficient for its coordinate lines to be geodesics. For example, polar coordinates in E² are orthogonal but the geodesic from (1,0) to (0,1) is not along the line of constant r=1.

Is the line of RW constant time a spacetime geodesic? If so, is there a proof I can look at? If not, is there an easy counterexample like the above one for polar coordinates?

 

[Calimero]

No, it is not a geodesic of spacetime for sure.

 

[George Jones]

No, it is not a geodesic of spacetime for sure.

Right, but it is a geodesic of space.

 

[George Jones]

Before discussing spacetime, let's discuss a more familiar example, the 2-dimensional surface of a sphere (like the Earth) in normal Euclidean 3-space. A geodesic on the surface of the Earth is not a geodesic in 3-space.

In what sense is a geodesics on the surface of the Earth a geodesic?

 

[andrewkirk]

Thanks for the answers. So the coordinate line is a geodesic of the 3D constant-time hypersurface but not a geodesic of spacetime. The globe example clarifies that nicely.

What troubles me a little about this is that I have been thinking of proper time and proper distance as analogous quantities that are both coordinate independent. For proper time this still seems to be the case - it is just the length of the timelike geodesic passing throught the two spacetime events. But the definition of proper distance now appears to be coordinate-dependent. The conceptual symmetry that I thought existed between proper time and proper distance is broken.

Is the symmetry recoverable, or should I throw it away?

 

[PAllen]

Thanks for the answers. So the coordinate line is a geodesic of the 3D constant-time hypersurface but not a geodesic of spacetime. The globe example clarifies that nicely.

What troubles me a little about this is that I have been thinking of proper time and proper distance as analogous quantities that are both coordinate independent. For proper time this still seems to be the case - it is just the length of the timelike geodesic passing throught the two spacetime events. But the definition of proper distance now appears to be coordinate-dependent. The conceptual symmetry that I thought existed between proper time and proper distance is broken.

Is the symmetry recoverable, or should I throw it away?

They are both the same. Proper time is only defined along some world line. It is invariant for that world line between two events. In addition to different non-geodesic world lines connecting two events, in GR (in general) there may be multiple timelike geodesics connecting two events. This happens in as simple a system as our solar system.

In SR, you can speak of the proper time between two causally connected events, or the proper distance between events with spacelike separation, because it is implicit (in most contexts) that you mean measured along the unique geodesic between them. In GR, it is simply not possible, in general, to make such a statement.

You can talk about the minimum proper distance, among all spacelike geodesics, between two events in GR - but this is not normally done, as it is not particularly useful. In this case, the geodesic in a hypersurface of constant cosmological time will not be such a distance.

A key reason why this attempt to generalize proper distance as a function of events to GR is not normally done, is that once it is true that two events don't determine a unique geodesic, why consider only geodesic paths? Then you confront the fact that for spacelike connected events, you can construct spacelike non-geodesic paths between them with arbitrarily small proper distance.

I should add that there is an asymmetry in that there is generally a unique longest proper time between two causally connected events in GR, and it corresponds to (one of) the geodesics. This asymmetry comes from the metric signature - there is one timelike dimension (there is a coordinate independent way to state this fact). The existence of more than one spatial dimension means that there are paths with arbitrarily large and arbitrarily small proper distance (the latter are simply near light like paths that are not light like geodesics; any two spacelike separated events can be connected with arbitrarily close to lightlike paths that are still spacelike everywhere).

[edit: to see the last point, imagine light cones from each event with spacelike separation. These light cones will generally intersect. You can choose a spacelike path that goes just 'outside' one light cone till the intersection with the other, then just outside the other light cone. While still being spacelike everywhere, you can come as close as desired to the light cones, producing a proper distance along the path arbitrarily close to zero.]

 

[andrewkirk]

Thanks Pallen for a very thought-provoking and helpful post.

My attempt to create a path of arbitrarily short length between two spacelike separated events is as follows:

Consider a spacetime diagram for 1D space, with t on the vertical axis and x (space) on the horizontal. Let events A and B have coordinates (0,0) and (0,1) and assume the spacetime is flat. Then the following path has length θ, which can be as small a positive number as we wish:

Start at (0,0), proceed on a spacelike, almost lightlike, path straight to $(\frac{1-\theta}{2},1-\theta)$, and on another spacelike, almost lightlike, path from there straight to
$(1-\theta,0)$, then on the straight spacelike path from there to (1,0).​

[This bit was edited to incorporate the advice of Pallen in the post below]

To get an arbitrarily long path is easier: just head in the wrong direction for as long as you wish, then turn around and head back and past the starting point until you get to the destination.

We can do the same thing for proper time by reflecting the path in the line t=x. This enables us to create an arbitrarily short path between two timelike separated events at (x,t) = (0,1) and (0,0). But it also seems to allow us to create an arbitrarily long path between the events, again by just reflecting the arbitrarily long path between (0,0) and (1,0) in the line t=x. Wouldn't that then contradict the suggestion that there is a maximal length path between timelike separated events?

We could overcome that by requiring the path to be constructed of a finite number of segments each of which is a possible worldline. That would disqualify the arbitrarily long path because it involves travel backwards in time. But it would also disqualify the second leg of the arbitrarily short path between (0,0) and (1,0). So we could no longer say there exist arbitrarily short paths between spacelike separated events.

In fact if we restrict ourselves to possible paths we get the asymmetry that there is no nonzero minimum on the length of possible paths between timelike separated events but there is a nonzero minimum on the lengths of paths between spacelike separated events, and that minimum is achieved for one or more possible paths. Conversely there is a maximum on the length of possible paths between timelike separated events but not between spacelike separated events.

Actually that is a sort of symmetry I suppose.

 

[PAllen]

Thanks Pallen for a very thought-provoking and helpful post.

My attempt to create a path of arbitrarily short length between two spacelike separated events is as follows:

Consider a spacetime diagram for 1D space, with t on the vertical axis and x (space) on the horizontal. Let events A and B have coordinates (0,0) and (0,1) and assume the spacetime is flat. Then the following path has length θ, which can be as small a positive number as we wish:

Start at (0,0), proceed on a lightlike path straight to $(\frac{1-\theta}{2},\frac{1-\theta}{2})$, and on another lightlike path from there straight to
$(1-\theta,0)$, then on the straight spacelike path from there to (1,0).​

To get an arbitrarily long path is easier: just head in the wrong direction for as long as you wish, then turn around and head back and past the starting point until you get to the destination.

We can do the same thing for proper time by reflecting the path in the line t=x. This enables us to create an arbitrarily short path between two timelike separated events at (x,t) = (0,1) and (0,0). But it also seems to allow us to create an arbitrarily long path between the events, again by just reflecting the arbitrarily long path between (0,0) and (1,0) in the line t=x. Wouldn't that then contradict the suggestion that there is a maximal length path between timelike separated events?

We could overcome that by requiring the path to be constructed of a finite number of segments each of which is a possible worldline. That would disqualify the arbitrarily long path because it involves travel backwards in time. But it would also disqualify the second leg of the arbitrarily short path between (0,0) and (1,0). So we could no longer say there exist arbitrarily short paths between spacelike separated events.

In fact if we restrict ourselves to possible paths we get the asymmetry that there is no nonzero minimum on the length of possible paths between timelike separated events but there is a nonzero minimum on the lengths of paths between spacelike separated events, and that minimum is achieved for one or more possible paths. Conversely there is a maximum on the length of possible paths between timelike separated events but not between spacelike separated events.

Actually that is a sort of symmetry I suppose.

A path on which you claim a proper distance should be spacelike everywhere. So your example is not considered admissible. But a small deviation from it could be space-like everywhere, and achieve the same result.

For causally connected events, it is easy to get arbitrarily small proper time along a pure time-like path between them. Arbitrarily long can only be done for a smooth path by having a bend that is space-like, leading to an imaginary contribution. Even if you make the bend sharp, it is still not a valid world line because you have two arbitrarily close events on the the alleged world line that are not causally connected (space-like separation). This is not allowed for a world line.

Thus, what I said is true:

- for causally connected events, there is a maximum proper time between between them, representing one of the geodesic paths between them. There is no minimum proper time between them (among non-geodesic, valid, world lines between them).

- for events with space-like separation, there would be minimum geodesic proper distance between them, but there would be other completely valid spacelike paths between them with arbitrarily close to zero proper distance. The claim that such a path 'goes back in time' is false - along an everywhere space-like path, all events may be validly considered simultaneous. You can choose a coordinate system in which this path is embedded in a simultaneity surface, representing constant coordinate time. There is no way to claim there is something wrong with such a path in a coordinate independent way except for the obvious statement that it is not a 4-geodesic (but it can be a 3-geodesic in a hyper-surface containing it).

[Edit: The existence spacelike paths with arbitrarily short proper distance between two given causally disconnected events is really a generalization of the concept of length contraction. It is thus, not surprising at all.]