# Distance meaning

1. Oct 13, 2004

### blue_sky

In full GR, if somebody ask me what is the distance between two bodies I do not think that the question make sense.
If I use a signal travelling at c when the signal hit the body, in general, it is not in the original point where he was when the question was asked and when the signal is back the body is in a different point too.

Am I correct?

blue

2. Oct 13, 2004

### Fredrik

Staff Emeritus
Distance in space is perfectly well defined. What isn't well defined is exactly what "slice" of spacetime we should call "space".

(What I called "space" in the first sentence should actually be called "a spacelike hypersurface of spacetime").

Last edited: Oct 13, 2004
3. Oct 13, 2004

### Stingray

It is possible to define the length of a curve in an invariant way. So take two points in spacetime, and find a geodesic between them. Then compute the length of the geodesic. The problem with this approach is that there is in general more than one geodesic connecting those points. So your distance function might be multi-valued. It is however, the natural generalization of what we do in flat spacetime.

The formal name for this distance function is Synge's biscalar or Synge's world function.

4. Oct 14, 2004

### blue_sky

I was referring to "spatial" distance while in your explanation i think you are referringt to the space-time distance. Am I right?

blue

5. Oct 14, 2004

### blue_sky

If what slice of spacetime we should call "space", can I say "spatial" distance is not well defined too?

blue

6. Oct 14, 2004

### Stingray

Yes, I meant spacetime distance. Points in GR are points in both space and time.

Everything in relativity should be thought of as occuring in spacetime, not space. So each body traces out a timelike line. There are several ways you can define distances, but it depends on what exactly you want to do.

The simplest definition supposes the existence of some given reference frame where there exists a particular time parameter. Find the coordinates of each body at a fixed time t, and then compute the geodesic distance between those points. This definition is not unique at all, so there's really no point in using it unless you are trying to understand the results of some (thought) experiment done in a known reference frame.

Next, you can pick a particular value of proper time for one of the bodies, t. Call its position in spacetime at that time z1(t). Now look at the set of (spacelike) geodesics which start at z1(t) and are orthogonal to that body's four-velocity (this is well-defined despite being nonlocal). At least one of these will intersect the other body's worldline. Compute the geodesic distance between that point of intersection and z1(t) to get a distance. This is really very similar to the first method. The difference is that I'm effectively defining a preferred coordinate system by using the rest frame of one of the bodies. This is a very natural thing to do, and is an invariant. There are cases, however, where the prescription fails to work for various reasons. It's usually ok though.

There are other definitions, but they get more complicated (for example, there is one based on null separated points between the two worldlines that is very useful when calculating electromagnetic fields).

7. Oct 14, 2004

### Fredrik

Staff Emeritus
Yes. Since the concept of "space" isn't well defined, the concept of "spatial distance" isn't either.

Last edited: Oct 15, 2004
8. Oct 15, 2004

### Garth

There is the further problem of relating the units of mass, length and time used in the metric and the field equation to actual measurements that might be made in a laboratory. In GR we assume that these units are constant, but that is an assumption that might not hold. Weyl conformal transformations of the metric re-interpret these units for laboratories separated across space-time curvature.
The key question is "How do you measure it?"
In the 1930's Milne introduced the concept of "radar distance", that defined distance by the time taken for a light signal to leave Earth and be reflected back by the object in question. Not bad seeing radar wasn't invented until a decade later!
This reduced distance measurements to own clock measurements, something we could measure if we could wait long enough!
Garth

9. Oct 15, 2004

### pervect

Staff Emeritus
What you have to do to define distance in GR is to set up a coordinate system that's good enough to separate out distance from time.

The easiest way of doing this is to define a vector that points in the time direction for every observer, a vector field.

Distance will then be perpendicular to time, everywhere.

You can then use standard rulers to measure distance, once you've set up the coordinate system.

In cosmology, I think people tend to use a coordinate system where the time direction is defined to be that which gives the appearance of isotropy of the universe.

10. Oct 15, 2004