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Kevin Wampler schrieb:
> clipka wrote:
>> Kevin Wampler schrieb:
>>
>>> I think that this becomes problematic for measuring between points
>>> which aren't connectible by a lightlike geodesic, but maybe there's
>>> some clever way around that (although I don't see how).
>>
>> Well, if two points cannot be reached from one another - is there
>> /any/ way to assign a distance to these points at all?
>>
>> So I think this is a non-issue.
>
> In this context a lightlike geodesic refers to a path in spacetime which
> light would follow, and I meant to imply that defining the distance
> between points which could only be connected by going *slower* than
> light would also be hard to define uniformly.
Maybe you'd first need to define "point": Are you talking about a point
in 3D space which extends along time - i.e. a line in 4D space - or an
actual 4D point that only exists at a certain time T?
In the former case, this is actually a non-issue: Either you can find
(true) points on that line that /can/ be reached at /exactly/ light
speed, or you cannot reach that other line at all.
In the latter case, you can define the spacelike distance between the
points as the minimum difference between the spacelike component of two
intersecting lightlike geodesics running through each of the points
respectively.
Or, to eliminate the problem of "far-away" spacetime being curved, you
can have a look at all possible sequences of points that can be
connected with lightlike geodesics, compute the vector sum of the
spacelike components, and define the spacelike distance as the minimum
length of any such vector sum.
In the end it will boil down to defining the spacelike distance as the
length of the spacelike component of the shortest "direct" geodesic
between the points.
(Note that of course this leaves the definition of distance subject to
your frame of reference, as it defines the orientation of the spacelike
components of spacetime, but that's a known effect in relativistic physics.)
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