In article ,
Mahdiarnt wrote:

Consider an observer. This is represented in the space-time manifold
by a time-like curve. Now consider of an event taking place at some
point on the same manifold, not necessarily on the observer's
world-line. I want to know about the measured distance of the event
in view of this single observer.

I am not sure of the answer.

Since you say you want the "measured" distance, how about telling
us how you imagine this quantity will be measured?

I don't see a unique, obviously best meaning to ascribe to the concept
of "distance from an event to an observer." There are multiple
different ways one can imagine defining something that could be described
in these words, which will in general give different answers.

But there are two quantities which seem
to have some relevance. (1) If the event is not too far from the
observer, there is (I thinjk) a unique space-like geodesic connecting
the event to the world-line of the observer.

That's certainly not true as written. There are infinitely many spacelike
geodesics joining a given event to a given world line. To see this,
draw the future and past light cones of the event. As you note below,
these intersect the world line in two distinct points. Now pick
any point on the world line between those two points. That point
is spacelike separated from the given event, and so can be joined to
it with a spacelike geodesic.

Perhaps you mean that there is a unique spacelike geodesic that is
perpendicular to the world line at the point where they intersect? I
think that's generically true, for sufficiently nearby events. The
length of this geodesic segment is
one reasonable way to define "the distance from an event to an
observer," although I'd prefer not to describe it as the "measured
distance" from event to observer, since it doesn't correspond in any
simple way to the results of a measurement. (The observer could make
a whole bunch of measurements and calculate this number, but only with
a fair amount of effort.)

(2) The light-cone of
the event interesects with the world-line of the observer at two
points. These points can be thought of as the events of respectively
sending and receiving a signal by the observer to and from the event.
The time elapsed between the sent signal and the received one as
measured by the observer's clock, i.e. the 4-dimensional length of
this portion of the observer's world-line, \int |ds^2|^(1/2) along it,
times c/2 is another candidate. This latter alternative very much
resembels what we do in our experiments.

I'd vote for this as a reasonable candidate for the "measured
distance" from an observer to an event, since it corresponds to one
fairly natural method of making such a measurement (it's essentially
radar ranging).

But I can imagine that someone could describe a different experimental
procedure that would also be reasonably described as a measurement
of distance from an observer to an event but that might yield
a different result. In other words, I personally don't feel
a pressing need to give a definition to this concept.

Even if neither of the above is correct, what is the significance of
them? And what is the true measured distance?

["What is truth?" said jesting Pilate, and would not wait for answer.]

There's a big difference between asking for the "measured" distance
and the "true" distance. Personally, I'd prefer not to call anything
the "true" distance. If I really had to pick something, I think I'd pick
the length of that perpendicular geodesic segment, since it seems
closest to the thing that we used to call the distance from a point
to a line in high-school geometry.

-Ted

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