This question is about the meaning of the length measurment of events,
from a particular observer, in terms of the metric tensor of the
4-dimensional space-time.

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. 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. The 4-dimensional length
of this geodesic, \int |ds^2|^(1/2) along it, is a candidate, althoug
I don't know exactly what this quantity means. (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.

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

Thanks for attention.

On Wed, 21 Jan 2004 17:42:03 +0000 (UTC),
(Mahdiarnt) wrote:

This question is about the meaning of the length measurment of events,
from a particular observer, in terms of the metric tensor of the
4-dimensional space-time.

Here is the way I view the situation. The fundamental physical
quantity that describes the geometry of space-time is the Lorentz
interval, which is invariant to all observers.

Think of the geometry of a ball. At any point on the ball, (a 2d
surface), there will be a "flat" plane tangent to the ball.
Similarly, at any point in space-time, there will be a "flat" 4
dimensional "tangent space" that's tangent to the manifold. This
tangent space will have the familiar metric from special relativity

ds^2 = dx^2 + dy^2 + dz^2 - cdt^2 (1)

Now, if we are given the Lorentz interval in special relativity, how
do we separate it into space and time components? Well, we have to
pick some specific observer, moving with some specific velocity. The
choice of this observer defines how the "time" arrow points. The
directions orthogonal to the direction chosen to be "time" are
"space".

In general relativity, the issue is much the same, but we have to
choose an entire coordinate system. The choice of coordinates is
arbitrary and can be made in any manner that's convenient. In special
relativity we can think of the coordinate axes as straight lines - in
general relativity, space-time itself is curved, so the co-ordinate
axes themselves are in general also allowed to be curved.

In special relativity we could make one choice for the direction of
time, and because of flatness we could translate this choice
everywhere. In general relativity, because of the curved nature of
the manifold, parallel transport of a vector representing time will
depend on the path along which the vector is transported. (Given the
metric, we can view a vector transported over an infinitesimal
distance as being "parallel transported" if, and only if, the vector
and the transported vector are the two opposite sides of a
parallelogram.) Thus we have to make a separate choice of the
direction that is timelike at every point in space-time to remove this
ambiguity. We are free, in general, to do this in whatever manner we
like, but usually some ways will be more convenient than others. One
measure of convenience might be the elimination of unnecessary
torsion.

Once we have made the choice of coordinate systems, the issues of
separating the lorentz interval into space and time components becomes
easy. We can imagine the "grid" structure of the coordinate system
drawn by lines of a constant coordinate. We require any specific set
of coordinates to map to only one point in space-time as a requirement
for a well-behaved coordinate system.

I think that I should correct myself. In the first alternative, the
space-like geodesic should be normal to the world-line of the
observer, too. Otherwise it will not be unique.

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

--
[E-mail me at , as opposed to .]

(Mahdiarnt) wrote in message . com...
I think that I should correct myself. In the first alternative, the
space-like geodesic should be normal to the world-line of the
observer, too. Otherwise it will not be unique.

I think it is important to undertand when spacetime is flat.

If there are a absence of gravitational waves, then this would be so.

If you look at Gaussian coordinates, you get a sense of the higher
geometires. IN undertanding the metric and the relationship of quark
to quark measures, become distictive in separation.

Hope that helps.

Sol

Dear Ted,

Before anything, I agree that the mentioned space-like geosdesic
should be perpendicular to the world-line of the observer, as I
corrected in a followup to my original post.

Your answer was illustrative. I learned that all that matters is what
we really measure; it's then our job to find what quantity in the
space-time we have actually measured. But what still bothers me, is
the implementation of the notion of distance in the results of general
relativity. For example, in the Schwarzschild solution, when we derive
the equations of geodesics, we arrive at an equation like that of the
Newtonian theory with a small correction. But the similarity to the
Newtonian case is achieved only if we interpret the r coordinate as
distance from the "sun" to the "planet," and the phi coordinate as the
angle. To me these are mere coordinates. Could you tell me the result
of what measurements are r and phi?

In article ,
Mahdiarnt wrote:

But what still bothers me, is
the implementation of the notion of distance in the results of general
relativity. For example, in the Schwarzschild solution, when we derive
the equations of geodesics, we arrive at an equation like that of the
Newtonian theory with a small correction. But the similarity to the
Newtonian case is achieved only if we interpret the r coordinate as
distance from the "sun" to the "planet," and the phi coordinate as the
angle. To me these are mere coordinates. Could you tell me the result
of what measurements are r and phi?

That's a very good question.

If we use Schwarzschild coordinates to describe the spacetime around a
black hole, then the coordinate r of a given point is not (in any
meaningful sense) the distance from the black hole to the point. It
is (1/2pi) times the circumference of a circle centered on the black
hole and passing though that point. And the coordinate phi is just
(1/r) times the arc length along such a circle.

In fact, I think that Taylor and Wheeler, in their introductory
book "Exploring Black Holes" go to the trouble of referring to r
as the "reduced circumference" coordinate rather than the
radial coordinate, in order to emphasize this distinction.

For both r and phi, the measurements are to be taken along curves at a
fixed time. This is a meaningful thing to say because the
Schwarzschild geometry is static. (That is, there's a preferred time
coordinate such that nothing about the metric changes in those
coordinates.)

That's the operational, in-principle-measurable description of
what the Schwarzschild coordinates r,phi mean.

Now, if we're using Schwarzschild geometry to describe spacetime
outside of the Sun rather than outside of a black hole (which is
perfectly correct, in the excellent approximation that the geometry
is static), then it does make sense to talk about the distance from
the center of the Sun to a given point. After all, because
we're assuming things are static, there's a preferred notion
of things being "at the same time," so there's a unique
spacelike geodesic that goes from the given point to the center
of the Sun at constant time. The length of that geodesic will not
be exactly the same as the Schwarzschild r coordinate of the point.
For a system like the Sun, for which the curvature is always pretty
weak (the Sun's quite far from being a black hole), the difference
would be tiny. For something like a neutron star, though, it
could be significant.

-Ted

--
[E-mail me at , as opposed to .]

Hello Ted:

To me these are mere coordinates. Could you tell me the result
of what measurements are r and phi?

That's a very good question.

If we use Schwarzschild coordinates to describe the spacetime around a
black hole, then the coordinate r of a given point is not (in any
meaningful sense) the distance from the black hole to the point. It
is (1/2pi) times the circumference of a circle centered on the black
hole and passing though that point. And the coordinate phi is just
(1/r) times the arc length along such a circle.

In fact, I think that Taylor and Wheeler, in their introductory
book "Exploring Black Holes" go to the trouble of referring to r
as the "reduced circumference" coordinate rather than the
radial coordinate, in order to emphasize this distinction.


I don't think this is a great question because it depends only on the
coordinate system chosen. One could just as well use isotropic
coordinates. It appears that theorists use the Schwarzschild
coordinates because they simplify calculations a bit. Experimentalists
use Taylor series expansions of the metric in isotropic coordinates.

Sorry to go off on a tangent here, but I was just looking at the
isotropic form, eq. 31.22 in MTW:

ds^2 = -((1 - M/2R)/(1 + M/2R))^2 dt^2

+ (1 + M/4R)^4[dr^2 + dr^2(dtheta^2 + sin^2 theta dphi^2)

If R = M/2, then the coefficient for dt^2 goes to zero, but the radial
term does not become underfined as is the case for this metric in the
Schwarzschild metric. What is the usual take on this?


doug
quaternions.com