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- I am wondering if the coordinate chart I derive here has appeared anywhere in the literature on the Bell Spaceship Paradox. I have been unable to find it.
A question that might occur to anyone reading about the Bell Spaceship Paradox is, can we construct a coordinate chart in which all of the Bell observers (i.e., observers following worldlines like those of the spaceships in the "paradox" scenario) are "at rest"? I put "at rest" in scare-quotes because, since the expansion scalar of the Bell congruence is positive, these "at rest" observers do not maintain constant proper distance from each other (which is, of course, the point of the "paradox"). However, with that caveat, it seems that we can indeed construct such a chart by a simple series of transformations from the standard Minkowski chart. I'll describe that construction below, but I have been unable to find any reference in the literature that gives it, and I am wondering if any other PF members have.
The construction starts from the specification of a "Bell observer" worldline in the Minkowski chart. We'll restrict to 1+1 spacetime dimensions to keep things simple. Using lower case ##t## and ##x## for the Minkowski chart coordinates, we have each Bell observer worldline specified as follows:
$$
\left( x - X \right)^2 - t^2 = \frac{1}{a^2}
$$
where ##X## is a parameter that is constant along each worldline and so serves to label the worldlines (the reason for the notation ##X## will become evident below), and we are using units in which ##c = 1##.
We now look for a coordinate transformation that will make each of these worldlines a curve of constant spatial coordinate in the new chart. The obvious ansatz to try given the worldline equation above is:
$$
t = \tilde{T}
$$
$$
x = X + \sqrt{\tilde{T}^2 + \frac{1}{a^2}}
$$
where ##\tilde{T}## and ##X## are the new coordinates. (The reason for the notation ##\tilde{T}## instead of just ##T## will be seen shortly.) This gives
$$
dx = dX + \frac{\tilde{T} d\tilde{T}}{\sqrt{\tilde{T}^2 + \frac{1}{a^2}}}
$$
and the line element then becomes
$$
ds^2 = - d\tilde{T}^2 + \left( dX + \frac{\tilde{T} d\tilde{T}}{\sqrt{\tilde{T}^2 + \frac{1}{a^2}}} \right)^2
$$
which simplifies to
$$
ds^2 = - \frac{1}{1 + a^2 \tilde{T}^2} d\tilde{T}^2 + \frac{2 a \tilde{T}}{\sqrt{1 + a^2 \tilde{T}^2}} d\tilde{T} dX + dX^2
$$
We can simplify this further with an obvious rescaling of the time coordinate:
$$
dT = d\tilde{T} \frac{1}{\sqrt{1 + a^2 \tilde{T}^2}}
$$
This gives ##a\tilde{T} = \sinh a T##, and the line element now becomes
$$
ds^2 = - dT^2 + 2 \sinh aT dT dX + dX^2
$$
It is probably helpful to clarify some points. First, the surfaces of constant ##T## in this chart are not orthogonal to the Bell observer worldlines (except at ##T = 0##, as is obvious from the above line element). Instead they are the same surfaces as the surfaces of constant Minkowski coordinate time. One of the consequences of the Bell spaceship paradox is that it is not possible to construct a chart in which all of the surfaces of constant time are orthogonal to the Bell observer worldlines. (You can construct a Rindler chart centered on one Bell observer worldline, and the surfaces of constant coordinate time in that chart will be everywhere orthogonal to that Bell observer worldline, but they won't be orthogonal to any other Bell observer worldline, except at ##t = 0##, and of course the chart will only cover one "wedge" of the spacetime.)
Also, after the rescaling of the time coordinate, ##T## is the same as proper time for all the Bell observers, and the surfaces of constant ##T## are flat. Distance in those surfaces, however, is not the same as proper distance between the Bell observers, because of the non-orthogonality described above. Note that, since the coefficient of the ##dT dX## term in the line element increases with ##T## (and does so exponentially), the non-orthogonality, and hence the mismatch between ##X## distance in the chart and proper distance between Bell observers, gets more and more severe with time.
Again, my question is: has anyone seen a reference to this chart, or a similar one, anywhere in the literature?
The construction starts from the specification of a "Bell observer" worldline in the Minkowski chart. We'll restrict to 1+1 spacetime dimensions to keep things simple. Using lower case ##t## and ##x## for the Minkowski chart coordinates, we have each Bell observer worldline specified as follows:
$$
\left( x - X \right)^2 - t^2 = \frac{1}{a^2}
$$
where ##X## is a parameter that is constant along each worldline and so serves to label the worldlines (the reason for the notation ##X## will become evident below), and we are using units in which ##c = 1##.
We now look for a coordinate transformation that will make each of these worldlines a curve of constant spatial coordinate in the new chart. The obvious ansatz to try given the worldline equation above is:
$$
t = \tilde{T}
$$
$$
x = X + \sqrt{\tilde{T}^2 + \frac{1}{a^2}}
$$
where ##\tilde{T}## and ##X## are the new coordinates. (The reason for the notation ##\tilde{T}## instead of just ##T## will be seen shortly.) This gives
$$
dx = dX + \frac{\tilde{T} d\tilde{T}}{\sqrt{\tilde{T}^2 + \frac{1}{a^2}}}
$$
and the line element then becomes
$$
ds^2 = - d\tilde{T}^2 + \left( dX + \frac{\tilde{T} d\tilde{T}}{\sqrt{\tilde{T}^2 + \frac{1}{a^2}}} \right)^2
$$
which simplifies to
$$
ds^2 = - \frac{1}{1 + a^2 \tilde{T}^2} d\tilde{T}^2 + \frac{2 a \tilde{T}}{\sqrt{1 + a^2 \tilde{T}^2}} d\tilde{T} dX + dX^2
$$
We can simplify this further with an obvious rescaling of the time coordinate:
$$
dT = d\tilde{T} \frac{1}{\sqrt{1 + a^2 \tilde{T}^2}}
$$
This gives ##a\tilde{T} = \sinh a T##, and the line element now becomes
$$
ds^2 = - dT^2 + 2 \sinh aT dT dX + dX^2
$$
It is probably helpful to clarify some points. First, the surfaces of constant ##T## in this chart are not orthogonal to the Bell observer worldlines (except at ##T = 0##, as is obvious from the above line element). Instead they are the same surfaces as the surfaces of constant Minkowski coordinate time. One of the consequences of the Bell spaceship paradox is that it is not possible to construct a chart in which all of the surfaces of constant time are orthogonal to the Bell observer worldlines. (You can construct a Rindler chart centered on one Bell observer worldline, and the surfaces of constant coordinate time in that chart will be everywhere orthogonal to that Bell observer worldline, but they won't be orthogonal to any other Bell observer worldline, except at ##t = 0##, and of course the chart will only cover one "wedge" of the spacetime.)
Also, after the rescaling of the time coordinate, ##T## is the same as proper time for all the Bell observers, and the surfaces of constant ##T## are flat. Distance in those surfaces, however, is not the same as proper distance between the Bell observers, because of the non-orthogonality described above. Note that, since the coefficient of the ##dT dX## term in the line element increases with ##T## (and does so exponentially), the non-orthogonality, and hence the mismatch between ##X## distance in the chart and proper distance between Bell observers, gets more and more severe with time.
Again, my question is: has anyone seen a reference to this chart, or a similar one, anywhere in the literature?