- #1
Figaro
- 103
- 7
Based on this lecture notes http://www.helsinki.fi/~hkurkisu/CosPer.pdf
For a given coordinate system in the background spacetime, there are many possible coordinate systems in the perturbed spacetime, all close to each other, that we could use. As indicated in figure 2, the coordinate system ##{\hat x^α}## associates point ##\bar P## in the background with ##\hat P##, whereas ##{\tilde x^α}## associates the same background point ##\bar P## with another point ##\tilde P##.
1) My understanding here is that both coordinate ##{\hat x^α}## and ##{\tilde x^α}## give the same point in the background with different coordinate value, but what does equation (4.2) ( ##\tilde x^α (\tilde P) = \hat x^α (\hat P) = \bar x^α (\bar P)##) mean? By this, does that mean there is an abuse of notation such that ##\bar x^α (\bar P)## is "THE" point?
The coordinate transformation relates the coordinates of the same point in the perturbed spacetime
##\tilde x^α (\tilde P) = \hat x^α (\tilde P) + ξ^α##
##\tilde x^α (\hat P) = \hat x^α (\hat P) + ξ^α##
2) What does he mean by this? Based on my search there is one statement saying,
A gauge transformation is different from a general coordinate transformation ##Q (x)## → ##\tilde Q (x)##. In a gauge transformation, ##\tilde Q## and ##Q## are both calculated at the same coordinate value corresponding to two different space-time points, while in a general coordinate transformation both Q and x are transformed so that we are dealing with the values of a quantity at the same space-time point observed in the two systems.
Based on my understanding ,in a gauge transformation we use ##\tilde x^α## and ##\hat x^α## to evaluate the same point in the perturbative spacetime, it yields different spacetime points in the "background spacetime"? For a general coordinate transformation if we transform ##Q → \tilde Q## but evaluate at the same point ##x##, the "TRUE" location of the point is not changed, just the coordinate system/observer's perspective just like in the passive rotations. But if we also transform ##x → \tilde x## then we not only change the perspective but also the "TRUE" position of the point, but in both observers's coordinate frame they see that the point has the same position with respect to their coordinate frame, is this correct? It seems to contradict the statement above that "we are dealing with the values of a quantity at the same space-time point observed in the two systems"
For a given coordinate system in the background spacetime, there are many possible coordinate systems in the perturbed spacetime, all close to each other, that we could use. As indicated in figure 2, the coordinate system ##{\hat x^α}## associates point ##\bar P## in the background with ##\hat P##, whereas ##{\tilde x^α}## associates the same background point ##\bar P## with another point ##\tilde P##.
1) My understanding here is that both coordinate ##{\hat x^α}## and ##{\tilde x^α}## give the same point in the background with different coordinate value, but what does equation (4.2) ( ##\tilde x^α (\tilde P) = \hat x^α (\hat P) = \bar x^α (\bar P)##) mean? By this, does that mean there is an abuse of notation such that ##\bar x^α (\bar P)## is "THE" point?
The coordinate transformation relates the coordinates of the same point in the perturbed spacetime
##\tilde x^α (\tilde P) = \hat x^α (\tilde P) + ξ^α##
##\tilde x^α (\hat P) = \hat x^α (\hat P) + ξ^α##
2) What does he mean by this? Based on my search there is one statement saying,
A gauge transformation is different from a general coordinate transformation ##Q (x)## → ##\tilde Q (x)##. In a gauge transformation, ##\tilde Q## and ##Q## are both calculated at the same coordinate value corresponding to two different space-time points, while in a general coordinate transformation both Q and x are transformed so that we are dealing with the values of a quantity at the same space-time point observed in the two systems.
Based on my understanding ,in a gauge transformation we use ##\tilde x^α## and ##\hat x^α## to evaluate the same point in the perturbative spacetime, it yields different spacetime points in the "background spacetime"? For a general coordinate transformation if we transform ##Q → \tilde Q## but evaluate at the same point ##x##, the "TRUE" location of the point is not changed, just the coordinate system/observer's perspective just like in the passive rotations. But if we also transform ##x → \tilde x## then we not only change the perspective but also the "TRUE" position of the point, but in both observers's coordinate frame they see that the point has the same position with respect to their coordinate frame, is this correct? It seems to contradict the statement above that "we are dealing with the values of a quantity at the same space-time point observed in the two systems"
