Response to #57: Part 2 of 2
Part-1 outlines the example, including reference diagrams, and its purpose; therefore it is the logical starting point of this discussion. Part-2 now considers what, if any, physical interpretations can be drawn. As previously introduced, the local speed of light is always assumed to conform to:
[1] c = s_B/t_B = s_A/t_A = 1
If we consider equation [1] with respect to observer (B), who is conceptually at infinity, i.e. no spacetime curvature, where [\gamma=1], we might consider [c] expressed in unit distant and time producing a unit value of [c]:
[2] c = s_B/t_B = 1/1 = 1
However, the suggestion is that if these unit measures were at [A] with [\gamma=8], equation [2] would become:
[3] c = s_A/t_A = 8/8 = 1
However, the values inserted in [3] are only perceived from the `
conceptual frame` and not by the local observer at (A). Locally, at (A), time t_A is still considered as unit time, as the
`physical` perception is that the rate of time is still 1 second per second. As such, the local values of equation [3] would be:
[4] c = s_A/t_A = 1/1 = 1
In sense, we are seeing the constancy of the speed of light in both the local frame and the conceptual frame, which might be generally written as either:
[5] c = dr/d\tau = 1 Local frame: min-r/min-t
[6] c = ds/dt = 1 Conceptual frame: max-r/max-t
What equation [5] implies is that if [c=1] in all local frames, then the local (proper) time is always dilated and the local perception of radial distance has also to be the coordinate-radius [dr] not [ds].
Interpretation?
The local observer doesn’t directly perceive the effects of relativity and therefore always measures the minimum value of space [dr] and time d\tau
In contrast, the conceptual frame corresponding to equation [6] is referencing the maximum values, [ds] and [dt]. However, both appear to support the constant speed of light [c=1] based on equations [3] and [4]. At this point, we might try to imply some physical interpretation to the frequency of the photon emitted and absorbed at any point. The following interpretation is forwarded on the assumption that modern relativity replaces the description of Newtonian gravity as a force with the concept of spacetime geometry. As such, gravitational redshift has to be physically interpreted in terms of time or space changes.
Interpretation?
When emitted from (A), the photon was said to be blue, i.e. high frequency, but this was only with respect to the local dilated time at (A). As the photon moves toward (B), the relative rate of time ticks faster, redshifting the frequency of the photon. In contrast, a photon emitted at (B) was said to be red with respect to its local time, so as the photon moves toward (A) the frequency is blueshifted.
While I understand the maths that have led to the conclusion outlined in the following quote, it is suggested this conclusion can be also be interpreted from the conceptual frame to be physically more meaningful.
From #57: So only the conclusion reached by assuming the speed of light is constant everywhere in a gravitational field gives an asymmetric result that ABA>BAB.
With reference to the figures in the attachment in part-1, it is difficult to conceive that ABA > BAB, as to the photon(s), it represents the same physical path. Physically , the only thing that changes is the relative rate of time perceived by the 2 observers in (A) and (B) who may be determining the distance based on their local assumption that [c=1] and that the roundtrip elapsed time is either [t_A or t_B]. It is clear that if [c=1] and ABA equals BAB, then the local observers would have to agree on the roundtrip time, which they do not given the relative time dilation. However, it will be suggested that we can use the conceptual frame to get a more meaningful resolution.
If we start with observer (B) and figure-1a, with each segment corresponding to [\gamma=1..8], we may also considered each segment to correspond to a unit distance traversed in unit time, such that:
[7] c = s_{BAB}/t_{BAB} = 16/16 = 1
However, we know that the observer at (A) must measure the roundtrip time against its local proper time, which the conceptual observer can see is related by:
[8] t_A = t_B*\gamma_A
However, locally, observer (A) must still determine [c=1] therefore:
[9] c_A = S_{ABA}/ t_B*\gamma_A = S_{BAB}/t_B = c_B = 1
If so, this suggests:
[10] S_{ABA} = S_{BAB}*\gamma_A
[11] S_{ABA} > S_{BAB}
This seems to confirm the conclusion reached in #57. However, how else might we physical interpret equation [6]?
Interpretation?
(A) and (B) disagree on the length ABA and BAB, not because the photon actually traversed 2 paths of different length, but rather the BAB has been calculated in terms of [dr], while ABA has been calculated in terms of [ds]. Therefore, from the conceptual frame, the suggestion is that ABA = BAB.
Finally, to be clear on one point, the conceptual frame is not inferring that any absolute frame exists; it is simply intended as a possible learning aid. While mathematically theorist may reject this approach, the level of debate over the meaning of relativity, in this forum alone, suggests that mathematical derivation alone is still subject to many different physical interpretations. As always, would appreciate any constructive feedback.
