GR Length Contraction & Shapiro Delay: Investigating Light Speed

[grav-universe]

In this thread in another forum, I calculated the coordinate speed of light as measured by a distant observer. As far as I understand it, the length contraction in GR is only in the radial direction, right? None tangent, correct? The time delay of light found, though, trying different examples, was always 20 microseconds smaller than the values measured by Shapiro. However, when I use the coordinate speed of light c' = c (1 - 2 G M / (r c^2)), with the same length contraction regardless of direction, it then matches the equation used for the Shapiro delay and the data measured. What does this mean?

 

[PAllen]

I am having trouble with attaching any real meaning to 'coordinate speed of of <anything> measured by a distant observer'. Can you clarify what you mean?

In GR, I would defined coordinate speed as follows: Assuming the coordinates define a simultaneity surface at some event on some world line, the I would define coordinate speed of that world line at that event as:

compute 4-velocity in said coords; compute 3-norm in simul surface using induced metric; compute dot product with timelike unit vector 4-orthogonal to said surface; divide former by latter.

[Edit: wording above corrected for not looking far enough down in linked thread]

 

[PAllen]

I would describe Shapiro time delay not as measurement of coordinate speed 'at a distance' but measurement of average speed from some distant source to you, based on the simultaneity surface specified by the coordinates. Specifically proper distance within the surface, divided by overall time for some signal to reach you (measured as deviation from assumption of c). There is a further assumption (very reasonable) that proper distances are sufficiently static over the period of interest.

 

[PAllen]

As to your actual question, rather than asking that someone read a whole, long thread, can you answer the following:

How do you see the relation between 'length contraction' and Shapiro time delay?

I always viewed it (primarily) as a question of lightspeed not distance variation; or (when pushing the idea of multiple valid coordinates) as an orbital bump with light speed preserved for one chosen observer. The idea of relating it to length contraction is new to me.

 

[timmdeeg]

I always viewed it (primarily) as a question of lightspeed not distance variation; or (when pushing the idea of multiple valid coordinates) as an orbital bump with light speed preserved for one chosen observer. The idea of relating it to length contraction is new to me.

The Shapiro delay is usally understood as resulting from gravitational time dilation or from extra length due to gravitational space-contraction. The latter can be thought as arising from the projection of equally spaced distances in the gravitational well to the light path, which is tangent to the well. There the projected distances are shorter, accordingly.
Quite amazing, I found an explanation in one of my textbooks, describing the Shapiro delay as the sum of both, gravitational time dilation and extra length. I am not really sure, if this is correct. In my opinion its the same physics expressed in either way.

 

[grav-universe]

Right, I am basing it upon coordinate light speed, which should be c' = c (1 - r_s / r) / sqrt[1 - (sin θ')^2 (r_s / r)] depending upon the angle of the light travel to the radial direction according to the distant observer, with light speed c (1 - r_s / r) radially and c sqrt(1 - r_s / r) tangent, but this dependence upon the angle is due to length contraction in the radial direction only and none in the tangent direction while a hovering observer at r measures isotropic c. This gives a time delay for light traveling from Earth to another planet and back of Δt = 2 (G M / c^3) [2 ln(4 (r_e / R) (r_p / R)) - sqrt(1 - (R / r_e)^2) - sqrt(1 - (R / r_p)^2)], or Δt = 4 (G M / c^3) [ln(4 (r_e / R) (r_p / R)) - 1] to first order, where R is the point of closest approach. But it is short of the delay measured by Shapiro of 4 (G M / c^3), always about 20 microseconds, as compared to the entire delay of about 240 microseconds to Mars and back for example. All data so far appears to be about this much short. However, using c' = c (1 - r_s / r) gives the correct result in each case and the correct calculation as well from what I could find of the equation used for the Shapiro delay, which is given as just Δt = 4 (G M / c^3) ln(4 (r_e / R) (r_p / R)).

 

[Bill_K]

Quite amazing, I found an explanation in one of my textbooks, describing the Shapiro delay as the sum of both, gravitational time dilation and extra length. I am not really sure, if this is correct. In my opinion its the same physics expressed in either way.

Trust your textbook. You might get away with conflating the two concepts in special relativity where the same factor γ is involved. But not in general relativity, where the time dilation is related to g_tt, and the length contraction is related to g_rr, and g_tt and g_rr are not related!

 

[timmdeeg]

Trust your textbook. You might get away with conflating the two concepts in special relativity where the same factor γ is involved. But not in general relativity, where the time dilation is related to g_tt, and the length contraction is related to g_rr, and g_tt and g_rr are not related!

Ah, thank you for this good explanation.

 

[PAllen]

And, what I overlooked just thinking about formulas for speed of light, and not trying to separate causes, is that for a diagonal metric, the coordinate speed of light is trivially √g(tt)/g(xx), if x is arranged to be the direction of light path.

[Edit: Note that most of the path is dominate by g(rr) for a path going close to the sun. ]

 

[PAllen]

Right, I am basing it upon coordinate light speed, which should be c' = c (1 - r_s / r) / sqrt[1 - (sin θ')^2 (r_s / r)] depending upon the angle of the light travel to the radial direction according to the distant observer, with light speed c (1 - r_s / r) radially and c sqrt(1 - r_s / r) tangent, but this dependence upon the angle is due to length contraction in the radial direction only and none in the tangent direction while a hovering observer at r measures isotropic c. This gives a time delay for light traveling from Earth to another planet and back of Δt = 2 (G M / c^3) [2 ln(4 (r_e / R) (r_p / R)) - sqrt(1 - (R / r_e)^2) - sqrt(1 - (R / r_p)^2)], or Δt = 4 (G M / c^3) [ln(4 (r_e / R) (r_p / R)) - 1] to first order, where R is the point of closest approach. But it is short of the delay measured by Shapiro of 4 (G M / c^3), always about 20 microseconds, as compared to the entire delay of about 240 microseconds to Mars and back for example. All data so far appears to be about this much short. However, using c' = c (1 - r_s / r) gives the correct result in each case and the correct calculation as well from what I could find of the equation used for the Shapiro delay, which is given as just Δt = 4 (G M / c^3) ln(4 (r_e / R) (r_p / R)).

I reviewed some of the derivations of of the delay as well as computing it myself. Amazing how many different approaches there are. What I see is that the standard quoted delay is relative to 'as if there were no gravity' (thus it is seen anywhere in the solar system), and measurements determine differences in delay for different paths. Thus a sun grazing path will show delay primarily due to sqrt(g(tt)/g(rr)), while a more tangential path will approach sqrt(g(tt)). But each is considered to have a delay, and the former is quoted relative to imputed delay as if sun wasn't there (or compared to light speed at infinity) . At least some of the the early measurements actually measured only the change per day in radar round trip time versus what is expected from mutual orbits with constant c assumed, with large daily difference only occurring plus or minus a few days from target planet opposite the sun from earth.

I am not an expert on this, but that is what I see in various sources on the web, plus the discussion in MTW.

 

[grav-universe]

I reviewed some of the derivations of of the delay as well as computing it myself. Amazing how many different approaches there are. What I see is that the standard quoted delay is relative to 'as if there were no gravity' (thus it is seen anywhere in the solar system), and measurements determine differences in delay for different paths. Thus a sun grazing path will show delay primarily due to sqrt(g(tt)/g(rr)), while a more tangential path will approach sqrt(g(tt)). But each is considered to have a delay, and the former is quoted relative to imputed delay as if sun wasn't there (or compared to light speed at infinity) . At least some of the the early measurements actually measured only the change per day in radar round trip time versus what is expected from mutual orbits with constant c assumed, with large daily difference only occurring plus or minus a few days from target planet opposite the sun from earth.

I am not an expert on this, but that is what I see in various sources on the web, plus the discussion in MTW.

Right, the path is mostly radial, but a calculation based only upon the radial speed will vary from that including the tangent component by 20 microseconds for the sun, a considerable difference, but matches the Shapiro data. Have you found any derivations on the web or a way to perform the calculation that accounts for the tangential speed component but somehow still manages to give the Shapiro result? I have been looking, but sources on the web seem limited for derivations of the Shapiro delay. I am also searching for Shapiro's original paper.

 

[PAllen]

Right, the path is mostly radial, but a calculation based only upon the radial speed will vary from that including the tangent component by 20 microseconds for the sun, a considerable difference, but matches the Shapiro data. Have you found any derivations on the web or a way to perform the calculation that accounts for the tangential speed component but somehow still manages to give the Shapiro result? I have been looking, but sources on the web seem limited for derivations of the Shapiro delay. I am also searching for Shapiro's original paper.

The answer appears to be that all computations of this assume either PPN coordinates (which are isotropic for a spherically symmetric source) or isotropic SC coords, with approximate metric: expanding isotropic SC metric components in powers of M/r and keeping terms only to M/r or (M/r)^2). It is claimed in MTW that ephemeris are (already as of 1970) quoted in isotropic coordinates. Of course, coordinates cannot effect a measurement. So, the implication is that the planetary position producing some delay corresponds to a different r value in the two systems, and you must convert before comparing predictions.

MTW pp. 1097, sec. 40.1 discusses these issues.

 

[Bill_K]

For the geodesic equations in the Schwarzschild metric there are three first integrals:

L = r²(dφ/ds)
E = (1 - 2M/r)(dt/ds)
F = (1 - 2M/r)^-1(dr/ds)² + r²(dφ/ds)² - c²(1 - 2M/r)(dt/ds)²

For a null geodesic, s is an affine parameter. We may choose s to agree with t as r → ∞, implying E = 1. Also F = 0,

⇒ (dr/ds)² = c² - (1 - 2M/r)L²/r²

Let b be the perihelion distance, i.e. r = b where dr/ds = 0. This implies

L² = b²c²/(1 - 2M/b)

Putting it all together,

dt/dr = dt/ds · ds/dr = (1/c)(1 - 2M/r)^-1[1 - (b²/r²)(1 - 2M/r)/(1 - 2M/b)]^-1/2

which may be integrated to find the elapsed time,

t = ∫ (..RHS..) dr

So far this is exact. To get an integral which may be evaluated analytically, expand in powers of M/r:

dt/dr = (1/c)(r/√(r² - b²))[1 + 2M/r - Mb²/(r² - b²)(1/r - 1/b) + O((M/r)²)]

The result is

ct = √(r² - b²) + 2M ln(r + √(r² - b²)) + M(r - b)/√(r² - b²)

The first term is the Euclidean straight-line distance. The second is the usually quoted Shapiro delay. The third is usually discarded, order of magnitude the time it takes light to cross the Schwarzschild radius. For the sun, 2M = 3 km, so 2M/c = 3 km/(3 x 10⁵ km/sec) = 10 microsecs.

This is for half the trip: from the planet to the perihelion. From planet to planet it will be double that.