Talk:Shapiro time delay

Removed text
I removed this text: Some have argued that the Shapiro Effect can account for the Hubble redshift without having to posit an expanding universe. It appears this has been thoroughly discredited. P. Riis 17:13, 21 Oct 2004 (UTC)

True. Kmarinas86 23:22, 8 November 2005 (UTC)

False: This argument has not been discredited, claims alluding to it's refute have been asserted by mainstream Astronomers with an agenda to prevent the immenent loss of their funding.'' The only force stronger than the force of gravity, is the force of greed. Denying clear evidence is a sure sign of religious bias, and a spit in the face of empirical science. unsigned 23:18, 7 November 2006 (UTC)

I submit that the above be ignored. The Shapiro delay cannot explain Hubble flow: it is very much too small. This is not a matter of something being discredited, but of it never being considered seriously in the first place. I suspect that 'unsigned' would call me "mainstream Astronomer", but making a statement on this one way or the other is most unlikely to affect anyone's funding, assuming that they support it. Michaelbusch 22:48, 6 November 2006 (UTC)

I submit that no evidence be ignored, open discussion and research is the way to get at the truth, not by censoring opinions that don't agree with one's own. The Shapiro effect is not too small when applied to photons traveling over vast distances, such as through interstellar space. the cumulative effect would account for observed redshift on the Hubble scale. incidentally the Mössbauer effect and Compton effect have also been proposed as alternative causes for redshift. As for funding, no Astrophysicist who has published and built his reputation around such research, having acquired government funding involving "astronomical" sums of money is going to drop his basket of eggs to chase a rainbow, that may or may not hold the key to the true nature of the cosmos. unsigned 23:22, 7 November 2006 (UTC)

I did not censor, merely suggesting that your previous comment isn't relevant to the article. I simply gave the reason for Shapiro delay not being relevant on the cosmological scale: there is very much less mass. Mossbauer fails because it works only for solid materials and for gamma rays. Compton fails because it doesn't work at low energies. This is all I will say on the subject, because this is an encyclopedia, not a pseudo-science discussion board. Michaelbusch 00:05, 7 November 2006 (UTC)

How much mass there is, or isn't is a matter of distance, the Hubble scale assumes a given distance to redshift ratio, however there is no guarantee this ratio is correct. With gravitational drag the greater the distance light travels, the greater the redshift. This would explain why light from galaxies beyond 7 x 10^9 lightyears is unable to reach us. There may be an absolute maximum distance light can travel before it's wavelengths are time stretched beyond what earth based telescopes can detect.

To state that a comment on the Shapiro delay accounting for stellar redshift is irrelevant to an article on Shapiro delay is a bit like saying the pope is irrelvant to the vatican. this is not pseudo science. Furthermore, neither you, nor I nor any living human is an authority on the true nature of the cosmos, and any claim on such absolute knowledge is a pretence. All theories are by definition "theoretical" and chances are we're all way off. you may have the last word on this discussion board, but your's is not the last word on the subject. unsigned 23:37, 7 November 2006 (UTC)

At the penalty of contradicting myself, one final statement on this thread: Shapiro delay as described in this article does not change the frequency of light at all, just changing the travel time. There are corrections to frequency which are related to gravitational Doppler shift, but those are very small indeed. Michaelbusch 00:22, 7 November 2006 (UTC)

The frequency of light need not change to produce redshift in a time dragged frame, the reciprocal of frequency is wavelength, and it's the wavelength which would be stretched. It's necessary to look at the big picture, rather than nitpick over it's small components. There is also no guarantee that the GR model of gravity is correct, for example in a plasma universe governed solely by electrodynamic laws, the physics on a cosmic scale is not the same.

note: no accusation is made towards any particular Astronomer regarding their motives, undoubtedly there are very many mainstream Astronomers who are absolutely sincere and engaged in honest research, my comment was only directed at those (who know who they are) who do in fact have an agenda. unsigned 23:49, 7 November 2006 (UTC)

The interaction of the photon and the electron is not elastic. The transfer of energy in the interaction of infrared light and matter can only happen if the photon-electron interaction is inelastic. As the Einstein quote points out, photons must slow in order for them to follow a curved path. The side of the photon nearer the gravity source must have a slower speed for the path to curve, but both sides of the photon slow. Additionally, in that the interaction of the photon and the electron is inelastic, there is a transfer of energy in the interaction of the photon and the mass in the vicinity, resulting in a redshift of the photon. This explains both the Shapiro effect and cosmic redshift. This argument supports the “tired light” hypothesis and a static universe. My Flatley (talk) 23:54, 10 May 2011 (UTC)

Missing data on Shapiro effect
If someone knows the numerical data it would be nice to place them here on the page. How much longer it takes the signal to make the round trip and how much speed of light changes in the gravitational field. My theoretical result (from the principle of conservation of energy) tells that dc/dx=g/(2c), where x is in directon opposite to the source of gravitational field. I wonder what's the observational result in the Shapiro effect. Jim 18:58, 2 September 2005 (UTC)

I would suggest writing Prof. Shapiro directly. here is his email address at the Center for Astrophysics - Harvard Smithsonian Astrophysical Observatory: ishapiro@cfa

Some problems with current version
Quick note: one problem is that "isolated mass" should be "nonrotating isolated mass" and should include discussion of the coordinate chart. The Schwarzschild exterior chart is a good choice here, but of course other charts are more useful for other purposes, and reader should not be confused by conflating a chart with the solution it describes, particularly since Schwarzschild charts are often useful for other static spherically symmetric solutions, such as Reissner-Nordstrom or Janis-Newman-Winacour.---CH 04:41, 19 December 2005 (UTC)

Also, the last subsection is particularly bad in terms of failing to explain that this stuff (e.g. energy, momentum of photon, or better yet, of laser pulse) is observer dependent. ---CH 04:44, 19 December 2005 (UTC)

Page name
Page moved from Shapiro effect to Shapiro delay per a request on WP:RM. Search on Google Scholar shows the latter term is preferred in academic circles (more strongly than on Google, though there too). Rd232 talk 12:31, 27 December 2005 (UTC)


 * Thanks, Rd232! In fact, Shapiro time delay is the most precise short term which is in standard use, so I

had planned to make that change but hadn't gotten to it. But Shapiro delay is probably good enough. ---CH 15:25, 24 January 2006 (UTC)

Time delay and collapsing objects
Whoever wrote that was apparently confusing the Shapiro time delay effect with gravitational redshift in an expanding null congruence from a spherically symmetric gravitating object, apparently even confusing freely propagating light with signals from the interior of a collapsing fluid ball. If that person is upset that I removed the section, I ask that he/she at least study D'Inverno's textbook (which should clarify these points) before trying again. TIA!

This article is still badly in need to a thorough rewrite to improve accuracy, emphasis, and readability, (but still not very on my long personal to do list, unfortunately). ---CH 15:25, 24 January 2006 (UTC)

The Formula was wrong
It was obviously wrong as it gave a negative delta t for not too large angles between the source and the mass. I've changed the definition of the unit vectors into what I remember to be correct, but this needs to be checked. Count Iblis 15:11, 20 August 2006 (UTC)


 * The formula looks entirely fictitious, at least concerning the term with the natural logarithm. According to Einstein in 1916 the total change in angle is $$2 \frac{R_{s}}{R}$$. From this result one would expect, according to Euclidean geometry, $$\Delta x \approx R_s$$. 84.59.51.31 09:37, 17 July 2007 (UTC)
 * No, really $$\Delta x $$ would be much smaller than $$R_s$$.

Nonsense

 * $$\Delta t = -\frac{R_s}{c} \log(1-\mathbf{R}\cdot\mathbf{x})$$

This equations predicts, the delay becomes infinite if the source is in same direction as the center of mass, obvious nonsense.


 * It is not nonsense. What happens is that the mass is treated as a point mass and the formula is not an exact formula that would still give the correct answer in case the light comes to within the order of the Schwartzchild radius of the gravitating mass. The Shapiro formula is obtained by treating the deflections due to gravity as a first order perturbation in the straight line propagation, and that approximation will break down if the source is exactly behind the mass.


 * So, you see that the formula diverges logarithmically for the angle going to zero. You can interpret that as the gravitational time delay diverging in the limit of infinite gravity. What happens in reality when the source is right behind the mass is different, of course. The light then bends around the mass due to graviational lensing and you then do have a finite delay. Count Iblis (talk) 14:06, 25 April 2010 (UTC)


 * Assume a distant star directly behind a second star (radius R) in distance D. The distance to Earth would be D again. Euclid says


 * $$ \Delta x = 2 \left( \sqrt{D^2 + R^2} - D\right) \approx {\frac{R^2}{D}} = D \times \left(\frac{R}{D}\right)^{2}$$

The equation is incorrect or at least incomplete (not specifying validity interval), because at the time of inferior conjunction (at closest approach of planet Venus to Earth), the ray does not pass arround the Sun at all, but the equation predicts large delay due to small angle between planet and Sun as seen from Earth, being completelly independent on distances of both observer and reflector (planet) and the ray-path from Sun, but this was not measured. In the paper there is a peak in time delay only at superior conjunctions, but no peak at inferior conjunctions. Also the equation gives incorrect results, when pointing away from Sun - at more than 90° between R and x, the delay would be negative?

Due to space curvature, the path is longer, but due to time dilation, the time for the ray to travel is actually longer too, so there would be no observed delay due to space-time curvature, if both space and time curvature radius are same? πα 2016-01-02

Graphical Representation
It could be helpful if someone would draw a sketch with the emitter the receiver and the interfering plantet with vectors and such.Agge1000 06:34, 15 November 2007 (UTC)

Potential versus field
Gravitational time dilation is a function of gravitational potential, not of gravitational field strength. Since the delay can be viewed as an effect of time dilation, it too is a function of potential and not field. The article as currently written is confusing on this point, and should be clarified. Howard Landman (talk) 14:56, 25 December 2012 (UTC)

Fabricated quotation
This page had a quotation from an article by Shapiro that did not appear in the article cited. That quotation was replaced with the closest matching sentences from the article. — Preceding unsigned comment added by PaunchyCyclops (talk • contribs) 17:59, 30 December 2014 (UTC)

Garbled equation?
I've added a citation-needed tag for the equation for the delay, \Delta t -- I don't think the expression given can be correct (since it indicates that when R and x are orthogonal -- that is, when the source and the mass are at right angles from the point of view of the observer -- the delay is zero, which won't be the case). I'm loth to change this myself, however, because it may just be that the explanation for the symbols has become garbled (what's there seems ... dubious), and it would be best for the original author of this text to fix it. NormanGray (talk) 22:30, 23 February 2015 (UTC)

Please justify this assertion
"moderate strength (say, of stars and planets, but not one of a black hole or close binary system of neutron stars)"... the predictions of Relativity are not subject to this limitation. Either this assertion needs to be justified or removed. The stretching of space between two points due to mass is proportional to the difference in potential energy of the two points. That's it. — Preceding unsigned comment added by McGinnis (talk • contribs) 01:38, 12 June 2016 (UTC)

I find nothing in the literature to suggest the Shapiro effect is constrained to low field conditions. In fact, it cannot be unless the basic logic of conservation of energy and causality, on which relativity depends, are wrong. Unless some sort of logical argument or observation can show the Shapiro effect is not universally applicable, this proviso of "moderate strength" should be removed. — Preceding unsigned comment added by McGinnis (talk • contribs) 18:34, 28 August 2016 (UTC)

Any justification for this interpretation? I don't think that whoever wrote this article really has any understanding of Relativity. In a gravity well, space is stretched radially. This is why light is delayed. The light does NOT travel more slowly. The light has a longer path to cover because of the space dilation. Orbital distances are different from what Newton would predict for Euclidean space. But, general relativity has shown us that spacetime about gravity wells is non-Euclidean. Michael McGinnis (talk) 23:02, 4 September 2016 (UTC)


 * I think whoever wrote the article did a good job, considering there is confusion even in the advanced literature on these topics. You say, "light does NOT travel more slowly." While it is true that the proper velocity of light, as measured locally along its path, is constant, the coordinate velocity of light is less than c in a gravitational well. For example, dr/dt<c for a photon emitted radially from a star. This coordinate velocity would be the velocity as judged from infinity. Hence the confusion. Indeed, the Shapiro quote at the top of the article says, "Because, according to the general theory, the speed of a light wave depends on the strength of the gravitational potential along its path . . . " so Shapiro himself said the speed of a light wave depends on potential, and hence varies. Shapiro no doubt meant coordinate speed. Please see section below entitled Is the Shapiro delay due to time dilation or path length increase? Very often in GR, there are multiple "correct interpretations", but we must specify with precise accuracy the meaning of every term we use. 71.32.47.71 (talk) 20:00, 26 December 2018 (UTC)Kathleen Rosser

Why not use a Relativistic explanation?
If the Shapiro Delay is to be interpreted in relativistic terms then the delay can only be interpreted as a lengthening of path. In Relativity, path length is ct where t is the time it takes light to traverse the path. If you want to interpret it as light slowing down, you aren't using Relativity. The reason spacetime is curved in a gravitational field is that it is dilated. As Shapiro shows, the path length is increased. Is this page going to receive review or do I need to rewrite it? Michael McGinnis (talk) 02:10, 30 July 2017 (UTC)


 * You say, "If the Shapiro Delay is to be interpreted in relativistic terms then the delay can only be interpreted as a lengthening of path." This is inaccurate. Time dilation and path length increase contribute roughly equally. Please see section below entitled "Is the Shapiro delay due to time dilation or path length increase?" 71.32.47.71 (talk) 13:40, 27 December 2018 (UTC) Kathleen Rosser

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Is the Shapiro delay due to time dilation or path length increase?
The beginning of the article states, "The time delay is caused by spacetime dilation, which increases the path length." The term spacetime dilation, although accurate, may be unclear (I've never seen it before). Also, the increase in path length is only a partial cause of the delay. The statement would more accurately read: "The Shapiro time delay is caused both by gravitational time dilation and by a relativistic increase in path length." The contributions are roughly equal. This can be seen from the coordinate speed of light in the Schwarzschild metric, which is dx/dt = &radic;(-g00/grr). To obtain travel time t, one solves this for dt and integrates over the path x, where the path is approximated as straight. The fact that g00, which represents the time dilation factor, and grr, which represents the path length factor, are both present means they both contribute. This is clearly laid out in the textbook Gravitation and Inertia by Ignazio Ciufolini and John Archibald Wheeler (Princeton Series in Physics, 1995), pp122-125 71.32.47.71 (talk) 20:35, 26 December 2018 (UTC) Kathleen Rosser

The straight path approximation is incorrect in the first order
The time delay can be split into two different parts:

1) The Shapiro time delay, caused by a variation in the speed of a photon if it is submerged in a gravitational potential. 2) The geometric delay, caused by the increased length of the total light path from the source to the target, which is due to gravitational deflection.

This article focuses primarily on Shapiro time delay and tries to downplay the relevance of geometric delay. It states that: "the contribution from the change in path, being of second order in M, is negligible". The straight path approximation is actually incorrect in the first order. — Preceding unsigned comment added by 2001:1C00:F20:8100:45EF:C2D7:33F9:FF75 (talk) 13:07, 19 November 2019 (UTC)

extra distance

 * which is the extra distance the light has to travel
 * There is NO extra distance "around"!!! The orbit is NOT stretched. Ra-raisch (talk) 20:16, 25 June 2020 (UTC)
 * If this way is supposed to calculate shorthand a stretching of space radially, this should be mentioned as well as the further procedure which is completely unexplained. Ra-raisch (talk) 19:10, 26 June 2020 (UTC)
 * After all, I guess, that the subtitle "Time delay due to light traveling around a single mass" is wrong, since light is not going "around" the mass. Maybe the correct title should read "Time delay due to light traveling near a single mass", or perhaps "Time delay due to light round-traveling to a single mass". Ra-raisch (talk) 22:46, 26 June 2020 (UTC)
 * The solution is visible in the article by Friedman as follows: Since the tangential speed of light c'=c°σ is reduced by one shapiro factor σ=√(1-rs/r), the time flow τ = σt also is reduced thus the distance covered is reduced by twice the factor s = c'τ = σ²c°t.
 * $$d\varphi/dt = d\dot\varphi/d\dot t = -\sigma^2cb/r^2$$ Ra-raisch (talk) 10:30, 27 June 2020 (UTC)


 * The explanation of Friedman, which is no longer cited in the article, seems absurd, because c' is not referred to τ but to t. Ra-raisch (talk) 19:35, 12 June 2023 (UTC)

Ref.4
Elena V. Pitjeva:Tests of General Relativity from observations of planets and spacecraft is no longer available but at Springer https://link.springer.com/article/10.1134/1.1922533 Ra-raisch (talk) 20:22, 6 October 2020 (UTC)

Capital O function definition
Interesting article, but in the Shapiro time delay equation, in the last term of the right hand side, the capital O function should be defined or named. I don't think we mean the Little-O or the Big-O functions. Since here the argument to the O function depends on the small constant (G^2/c^6), the + capital O function perhaps means 'plus a small remainder term that depends on': ((G^2/c^6)*M^2) ? Thanks (Eromana (talk) 21:55, 14 October 2020 (UTC))
 * This of course is the Landau symbol of order of error Big_O_notation. Ra-raisch (talk) 09:47, 2 September 2021 (UTC)

Section needs to be fixed «Time delay due to light traveling around a single mass
This section has a formula: $$\Delta t = -\frac{2GM}{c^3} \ln(1 - \mathbf{R}\cdot\mathbf{x}),$$ where R is the unit vector pointing from the observer to the source, and x is the unit vector pointing from the observer to the gravitating mass M.

This formula does not describe the entire time dilation when receiving an echo from a certain planet behind the Sun on the Earth, but only one part of this delay. For example, the formula would be suitable in the case when there is a reflector near the Sun, from which the signal is reflected and returned to the Earth. If there is a planet behind the Sun that reflects the signal back to the Earth, then the total time will be obtained by adding a second analogous term, where the unit vectors should be taken not from the observer, but from the planet. Fedosin (talk) 14:25, 6 March 2023 (UTC)

The meaning of this formula is as follows. Consider the scalar product of unit vectors $$ \mathbf{R}\cdot\mathbf{x} $$. By definition, the scalar product of vectors is equal to the product of the absolute values of the vectors and the cosine of the angle between the vectors. There are unit vectors here, so their length is 1. In the scalar product of these vectors, only the cosine of the angle between the vectors remains. It turns out the formula
 * $$\Delta t_1 = -\frac{2GM}{c^3} \ln(1 - cos \alpha ),$$

where the angle $$ \alpha $$ is the angle between the direction from the Earth to the planet behind the Sun and the direction from the Earth to the Sun.

Let's expand the cosine in the first approximation: $$ cos \alpha \approx 1-\frac {\alpha^2}{2}$$. This gives:
 * $$\Delta t_1 = -\frac{2GM}{c^3} \ln(\frac {\alpha^2}{2}) =\frac{2GM}{c^3} \ln(\frac {2}{\alpha^2}).$$

Since the angle $$ \alpha $$ is small, it can be estimated as $$ \alpha \approx \frac {b}{r_E}$$, where $$ b$$ is the shortest distance from Sun to the line connecting the Earth and the planet behind the Sun; $$ r_E $$ is the distance from the Earth to the Sun. Hence it follows that
 * $$ \Delta t_1 =\frac{2GM}{c^3} \ln(\frac {2 r^2_E }{b^2}).$$

For the movement of the signal from the Sun to the planet and back will be:
 * $$ \Delta t_2 =\frac{2GM}{c^3} \ln(\frac {2 r^2_P }{b^2}),$$

where $$ r_P $$ is the distance from the planet to the Sun. The total time dilation in the first approximation will be
 * $$ \Delta t=\Delta t_1+\Delta t_2 =\frac{4GM}{c^3} \ln(\frac {2 r_E r_P }{b^2}).$$

However, according to Ref.: Logunov A.A. The Theory of Gravity. Moskva, Nauka (2001). ISBN 978-5020226999. arXiv:. https://arxiv.org/abs/gr-qc/0210005 ; See Eq. (12.52), the leading term in time dilation is equal to
 * $$ \Delta t \approx \frac{4GM}{c^3} \ln(\frac {4 r_E r_P }{b^2}).$$

It turns out that the formulas differ, Logunov's expression under the sign of the logarithm is 2 times larger. Therefore the formula
 * $$\Delta t = -\frac{2GM}{c^3} \ln(1 - \mathbf{R}\cdot\mathbf{x})$$

is inaccurate. And there is no reference to this formula yet.

I propose to remove this formula and replace it with the Logunov formula in the form
 * $$ \Delta t \approx \frac{4GM}{c^3} \ln(\frac {4 r_E r_P }{b^2}),$$

which describes in the first approximation the delay of the radar echo from the planet behind the Sun. Here we mean that the delay arises from the curvature of space-time by the gravitational field of the Sun. In this case, not only the length of the signal path changes, but also its speed near the Sun. — Preceding unsigned comment added by Fedosin (talk • contribs) 14:54, 6 March 2023 (UTC)

In Ref. Asada H. Gravitational time delay of light for various models of modified gravity. Physics Letters B, Vol. 661 Issue 2/3, pp 78-81 (2008). https://dx.doi.org/10.1016/j.physletb.2008.02.006 in Eq. 5 we can find more accurate formula for the main term of delay
 * $$ \Delta t \approx \frac{4GM}{c^3} \ln(\frac { (r_E +\sqrt { r^2_E -b^2}) (r_P +\sqrt { r^2_P -b^2})}{b^2}).$$

Usually $$ r_E $$ much more than $$ b $$ and the same $$ r_P >>b $$. — Preceding unsigned comment added by Fedosin (talk • contribs) 05:40, 7 March 2023 (UTC)

— Preceding unsigned comment added by Fedosin (talk • contribs) 14:32, 6 March 2023 (UTC)