Shapiro time delay vs cosmic expansion as sources of interstellar redshift?

Question

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1. What the student is really asking

Student’s wording	What it really means
“The radar pulses that passed the Sun took ~200 µs longer than expected. Does this mean that the speed of light itself changed for the observers on Earth?”	If a signal takes longer to go from A to B, does that imply that the local speed of the photons was different from the invariant value c?
“If the speed can change, does Einstein’s 2nd postulate (c is the same in every inertial frame) have to be abandoned?”	Is the post‑postulate falsified by the Shapiro delay?
“Should we think of c as only a defined number, not a physical constant, because gravity and energy everywhere can tweak it?”	Is the constancy of c only a convention, not a law of nature?
“Does the Shapiro effect have any bearing on the interpretation of cosmological red‑shift (expansion of the Universe)?”	Can the extra travel time/red‑shift produced by a gravitational field be confused with the red‑shift we attribute to the expanding Universe?

The answer requires a clear distinction between (i) the locally measured speed of light, which is always exactly c, and (ii) the coordinate speed of light that depends on the choice of space‑time coordinates (e.g., the Sun‑centered Schwarzschild coordinates used to describe the experiment).

2. Step‑by‑step explanation

2.1. What the Shapiro delay measures

Set‑up – A radio pulse is sent from Earth, skims the Sun at a distance r ≈ solar radius, is reflected by a spacecraft (or a planet) and returns to Earth.
Naïve expectation – If space were flat and empty, the round‑trip travel time would be
[ t_{\rm flat}= \frac{2\,L}{c}, ]
where L is the (Euclidean) Earth–spacecraft distance.
Observed result – The round‑trip time is longer by
[ \Delta t \simeq \frac{2GM_{\odot}}{c^{3}}\, \ln!\Bigl(\frac{4r_{E}r_{S}}{b^{2}}\Bigr) , ]
the Shapiro (gravitational) time delay, where
- G – Newton’s constant,
- M_{\odot} – mass of the Sun,
- r_E, r_S – distances of Earth and spacecraft from the Sun,
- b – impact parameter (closest approach).
For the Venus experiment Δt ≈ 200 µs, for the Cassini‑Saturn experiment Δt ≈ 240 µs.
Physical origin – In General Relativity (GR) the presence of mass curves space‑time. Light follows a null geodesic, i.e. a path for which the space‑time interval ds = 0. Because the coordinate t (the time measured by a distant observer) runs more slowly deeper in the gravitational potential, the coordinate dt required to traverse a given coordinate dr is larger. This manifests as an extra travel time when the ray passes near the Sun.

2.2. Local vs. coordinate speed of light

Quantity	Definition	Value in the Shapiro experiment
Locally measured speed	What an observer at the point where the photon passes measures with a ruler and a clock that are both in the same gravitational potential.	Exactly c = 299 792 458 m s⁻¹ (by construction of the metric).
Coordinate speed	Rate dr/dt in a chosen set of coordinates (e.g., Schwarzschild coordinates where t is the time kept by a far‑away observer).	Slightly less than c near the Sun: (v_{\rm coord}=c\,(1-2GM/rc^{2})).

Why the difference matters

The postulate “the speed of light in vacuum is the same in all inertial frames” is a local statement. It says that any freely‑falling observer (i.e. an observer in an infinitesimally small region where gravity can be ignored) will always measure the speed of a light pulse as c.
The Shapiro delay does not involve a local measurement of speed; it involves the integrated coordinate time taken for the photon to travel a macroscopic curved path. The integration of the slower coordinate speed over the portion of the path near the Sun yields the extra delay.

Thus the delay does not contradict Einstein’s second postulate.

2.3. Does the experiment “slow down” the photon?

No. In the local inertial frame of an observer comoving with the photon (or, more realistically, an observer at the same point with a small laboratory), the photon still moves at c. The elapsed coordinate time measured by a distant Earth clock is larger because the clock ticks slower in the deeper potential. The photon’s world‑line is the same null line; only the coordinate mapping stretches the time axis.

2.4. Is “c is only a defined constant” a valid reinterpretation?

In the International System of Units (SI) the meter is defined by the value of c: 1 m = (1 c s). This definition makes c an exact number by convention; it is not measured each time.
The physical constancy is that any locally measured speed of light in vacuum equals this exact number. GR predicts that in any freely‑falling laboratory, even in the presence of strong gravitational fields, the measured speed will be c. The experiment merely confirms the gravitational time‑dilation part of the metric, not a variation of the local speed.

Hence the experimental outcome supports, rather than undermines, the idea that c is a fundamental constant.

2.5. Relation (or lack thereof) to the cosmological red‑shift

Phenomenon	Origin	Mathematical form	Observed effect
Gravitational (Shapiro) delay + red‑shift	Static space‑time curvature around a mass (Schwarzschild metric).	Frequency shift: (\displaystyle \frac{\nu_{\rm rec}}{\nu_{\rm em}} = \sqrt{\frac{1-2GM/rc^{2}}{1-2GM/r_{\infty}c^{2}}}).	Photons climbing out of a potential lose energy → gravitational red‑shift (tiny, ≈10⁻⁶ for the Sun).
Cosmological expansion red‑shift	Dynamical Friedmann‑Lemaître‑Robertson‑Walker (FLRW) metric with a scale factor a(t).	(\displaystyle 1+z = \frac{a(t_{\rm now})}{a(t_{\rm em})}).	Photons are stretched as the Universe expands → Hubble law (z ≈ H₀ d for nearby objects).

Key points:

Different metrics – The Sun’s field is static; the Universe’s metric is time‑dependent. The Shapiro effect can be expressed as a coordinate delay in a static geometry, while cosmological red‑shift is a global scaling of wavelengths over billions of years.
Magnitude – Gravitational red‑shift from the Sun is of order 10⁻⁶, whereas typical extragalactic red‑shifts are z ≈ 0.1–10 (10 % to many hundred percent). The two are incomparable.
Observables – The Cassini experiment measured phase changes in the radio carrier, which are interpreted as a Shapiro‑induced extra time, not as a change of the fundamental cosmic scale factor.

Consequently the Shapiro experiment does not provide an alternative explanation for the cosmological red‑shift. The latter remains best described by the expansion of space (or, equivalently, by the FLRW metric).

3. Final answers to the three questions

Question	Answer (short)
Q1. Does the observed delay mean the speed of light changed, violating Einstein’s 2nd postulate?	No. The locally measured speed of light is always c. The delay is caused by the coordinate time running slower in the Sun’s gravitational potential, not by a change in the intrinsic photon speed.
Q2. Must we abandon the axiom and treat c as merely a defined convention because gravity affects it?	No. The constancy of c is a local physical law that holds in all (locally inertial) frames, even in strong fields. The definition of the meter via c reflects this constancy; the experiment confirms the GR prediction of gravitational time dilation, not a failure of the axiom.
Q3. Does the Shapiro delay/red‑shift undermine the interpretation of cosmic expansion?	No. The Shapiro effect is a small, static‑field phenomenon (Δt ∼ 10⁻⁴ s, red‑shift ∼ 10⁻⁶) distinct from the large, dynamic red‑shift caused by the expanding Universe (z ≳ 10⁻²). The two have different physical origins and are treated with different metrics.

4. Common Mistakes (and how to avoid them)

Mistake	Why it’s wrong	How to avoid it
*Confusing coordinate* speed with local speed.** Believing the photon “actually slowed down”.	Coordinate speed depends on the choice of time coordinate; it can be < c even though locally the speed is always c.	Remember the equivalence principle: in a sufficiently small freely‑falling lab, the metric is locally Minkowskian and the measured speed is exactly c.
Claiming the Shapiro delay disproves the constancy of c.	The constancy of c is a local postulate, not a claim about integrated travel times over curved space‑time.	Write out the null condition (ds^{2}=0) in the relevant metric; you will see that it forces the local speed to be c regardless of the metric components.
Equating the tiny gravitational red‑shift near the Sun with the large cosmological red‑shift.	They arise from different metrics (static Schwarzschild vs. dynamic FLRW) and differ by many orders of magnitude.	Compare the formulas: gravitational red‑shift ∝ GM/(rc²) ~10⁻⁶, while cosmological z ∝ H₀ d/c for distant galaxies, which can be ≳0.1.
Thinking the experiment measured “c slowed to 299 792 458 m/s – 0.0002 %”.	The experiment measured Δt relative to a Newtonian‑flat‑space prediction, not a change in the fundamental value of c.	Recall the SI definition of the metre: the numerical value of c is exact; the experiment merely tests the GR correction to the coordinate time of flight.
Ignoring the role of the observer’s clock.	Gravitational time dilation means the Earth clock runs at a different rate than a clock near the Sun; forgetting this leads to the impression that the photon took longer because it moved slower.	Write the proper time interval measured by the Earth observer: (\Delta \tau_{\rm Earth}= (1+Φ/c^{2})\,\Delta t). The factor (1+Φ/c²) is the source of the extra delay.
Assuming that any “delay” automatically means the vacuum isn’t empty.	The delay is caused by the curvature of space‑time, not by a medium with refractive index > 1.	Recall that in GR the “vacuum” can still have a curved geometry; the speed of light in that geometry remains c locally.
Treating the Shapiro delay as a test of the speed of light rather than of the metric.	The experiment is a null‑geodesic test of the Schwarzschild part of the metric (the (g_{tt}) component).	Identify what the measurable quantity is: the round‑trip phase of a radio carrier, which depends on the integrated metric coefficient, not on c itself.

Original question: Shapiro time delay vs cosmic expansion as sources of interstellar redshift? on Physics Stack Exchange, licensed CC BY-SA.

1. What the student is really asking

2. Step‑by‑step explanation