The delay that is not a bend

The deflection of starlight past the Sun was measured in 1919 and made general relativity famous. A second effect of the same metric, on the same rays, went unnoticed for forty-five years: light passing a mass takes longer to get there than it would if the mass were absent, and the extra time is measurable with a radar set.

Irwin Shapiro proposed the measurement in 1964, calling it “a fourth test of general relativity”, and it was made with radar echoes from Venus and Mercury within a few years. It is now the best-measured of the classical tests by a wide margin.

Where the delay comes from #

The Schwarzschild metric outside a spherical mass has two departures from flatness that matter at leading order — one in the time part and one in the radial part — and both are the same small number:

$$\varepsilon = \frac{2GM}{rc^2}.$$

Where the delay comes from is the metric’s time factor, and it is the same one that makes a clock lower down run slow. At the Sun’s surface it departs from unity by about four parts in a million — small enough that nothing about the Sun looks unusual, and large enough that a radar signal grazing it arrives a couple of hundred microseconds late over a round trip to Venus.

For a distant observer using their own coordinates, light travelling radially covers coordinate distance at a rate

$$\frac{\mathrm{d}r}{\mathrm{d}t} = c\left(1 - \frac{2GM}{rc^2}\right),$$

so a coordinate journey takes longer near the mass than far from it. The word “so” there is doing something that deserves care, and the next section is about it. Taking the arithmetic at face value first, the extra time accumulated along a path with closest approach $b$ is

$$\Delta t = \frac{4GM}{c^3}\ln\frac{4r_1r_2}{b^2}$$

for a round trip with both endpoints far outside $b$. For Earth and Venus at superior conjunction, with the path grazing the Sun, that is 233 microseconds.

That the same component produces both effects is worth stating rather than assuming. A clock lower down runs slow relative to a distant one, and a signal passing lower down takes longer as measured by that distant one — these are the same statement about the same metric component, read once as a rate and once as a time of flight.

Two features of that formula are worth dwelling on. The impact parameter appears only inside a logarithm, and squared, so halving $b$ adds $4GM\ln 4/c^3$ to the delay — about 27 microseconds — regardless of what $b$ was. Getting from ten solar radii to one adds the same amount as getting from a hundred to ten. The endpoints appear inside the same logarithm, so moving the reflector from Venus to Jupiter adds another fixed amount rather than scaling anything, and the delay is remarkably insensitive to the whole geometry — which is exactly what makes it measurable over months as a planet’s alignment changes.

The other feature is the prefactor. $4GM/c^3$ for the Sun is 19.7 microseconds, and everything else in the expression is a logarithm of order ten. So the delay is always tens to hundreds of microseconds for any solar-system geometry, which is a comfortable interval to measure and an uncomfortable one to explain away: it is far larger than any plausible clock error and far smaller than anything that could be confused with a mis-modelled orbit.

The speed that is not a speed #

A coordinate speed less than $c$ invites the statement that gravity slows light down, and that statement is about a coordinate system rather than about light.

Any observer who measures the speed of light passing them, with their own clock and their own ruler, gets $c$ exactly, wherever they are. What varies is the relation between their clock and a distant one, and between their ruler and a distant one’s coordinate labels. The delay is the accumulated mismatch, and calling it a slowing is a description of a bookkeeping convention.

The same care applies to a familiar analogy. Writing the metric’s effect as an effective refractive index

$$n(r) \approx 1 + \frac{2GM}{rc^2}$$

reproduces both the delay and the deflection correctly to first order, and it is a genuinely useful device.

A medium whose refractive index varies bends light and delays it together, which is exactly what the effective index of a gravitational field does. The analogy is close enough to compute with: air near a hot road has a lower index below than above and its rays curve, and the curving and the delay are two consequences of one profile — the same statement Fermat’s principle makes about a path with no boundary in it. Gravity does the same with a profile fixed by mass instead of by temperature.

Fermat’s principle sharpens the analogy into a derivation. The path a signal takes is the one whose travel time is stationary under small variations, and the gravitational deflection follows from applying that to the effective index — so the bending is not a separate phenomenon from the delay but a consequence of it. The ray goes where it goes because of how the delay varies across the beam.

There is a limit in which the analogy fails outright and it is worth naming, because it is the same limit in which the delay’s interpretation becomes unambiguous. A refractive medium delays light because the light interacts with matter, so a stronger effect means more interaction and, in every real medium, some absorption and some dispersion. The gravitational delay has neither. Nothing is absorbed, and a gamma ray and a radio wave are delayed by exactly the same amount — which is what the achromaticity later in this essay rests on, and what no material medium can imitate.

Why it is a different test #

The deflection and the delay are not two consequences of one number. They weigh the metric’s two departures differently.

The delay, to leading order, comes from the time term alone — the one a tower of clocks measures directly. So a theory that got the time curvature right and the space curvature wrong would predict the right delay and the wrong deflection; a theory that got their sum right and the split wrong would predict the right deflection and the wrong delay. The standard way of comparing theories introduces a parameter for the ratio, which is exactly 1 in general relativity, and the sharpest limit on it comes from a delay measurement rather than a deflection one: the Cassini spacecraft’s radio link, in 2002, constrained it to $1 \pm 2.3\times10^{-5}$.

There is a second reason the delay is the sharper instrument, and it is about what has to be known independently. An astrometric measurement of the deflection needs the star’s undeflected position, which cannot be observed while the Sun is in the way, so it is inferred from a photograph taken six months earlier through a different part of the instrument at a different temperature. A timing measurement needs the spacecraft’s orbit, which is known from the same tracking data being used to make the measurement, and the delay’s signature in time — a sharp peak at conjunction on top of a smooth orbital variation — is separable from an orbital error in a way that an astrometric offset is not.

It is also worth being clear that the split between “time curvature” and “space curvature” is itself coordinate-dependent, and that the parameter constrained by these measurements is defined within a particular family of coordinate choices. What is not coordinate-dependent is the set of predictions — the deflection angle a telescope measures and the echo delay a clock measures — and the parameter is a way of labelling theories by what they predict for those two, not a physical quantity in its own right. The value of that labelling is entirely practical: it turns “is the theory right?” into two numbers that different experiments constrain differently.

The competing delay #

The Sun is not surrounded by vacuum. Its corona is a plasma, and a plasma has a refractive index below one that depends on the frequency, so it delays a radio signal too.

The two are separated by the one thing they do not share. The gravitational delay is achromatic: it is the same at every frequency, because the metric knows nothing about the colour of what is travelling through it — where the plasma’s own dispersion is the whole of how it is detected. The plasma delay goes as the inverse square of the frequency. So a link operating at two or three well-separated frequencies can measure the plasma contribution and subtract it, leaving the gravitational one — which is why the Cassini measurement was made with three radio bands rather than one, and why its precision exceeded the previous best by two orders of magnitude.

That contrast is worth stating as a general point about instruments. A systematic error with a different functional dependence from the signal is not an error at all once the dependence is measured, and the whole design of the experiment was an exercise in arranging for the two to depend differently on something that could be varied.

Taken to the regime where it stops being a correction, the same delay becomes the whole story. Signals from a clock falling towards a horizon arrive later and later at a distant receiver, without limit, so the infall is never seen to complete. That is this essay’s effect with the small parameter no longer small, and it is worth ending on because it makes clear that the delay is geometric rather than a property of any medium.

Weighing a neutron star with a delay #

The delay’s most productive use is not in the solar system at all. It is the only method that measures the mass of a neutron star without assuming anything about how neutron stars are built.

Take a pulsar in a binary whose orbit is seen nearly edge-on. Once per orbit the pulsar passes behind its companion, and its pulses have to cross the companion’s gravitational field on the way out — so they arrive late, by exactly the expression this essay has been computing, with the impact parameter sweeping through a minimum as the pulsar goes round.

The shape of that delay against orbital phase carries two numbers. Its amplitude is proportional to the companion’s mass, through the same $GM/c^3$ prefactor; its sharpness — how narrow the peak is around conjunction — depends on how close the line of sight comes to the companion, which is the orbit’s inclination. So one curve, fitted to pulse arrival times over many orbits, returns both the companion’s mass and the geometry.

That is a remarkable thing to be able to do. Every other route to a stellar mass involves a model: a spectrum interpreted through an atmosphere, a light curve interpreted through a stellar structure, a velocity interpreted through an assumed inclination. The Shapiro delay involves only the metric, and the metric is what is being tested elsewhere in this essay rather than assumed here.

The results have settled a question that could not be settled any other way. Neutron stars measured this way come out at around two solar masses in the heaviest cases, and that number is a hard constraint on what matter does at nuclear density: any proposed equation of state that cannot support two solar masses against collapse is excluded, by a timing measurement made with a radio telescope.

The correction every navigator applies #

Long before it was a test of anything, the delay became a term in a piece of routine software, and it is worth recording because it marks the transition from a result to a tool.

Every spacecraft’s position is determined from the round-trip time of a radio signal, and every such signal passes through the Sun’s field — not grazing it, usually, but through it. The delay is tens of microseconds for a distant spacecraft even well away from conjunction, which is kilometres of apparent range. A navigation solution that omitted it would place the spacecraft in the wrong place by more than its own trajectory uncertainty.

The same term appears in the analysis of very long baseline interferometry, where the arrival times of a radio wavefront at two telescopes are differenced to microscopic precision. The two paths pass through slightly different parts of the Sun’s field — and the Earth’s, and the planets’ — so the differential delay is a real term in the model, at the level of a hundred picoseconds. Leaving it out would corrupt the reference frame that everything else is measured against.

There is a small irony in that. The measurements that constrain general relativity most tightly are made with instruments whose ordinary operation already assumes it, so the theory is being used to reduce the data that tests it. That is not circular — the test is a residual after the model is fitted, and a wrong theory would leave a signature in the residual with a distinctive dependence on geometry — but it does mean the delay has stopped being an exotic prediction and become part of what “the distance to the spacecraft” means.

Why forty-five years #

The gap between 1915 and 1964 is worth a sentence, because the reason is not conceptual.

Nothing about the calculation is hard. The delay follows from the same metric and the same first-order expansion as the deflection, in a few lines, and anybody who could do one could have done the other. What was missing was any way to look.

Measuring it needs two things that did not exist before the war. It needs a transponder — something at the far end of the path that returns a signal, since a passive reflection from a planet at conjunction is hopelessly faint — or else a radar powerful enough to get an echo from Venus, which took until 1961. And it needs timing at the microsecond level over intervals of many minutes, which needs an oscillator far better than a pre-war laboratory had.

So the effect waited for radar and for atomic frequency standards, and arrived within a few years of both. It is a fair example of a prediction whose delay in being tested says nothing about the theory and everything about what could be built — and of a case where the person who proposed the measurement was proposing an experiment rather than a calculation.

Where the model stops #

Everything is first order in $GM/rc^2$. For the Sun that quantity is $4\times10^{-6}$ at the limb, so the second-order correction is of order a part in a hundred thousand of the delay — a few picoseconds — which is below current measurements and not by an enormous margin.

The Sun is taken as spherical and static. Its rotation adds a frame-dragging contribution that depends on which way round the light passes, and its oblateness adds a quadrupole term. Both are far below the current sensitivity for the Sun and neither is negligible for a pulsar in a tight binary.

The plasma model is a model. The coronal electron density used in the figures is a standard three-term empirical fit, and the real corona is neither spherically symmetric nor steady — it has streamers, it varies through the solar cycle, and a coronal mass ejection crossing the line of sight changes the column density abruptly. That is precisely why the frequency dependence is the thing to exploit rather than the model, and why a measurement that had to trust a coronal model would be far weaker than one that measures and removes it.

And the path is taken as a straight line. The delay integral above is evaluated along the undeflected path, which is consistent to first order, since the deflection is itself first order and its effect on the path length is second. Doing better means solving the null geodesic equation, which is what the analyses of real measurements do.

The observation belongs to somebody else. The radar echoes from Venus and Mercury, the Viking landers’ transponders, the Cassini link and the pulsar timing that now provides the strongest constraints in strong fields are measurements of the sky, and the collections that own the sky treat them as such. What is derived here is the metric’s prediction and the shape of the curve; the error bars are theirs.

One last boundary is worth marking because it is where the whole calculation would have to be redone. Every expression here treats the mass as a fixed background and the light as a test signal that does not affect it, which is exact to any precision anybody will reach in the solar system. It stops being adequate when the two masses in a system are comparable and both are moving fast, which is the case for a binary pulsar — and there the delay is not a small correction to be measured but one of several relativistic terms that have to be fitted together, each with its own dependence on the orbital phase.

What the pictures cannot show #

The delay figure draws microseconds against solar radii, and both the microseconds and the geometry are far too small and far too large respectively to be drawn together. A picture with the Sun, the two planets and the path to scale would be a page of white with two dots on it, and the deflection at the limb would be a thousandth of the width of the line drawing the ray.

The accumulation figure is drawn on a logarithmic axis because on a linear one it would be a step at the origin, and that choice of axis is what makes its content visible. The content is real and the visibility is a decision: half of the delay comes from beyond twenty-two solar radii, and no linear plot of the same function would let anybody see it.

Where the ladder goes next #

This ladder began with the bend Newton got half right, where the factor of two between the two predictions is the whole content. This rung reads the same metric as a time of flight rather than as an angle. The rungs above it: the deflection near a compact object, where the weak-field formula fails and photon orbits exist; the delay in a binary pulsar, where the companion’s mass is measured from it directly; gravitational lensing time delays between multiple images of one source, which measure the expansion rate of the universe and are somebody else’s observation; and the strong-field regime, where the logarithm becomes a divergence.

The habit worth carrying away is about how an effect is distributed along a path. A logarithm has no scale, so an effect governed by one is not localised anywhere, and the instinct to look for where a phenomenon “happens” fails here completely. Half the delay comes from a region where the field is weak enough to be neglected in any local argument, and the total depends on where the experiment’s endpoints are — which is an unusual property for a measurement and the reason the delay’s geometry has to be modelled rather than approximated.