The orbit that ages less than a throw

The longest way round is the shortest clock established the principle that replaces Newton’s first law in relativity: of the routes between two events, the one a freely falling clock takes is the one along which its own time is stationary, and in flat spacetime that stationary route carries the most time of all. A thrown ball follows a parabola because the parabola has the most time on it.

The principle has a qualifier that is easy to read past, and the cleanest way to see what it costs is to find two freely falling clocks that disagree. They are not hard to find. Put one clock in a circular orbit around the Earth. Throw another straight up from the same point at the same moment, fast enough that it falls back to that point exactly one orbital period later, when the orbiting clock comes round again. Both clocks are in free fall from the first event to the second. Both follow paths along which proper time is stationary. They cannot both carry the most time, and they do not.

Two free falls between the same two events #

The meeting point is taken two Earth radii from the centre — an altitude of 6,371 kilometres, well clear of the atmosphere — and the Earth is treated as a non-rotating point mass for everything that happens outside it. At that radius a circular orbit takes 3.98 hours. A clock thrown straight up has to leave at 5.87 kilometres per second to climb to 4.46 Earth radii and fall back in exactly that time.

The two eccentric orbits are there for a reason that becomes important later. The period of a Kepler orbit depends only on its semi-major axis, so every orbit with a semi-major axis of two Earth radii takes 3.98 hours whatever its shape. Any such orbit that passes through the starting point returns to it one period later. The circular orbit is one member of a whole family of free falls that leave and rejoin at the same two events.

The throw is not in that family. It has a larger semi-major axis and would take longer than 3.98 hours to complete a full radial orbit, but it never completes one: it reaches the same point on the way down after exactly the right time, which is a coincidence arranged by choosing its speed.

The rate of a clock against a distant reference, to the precision that matters near the Earth, is $1 + \Phi/c^2 - v^2/2c^2$, with $\Phi = -GM/r$ the gravitational potential and $v$ the clock’s speed. The first correction is the clock that runs slow lower down; the second is the clock that has to slow when it moves. Integrating both along each path gives what each clock loses over the trip, and the figure prints the totals.

The orbiting clock loses 7.47 microseconds. The thrown clock loses 3.37. When the two meet again the thrown clock is 4.10 microseconds ahead, which is a fractional difference of about three parts in ten thousand million over the trip and would be unambiguous to any modern atomic clock. And a third clock, held at the starting radius by a rocket firing continuously — the one clock of the three that is not in free fall — loses 4.98 microseconds, which puts it ahead of the orbit as well.

That last comparison is the one that seems to contradict the principle outright. The held clock accelerates the whole time. The orbiting clock accelerates not at all, in the only sense of acceleration a clock can feel. If free fall carried the most time, the orbit would beat the rocket, and it loses to it by 2.49 microseconds.

Where along each path the time goes #

The totals hide a great deal of structure, and following each clock’s loss as it accumulates shows where the difference is made.

The circular orbit and the held clock both lose at constant rates, because nothing about either changes along the way. The orbit’s rate is exactly one and a half times the held clock’s: it sits at the same depth in the potential, paying the same toll for that, and it also moves, paying a second toll for speed on top. A circular orbit’s speed is set by its radius, $v^2 = GM/r$, so the speed toll is exactly half the depth toll and the orbit’s total is three halves of the held clock’s.

The thrown clock does something neither can. It leaves at 5.87 kilometres per second from the same depth, so for its first few minutes it loses time faster than either. But it climbs, and as it climbs both tolls fall together: it is higher in the potential and it is slowing down. Near the top of its path, at more than twice the starting radius and almost stationary, its clock runs very nearly at the distant clock’s rate. It spends a long time up there — a thrown object is slow at the top of its arc, and the top of this arc is more than two Earth radii above the orbit — and that time is almost free.

The eccentric orbit is the curve worth staring at. It dives to perigee and pays heavily there, then swings out and pays little, and its total lands exactly on the circular orbit’s.

That every orbit of the same period pays the same is not a coincidence of the two eccentricities drawn; it is checked by quadrature along each orbit against a closed form, and it holds for every shape. The reason is a theorem about Kepler orbits: the average of $1/r$ over a full period is exactly $1/a$, the reciprocal of the semi-major axis, whatever the eccentricity. The depth toll is the average of $GM/r$, so it depends only on $a$; and by the conservation of energy the speed toll is the average of $GM/r - GM/2a$, so it does too. The factor of two between them is the orbital form of the virial theorem, which fixes the ratio of kinetic to potential energy for any bound inverse-square motion averaged over time.

Stationary is not the same as greatest #

The resolution of the apparent contradiction is in the eccentric orbits, and it has a name borrowed from optics.

A family of free paths that leave one event and come back together at another, all carrying exactly the same proper time, is the spacetime version of a bundle of rays that leave a point and reconverge at a focus. Oscillators launched together from the bottom of a well did the same thing at half a period, and there the action as a function of the endpoint stopped having a value. Here the orbits of one period all rejoin at the return event, and the return event is what geometry calls a conjugate point of the start, along each of them.

What a conjugate point does to an extremal principle is well understood, and it is the same thing in mechanics, in optics and in spacetime. A path that has not yet passed a focus is a genuine local extremum. A path that reaches or passes one is stationary and no longer extremal. Least action is not least past the first focus of a mechanical path; a ray reflected past the focus of a mirror takes the longest time rather than the shortest; and a timelike geodesic in spacetime carries the most proper time among its neighbours only up to its first conjugate point. The circular orbit reaches its conjugate point exactly at the return event. It is stationary there, and it is not a maximum — the equal times along its eccentric neighbours are the degenerate direction in which it has stopped being one.

That is the qualifier in the principle, and it is the whole of the resolution. The free path is where proper time is stationary. In flat spacetime, and in a gravitational field over short enough times, free paths have no conjugate points and “stationary” means “greatest”. An orbit is not short: a free path that goes all the way round a planet has refocused by the time it gets back.

The held clock and the thrown clock need a separate sentence, because neither is a small deformation of the orbit and a statement about neighbours says nothing about them. They show that the orbit is not the global maximum either, which is a stronger statement than a conjugate point alone makes. The thrown clock carries more time than any other path drawn here, and among the free paths that join these two events it has no competitor: a free path from the starting point back to itself must be either a Kepler orbit that returns in that time or a radial fall that reverses at the right moment, which is this throw — and, as the enumeration below shows, none of the orbits can compete with it.

The comparison has been published as a paradox, under the heading that the accelerated twin comes back older, and it is worth being clear that nothing in it is paradoxical. The twin who comes back younger came back younger because the travelling route through spacetime was a detour from the free one. The orbiting twin is younger for the same reason, and the free route that should have been compared with is not the orbit but the throw.

The same ordering without approximation #

Everything so far has used the weak-field rate, correct to first order in $GM/rc^2$, which near the Earth is less than a part in a thousand million and leaves nothing to worry about. A reasonable suspicion is that an effect of a few microseconds might be an artefact of truncating an expansion, or might reverse somewhere the field is strong.

The figure repeats the comparison around a black hole, with no expansion of any kind: the circular orbit’s rate is $\sqrt{1 - 3GM/rc^2}$, the held clock’s is $\sqrt{1 - 2GM/rc^2}$, and the thrown clock’s time comes from integrating the radial geodesic equations, with its launch chosen so that it returns after exactly one orbital period of coordinate time. The radius runs from 60 down to 6.5 in units of $GM/c^2$ — just outside the innermost orbit that can be stable at all.

Far out, the three losses multiplied by the radius level off at 3/2, 1 and 0.68. The first two are the weak-field ratios above, and the third matches the Earth calculation’s 0.68, which is an independent check between a velocity-Verlet integration in Newtonian gravity and a quadrature of the Schwarzschild geodesic. Close in, all three curves rise and separate, but they never cross. At 6.5 the orbiting clock keeps 73.4 per cent of the distant clock’s time, the held clock 83.2 per cent and the thrown clock 88.4 per cent. The ordering is a property of the geometry and not of the expansion used to compute it.

What the calculation takes for granted #

The Earth does not rotate. A rotating planet drags the local inertial frames around with it and gives prograde and retrograde orbits slightly different rates, an effect many orders of magnitude smaller than the ones computed here near the Earth. It changes none of the microseconds as printed and none of the ordering. The two clocks that flew in opposite directions is the case where the rotation matters, because there the reference is the ground.

The Earth is a point mass. Its real field has a quadrupole part from its oblateness, which makes orbits precess and changes their periods by parts in a thousand at low altitude. At two Earth radii, and for the purpose of comparing clocks that share the same field, it shifts every path together.

The eccentric orbits are idealised. The orbit of eccentricity 0.4 comes within 1.2 Earth radii of the centre, above the surface but inside the region where the atmosphere and the oblateness would both disturb it. The equal-time property belongs to exact Kepler orbits, and a real one would share it only approximately.

Proper time is assumed to be what the clock reads. Every number here is the integral of the metric along a path, and a real clock reads that only if its rate depends on nothing but where it is and how fast it moves — the clock hypothesis, which the held clock, accelerating at a quarter of a gravity for four hours, leans on more than the others.

Worldlines that meet again in four dimensions #

The figures draw radius against time and paths in space, and neither is a picture of spacetime. The claim that the orbits reconverge at a focus is a statement about worldlines in four dimensions, and the drawing that would show it — a helix around a time axis, with its eccentric neighbours spiralling round it and pinching back onto it after one turn — cannot be drawn with the orbital distances and the light-hours of the trip on the same scale. The focus is argued from the equal times rather than seen.

The strong-field figure also quietly changes the question. “One period later” is measured in the distant observer’s coordinate time, because that is the only clock the three share before they meet; near a black hole the three clocks disagree by a quarter about how long the trip took, and a comparison phrased in any one of their own times would put the three curves in different places. The ordering at the meeting is frame-independent. The horizontal axis is not.

The most any path can carry #

The held clock beats the orbit and loses to the throw, and it is natural to ask whether some cleverer powered trip — a rocket that climbs higher than the throw and returns faster — could beat the throw in turn. That question has a definite answer, and it comes from a theorem rather than from trying trips.

In a spacetime without pathologies — one in which every event has a well-defined past and future and nothing loops back on itself, which the region outside a planet or a black hole is — any two events that a clock can connect are connected by a path of greatest proper time, and that path is always a free fall. It is a result of the global geometry of spacetimes, due to Avez and Seifert in the 1960s. It turns the search for the most time into an enumeration: find every free path between the two events, and the longest of them is the longest of all paths.

Between these two events the free paths are few. A free path from the starting point back to itself after 3.98 hours is either a Kepler orbit that completes a whole number of revolutions in that time, or a radial fall that reverses and returns. Orbits of one revolution all lose 7.47 microseconds. Orbits of two or more revolutions have smaller semi-major axes, sit deeper and move faster, and lose more. The radial fall that returns in time is the throw. So the throw carries the most proper time of any path between the two events, powered or not, and no rocket trip can beat it.

That makes the held clock’s position between the two a clean statement rather than an accident. It is not free, so it cannot be the maximum; it is not the orbit, so it is not bound by the orbit’s refocusing; and it lands between them.

A thrown clock is not only a thought experiment. In 1976 a hydrogen maser was launched almost vertically on a rocket to an altitude of about ten thousand kilometres and fell back into the ocean less than two hours later, its frequency compared with a clock on the ground throughout. The mission measured the gravitational shift of its rate to about a part in ten thousand, and in the language of this essay it was a throw — a clock in free fall along a radial path, gaining time on the ground clock at the top of its flight exactly as the thrown clock here gains on the orbit.

Still open: a single clock on two paths at once #

Every clock here takes one path. Quantum mechanics permits something the geometry was not built for: a single clock — an atom with an internal ticking state — put into a superposition of two paths through spacetime that accumulate different proper times, then recombined. If proper time is what the clock’s internal state records, the two branches arrive with different readings, and the difference is information about which path was taken. The interference between the branches should then fade in proportion to how distinguishable the two readings are, even though nothing else distinguishes the paths.

That prediction was set out in detail in 2011, and it is a test of the connection between proper time and quantum states that no experiment on classical clocks can make. Atom interferometers have since measured gravitational phase differences between paths separated in height, but a loss of interference caused by a proper-time difference recorded in the atom’s own internal clock requires a time difference large compared with the clock’s period, and reaching it with atoms held in superposition for long enough is beyond present experiments. Whether it appears exactly as predicted is not yet known.

The habit worth carrying away is to ask of any extremal principle whether a focus lies between the ends. Stationary means extremal only until the paths refocus, and they refocus sooner than intuition expects: at half a period for an oscillator, at the focus of a mirror for light, and after one revolution for an orbit. A freely falling clock takes the path that is stationary, and the orbit is the clearest case where that path is not the one with the most time on it.