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The Twin Paradox: The "General Relativity" Explanation

Calling this the "General Relativity Explanation" raises some people's hackles; I'll say why below.  That's why "General Relativity" is in quotes, and why "gravitational" is in scare-quotes for much of this entry.

Some background

This explanation relies on a couple of assertions that are true, according to GR.  (We postpone asking what SR has to say about these assertions.)

• Free choice of reference frames: You can describe the physics of a situation in pretty much any reference frame you like, but some frames demand the introduction of force fields that don't show up in other frames.  You can call these "pseudo-force fields", or even "gravitational fields".
• Uniform "gravitational" time dilation: Say you have two identically constructed clocks.  One is deep down in a uniform "gravitational" potential well (or "pseudo-potential" well, if you prefer); the other is higher up.  If the two clocks compare rates by sending light signals back and forth, then both will agree that the lower clock runs slower than the higher clock.

Older books called our first assertion the General Principle of Relativity, but that term has fallen into disuse.  It may remind you of the Principle of Equivalence, but that's really something different.  The short and sloppy version of that says you can't distinguish a gravitational field due to matter from a pseudo-force field.  We won't need the Principle of Equivalence here, so I won't bother with a more careful statement.  Even so, you can probably see the connections: without Equivalence, uniform "gravitational" time dilation has nothing whatever to do with gravity.  (Call it "pseudo-force time dilation" instead.)

OK, now for the twin paradox

Our usual version, that is.  We'll pick a frame of reference in which Stella is at rest the whole time!  When she ignites her thrusters for the Turnaround, she is forced to assume that a uniform "gravitational" field suddenly permeates the universe; the field exactly cancels the force of her thrusters, so she stays motionless.

Not so Terence.  The field causes him to accelerate, but he feels nothing new since he's in free-fall (or rather the Earth as a whole is).  There's an enormous potential difference between him and Stella: remember, he's light-years from Stella, in a uniform "gravitational" field!  Stella's at the bottom of the well, he's at the top (or they would be, if the well weren't bottomless and topless).  So by uniform "gravitational" time dilation, he ages years during Stella's Turnaround.

Short and sweet, once you have the background!  As an added bonus, the "GR" explanation makes short work of Time Gap and Distance Dependence Objections.  The Time Gap Objection invites us to consider the limit of an instantaneous Turnaround.  But in that limit, the "gravitational" field becomes infinitely strong, and so does the time dilation.  So Terence ages years in an instant --- physically unrealistic, but so is instantaneous Turnaround.

The Distance Dependence Objection finds it odd that Terence's Turnaround ageing should depend on how far he is from Stella when it happens, and not just on Stella's measurement of the Turnaround time.  No mystery: uniform "gravitational" time dilation depends on the "gravitational" potential difference, which depends on the distance.

You may be bothered by the Big Coincidence: how come the uniform "gravitational" field happens to spring up just as Stella engages her thrusters?  You might as well ask children on a merry-go-round why centrifugal force suddenly appears when the carnival operator cranks up the engine.  There's a reason such forces have had to endure the derisive prefix "pseudo" in so many books.

You may find uniform "gravitational" time dilation, the second assertion, a mite too convenient.  Where did it come from?  Is it just a fudge factor that Einstein introduced to resolve the twin paradox?  Not at all.  Einstein gave a couple of derivations for it, having nothing to do with the twin paradox.  These arguments don't need the Principle of Equivalence.  I won't repeat Einstein's arguments (chase down some of the references if you're curious), but I do have a bit more to say about this effect in the section titled Too Many Explanations.

Gravitation time dilation without scare quotes (i.e., fields due to matter) is a different story.  These fields are never uniform, and the derivations just mentioned don't work.  The essence of Einstein's first insight into General Relativity was this: (a) you can derive time dilation for uniform "gravitational" fields; (b) the Principle of Equivalence then implies time dilation for gravitational fields (no scare-quotes).  A stunning achievement, but irrelevant to the twin paradox.

What is General Relativity?

Einstein worked on incorporating gravitation into relativity theory from 1907 to 1915; by 1915, General Relativity had assumed pretty much its modern form.  (Oh, the mathematicians found some spots to apply polish and gold-plating, but the conceptual foundations remain the same.)  If you asked him to list the crucial features of General Relativity in 1907, and again in 1915, you'd probably get very different lists.  Certainly modern physicists have a different list from Einstein's 1907 list.

Here's my version of Einstein's 1907 list (without worrying too much about the fine points):

General Principle of Relativity

All motion is relative, not just uniform motion.  You will have to include so-called pseudo-forces, however (like centrifugal force or Coriolis force).

Principle of Equivalence

Gravity is not essentially different from any pseudo-force.

Here's the modern physicist's list (again, not sweating the fine points):

Spacetime Structure

Spacetime is a 4-dimensional Riemannian manifold.  If you want to study it with coordinates, you may use any smooth set of charts (aka local coordinate systems).  (This free choice is what has become of the General Principle of Relativity.)

Principle of Equivalence

The metric of spacetime induces a Minkowski metric on the tangent spaces.  In other words, to a first-order approximation, a small patch of spacetime looks like a small patch of Minkowski spacetime.  Freely falling bodies follow geodesics.

Gravitation = Curvature

A gravitational field due to matter exhibits itself as curvature in spacetime.  In other words, once we subtract off the first-order effects by using a free-falling frame of reference, the remaining second-order effects betray the presence of a (true) gravitational field.

So modern usage demotes the uniform "gravitational" field back to its old status as a pseudo-field, with all the pejorative connotations of the prefix "pseudo".  And the hallmark of a truly GR problem (i.e., not SR) is that spacetime is not flat.  By contrast, the free choice of charts --- the modern form of the General Principle of Relativity --- doesn't pack much of a punch.  You can use curvilinear coordinates in flat spacetime.  (If you use polar coordinates in plane geometry, have you suddenly departed the kingdom of Euclid?)

The usual version of the twin paradox qualifies as a pure SR problem, by modern standards.  Spacetime is ordinary flat Minkowski spacetime.  Stella's frame of reference is just a curvilinear coordinate system.

The Spacetime Diagram Explanation is closer to the spirit of GR (vintage 1916) than the so-called "GR" explanation.  Spacetime, geodesics, and the invariant interval: that's the core of GR.