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Original by Michael Weiss.

# The Twin Paradox: The Equivalence Principle Analysis

## Some background

Although most scenarios in Special Relativity are most easily described using inertial frames, there is no reason why these frames absolutely must be used.  The Equivalence Principle analysis of the twin paradox simply views the scenario from the frame in which Stella is at rest the whole time.  This is not an inertial frame; it's accelerated, so the mathematics is harder.  But it can certainly be done.  When the mathematics is described fully, what results is that we can treat a uniformly accelerated frame as if it were an inertial frame with the addition of a "uniform pseudo gravitational field".  By a "pseudo gravitational field", we mean an apparent field (not a real gravitational field) that acts on all objects proportionately to their mass; by "uniform" we mean that the force felt by each object is independent of its position.  This is the basic content of the Equivalence Principle.

The Equivalence Principle analysis of the twin paradox does not use any real gravity, and so does not use any General Relativity.  (General Relativity is the study of real gravitational fields, not pseudo ones, so it has nothing to say about the twin paradox.)  Nevertheless, what General Relativity does say about real gravitational fields does hold in a restricted sense for pseudo gravitational fields.  The one thing we need here is that time runs slower as you descend into the potential well of a pseudo force field.  We can use that fact to our advantage when analysing the twin paradox.  But it needs to be emphasised that we are not using any actual General Relativity here, and no one ever needs to, to analyse the paradox.  We are simply grabbing a result about real gravitational fields from General Relativity, because we know (from other work) that it does apply to a pseudo gravitational field.

We begin with a couple of assertions that belong in the realm of General Relativity.  (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 "pseudo-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", 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.  This can be rephrased as "Time runs slower as you descend into the potential well of a uniform pseudo-force field."

Older books called our first assertion the General Principle of Relativity, but that term has fallen into disuse.

## 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 can assume that a uniform pseudo-gravitational field suddenly permeates the universe; the field exactly cancels the force of her thrusters, so she stays motionless.

(Of course, the frame in which Stella is always at rest in the scenario we have described is not uniformly accelerated, so the simpler description of a uniform pseudo gravitational field does not quite apply.  But we can consider that description to apply during the period of Stella's turnaround.)

Terence, on the other hand, does not stay motionless in Stella's frame.  The field causes him to accelerate, but he feels nothing new since he's in free fall (or rather, Earth as a whole is).  There's an enormous potential difference between him and Stella: remember, he's light years from Stella, in a pseudo gravitational field!  Stella is far "down" in the potential well; Terence is higher up.  It turns out that we can apply the idea of gravitational time dilation here, in which case we conclude that Terence ages years during Stella's turnaround.

Short and sweet, once you have the background!  But remember, this is not an explanation of the twin paradox.  It's simply a description of it in terms of a pseudo gravitational field.  The fact that we can do this results from an analysis of accelerated frames within the context of Special Relativity.

As an added bonus, the Equivalence Principle analysis 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 pseudo 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 pseudo-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 pseudo-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 why such forces carry the prefix "pseudo".

Real (not pseudo) gravitational time dilation (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 pseudo-gravitational fields, and (b) the Principle of Equivalence then implies time dilation for gravitational fields.  A stunning achievement, but irrelevant to the twin paradox.

You may find pseudo gravitational time dilation 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 Analyses.

## 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.  (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 one 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.
The General Principle of Relativity plays a key role in the Equivalence Principle analysis of the twin paradox.  And this principle gave General Relativity its name.  Even in 1916, Einstein continued to single out the General Principle of Relativity as a central feature of the new theory.  (See for example the first three sections of his 1916 paper, "The Foundation of the General Theory of Relativity", or his popular exposition Relativity.)

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 local coordinate systems (also called "charts").  (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 freely falling frame of reference, the remaining second-order effects betray the presence of a true gravitational field.
The third feature finds its precise mathematical expression in the Einstein field equations.  This feature looms so large in the final formulation of GR that most physicists reserve the term "gravitational field" for the fields produced by matter.  The phrases "flat portion of spacetime", and "spacetime without gravitational fields" are synonymous in modern parlance.  "SR" and "flat spacetime" are also synonymous, or nearly so; one can quibble over whether flat spacetime with a non-trivial topology (for example, cylindrical spacetime) counts as SR.  Incidentally, the modern usage appeared quite early.  Eddington's book The Mathematical Theory of Relativity (1922) defines Special Relativity as the theory of flat spacetime.

So modern usage demotes the uniform "gravitational" field back to its old status as a pseudo-field.  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, you certainly have not 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 Analysis is closer to the spirit of GR (vintage 1916) than the Equivalence Principle analysis.  Spacetime, geodesics, and the invariant interval: that's the core of General Relativity.

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