 ## Against Gamma: The Trouble with Time Dilation

For people new to special relativity, and people trying to understand it from simple "lay" explanations, "time dilation" can be a major stumbling block.  As generally "explained", time dilation means time slows down for objects which are moving quickly.

But this explanation is somewhere between misleading and completely false.

Consider two spaceships.  On Ship A, an astronaut watches Ship B zoom by at 0.866 C, and observes that Ship B's clocks are running slow by a factor of 2.  That's the so-called "time dilation".  Unfortunately, B sees the same thing happening with Ship A's clocks.  What's even worse, an observer floating in space, moving at the average of the speeds of the two ships, will assure us that the clocks on Ship A and Ship B are ticking at exactly the same rate.

So, what's "really" going on?

First, let's look at  γ.  It's a useful value, and for simple cases it is typically defined as: We can then very simplistically say, "γ is the time dilation factor!", and in fact that's just what is usually said.

But there's a big problem with this.  Look at the part of the Lorentz transform which gives us the time coordinate in a spaceship moving at velocity v along the X axis: Do you see the problem?  It's that second term.  Look at the derivative of τ with respect to t: This is "time dilation": 1/γ < 1.  It is the derivative of τ along the path the spaceship is following.
BUT! This is time contraction.  It's the derivative in the direction of time rather than in the direction the spaceship is traveling.  It has a physical meaning, too:  If a single observer in A's frame watches a series of clocks go by, where each of them is moving at velocity v and they're all synchronized to B's clock in B's frame of reference, then that observer will see the time shown on the B clocks advancing more quickly than time in A's frame.  Turn it around, look at it from the point of view of the line of clocks in B's frame which are passing the A observer, and it's just "time dilation" of a single observer in A's frame as viewed from B's frame.

Confused?  If you're not, perhaps you should read that again!

In other words, in the spaceship's frame, time is not "going slower" than time in the stationary observer's frame in any global sense.  Rather, time along the path the spaceship is following seems to go more slowly.

Or we could say, along the path Ship B is following in A's frame of reference, less time elapses for B than for A.

Or we could say,  Ship B is taking a shortcut through A's frame of reference.

And at the same time, Ship A is taking a shortcut through B's frame of reference.

Perhaps that seems confusing, and it's hard to quantify it.  But it's certainly closer to "the truth" than the bland assertion that B's clock is running slower than A's!

### An Example

Consider a revolving clock, moving on a circular path at v.

Viewed from a point in the center of the circle, the clock is moving continuously, and it "runs slow".  Geometrically, from the point of view of an observer in the center of the circle, it's just like the case of Ship B passing ship A moving linearly.

But when we turn it around, and view the stationary observer in the middle from the moving clock, we see the stationary observer's clock running fast!  Why is that?

Consider carefully the coordinates of the moving clock.  If we want to keep the "apparent motion" of the observer in the center of the circle fixed so that it moves along the X axis, then the moving coordinate system must be rotating..  Furthermore, at the point in the middle of the circle, the coordinates are rotating at exactly the right rate to keep the the stationary observer from moving either way along the X axis.

In other words, in the rotating coordinates which keep the instantaneous motion of the clock in the middle lined up with the X axis of the revolving clock, that clock does not move through space.  Since it's not moving, it's also not taking the "shortcut" the moving ships could take, above, and instead, we just see the bare "time contraction" term:  The stationary clock goes faster, from either point of view.

This is an incomplete description (you could turn it around and claim the revolving clock must be time contracted, too) but I think it's a step in the right direction.  I will say a bit more about it when we consider the revolving clock in detail.

### Another View:  "Time Dilation" as a Consequence of the Metric

Everything I've said up to now has been just an intuitive description.  All intuitive descriptions of spacetime behavior are incomplete, and may be misleading.  However, there are also some factual statements we can make about the behavior of the metric which may help with understanding "time dilation".

The distance between two points -- or two events -- is determined by the "metric".  In the ordinary Cartesian space of our everyday experience the metric is given by Pythagoras's theorem: All points located a particular distance from the origin form a circle.  With the metric of special relativity, on the other hand, a set of points which are equidistant from the origin forms a hyperbola.  And that difference is the source of the "time dilation" effect.

### The Arithmetic of the Metric

With 1 space dimension and 1 time dimension, in an inertial frame, the elapsed time experienced by an object that moves along a straight line between two events is Figure 1: For two events at the same location in space, that's just Δt. But if the space coordinate changes, then the proper distance must decrease.  The proper distance from the origin to a point on the T axis is just T -- the coordinate time.  However, the farther an event is from the T axis, the smaller is the proper distance from the origin to the event, because the Δx2 term is subtracted from the square of the T distance in the metric.

This is completely contrary to our everyday experience.  In normal "Cartesian" space, the length of the hypotenuse of a right triangle at least as long as each of the other sides.  Furthermore, the distance between two points in our everyday world can never be zero or negative, while it can be in SR 4-space.  We summarize this by saying the metric in Cartesian space is positive definite. The metric in relativity is not.  Attempts at applying everyday intuition to relativistic situations tend to result in frustration.  Because, in special relativity:

The farther away an event is physically, the closer it is in proper time.

The faster you go, the less (proper) distance you cover.

### A Hyperbolic View of What's Happening

A set of points that are equidistant from the origin using the SR metric forms a hyperbola.

Figure 2: A point which lies on the T axis represents an event in the rest frame of the origin.  It's in the same spacial location as the origin.  The set of points which are equidistant from the origin with such a point forms a hyperbola which bends "up" -- toward higher T values -- as it moves away from the T axis (see Figure 2).  All points on the green hyperbola are the same proper distance, ΔT, from the origin.  All points on the yellow hyperbola are also equidistant from the origin; their distance appears to be roughly (3/4)ΔT on the diagram.  All points on the red hyperbola are at  a proper distance of roughly (1/2)ΔT from the origin.

A "future" event which is distant in space from the T axis lies on a hyperbola which intersects the T axis at a lower T value.  Such an event is no "farther away" from the origin than a point on the T axis which has a smaller time coordinate.

So by moving away from the T axis, an object moves onto a different (and "closer") hyperbola of equidistant points.  Objects which are moving rapidly away from the T axis move to "closer" hyperbolas at the same time they are moving to larger T values; hence, they don't get very far from the origin very fast.  If an object moving to the right in Figure 2 moves ΔX spatially while ΔT (coordinate) time passed, we can see from the diagram that only about (1/2)ΔT amount of proper time would pass for the object.

The Lorentz transform performs a hyperbolic rotation, which rotates the points through which a moving object passes along their hyperbolas onto the T axis.  For most of us, this is an unfamiliar operation -- we're used to circular rotations, and we're used to a metric where all points on a circle are equidistant from the origin.

### Confusion Alert:  Why does motion toward the T axis cause time dilation?

The "proper time", or subjective elapsed time, which is experienced by an object must be measured between two events experienced by the object.  In the above discussion, I tacitly assumed the object we were discussing had started at the origin.

The time which passes for you must be measured from a moment (and location) where you are, up to another moment (and location) where you are.  This is simple; even obvious.  But the consequence of this simple observation may be confusing.

To find the elapsed time experienced by an object which starts far from the T axis and moves toward the axis, we need to measure the proper time between the object's starting point and its ending point -- and its starting point wasn't at the origin.  So, to find its (proper, subjective) elapsed time, we need to draw a line from its starting point -- off to one side -- to its ending point, and find the length of that line.   And what we'll find if we go through the exercise is that curves of equal distance from its starting point form hyperbolas centered on its starting point.

So the result is just the same as if we moved the origin to the point where the object started -- and in fact that's what we generally do, if it's at all possible, because it simplifies the math without affecting the outcome.

Page created in 2004.  Minor update to diagram, 11/14/06