Against Gamma:The Trouble with Time Dilation |

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

So, what's "really" going on?

First, let's look at

We can then very simplistically say, "

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

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

This is

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

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,

And at the same time,

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!

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

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

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

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.

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

For two events at the same location

The farther away an event isphysically, the closer it is inproper time.

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

A "future" event which is distant

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

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

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

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.