# Whats the bottom line?

1. Dec 22, 2006

### Bos

So whats the bottom line with the "thrown watch" problem? Which clock is running faster, or are they both going at the same pace and reading the same time? Does it matter how fast the thrower throws the watch in the air or does it have to accelerate at exactly 9.8 m/s2. Does it matter how high it is thrown?

2. Dec 22, 2006

### Chris Hillman

The details do of course depend upon the details of how high you threw the one watch, what spacetime model you are using (we were using the Schwarzschild vacuum solution of the EFE), and your state of motion in this model (we were assuming a static observer in the exterior region), and so on.

3. Dec 23, 2006

### pervect

Staff Emeritus
I suspect that something from the last thread confused you, but I'm not quite sure what. You may need to ask a more specific question.

The bottom line is this - for short trips, the thrown watch will always read the longest time. This is a consequence of the theorem that Stingray mentioned. In these simple cases, there will be one and only one way you can throw a watch and have it return to the surface. This will basically be to throw the watch "up" (ignoring Coriolis forces, if you must include those the direction of the throw will be slightly off from straight up).

For long trips, you can have multiple thrown watches re-unite at the same point in space-time. I.e. you can throw a watch so that it is in low Earth orbit, you can also throw it straight up. In these situations, you have what Stingray called "conjugate points" in https://www.physicsforums.com/showpost.php?p=1193340&postcount=9

When you have conjugate points (multiple ways of throwing a watch) you need to perform a more detailed analysis to answer the question.

In such cases, the direction of the throw is very important. As I mentioned (and I presented a terse calculation to prove it), if you throw a watch so that it enters low Earth orbit at sea level, it will NOT read a longer time interval than a watch that stays stationary on the Earth's surface when the watch returns to its starting point.

I thought the calculation I presented to demonstrate this was pretty self-explanatory, but one does need to know some basics (how to compute the Lorentz interval, and that the Lorentz interval is equal to proper time, the time elapsed on a clock in its own frame) to follow the calculation.

Here is an analogy that might help.

The shortest distance between two points is a straight line. But at least on a curved surface, the reverse is not true - a straight line is not necessarily the shortest distance between two points.

Suppose you are sitting in town Alpha, and want to get to town Bravo. In between is a mountain.

alpha____^^^^MOUNTAIN^^^^____bravo

One straight line path leads right over the mountain. This is a straight line, and it connects the two points (the two towns), but it isn't the shortest distance between them. This can't happen on a flat plane, but it can happen on bumpy (i.e. curved) terrain.

The shortest distance between two points will always be a straight line, but in the case of multiple straight lines between two points (something possible in curved geometries but not on planes) not all of them will be the path of shortest distance. In fact, some straight line paths can really be quite long, as the mountain example shows.

When we find the true shortest path between alpha and bravo, we will find that it is a straight line (or the eqivalent of a straight line on a curved surface, a geodesic). But when there are multiple straight line paths, we have to pick the right one.

The same thing happens in space-time, except that we talk about the longest proper time (i.e. the longest clock reading, not the shortest distance) between two points. In the case where we have multiple paths, just having a straight line path is not a guarantee that we've picked the optimum (in this case, the longest time interval rather than the shortest distance) path.

We can also have situations on a curved surface where a straight line path is longer than some non-straight path. This is somewhat analogous to the low earth orbit (LEO) case I talked about. In our example, some non-straight path that avoids the mountain might still be better than going over the mountain.

Note that while distances are minimized by straight line paths or by geodesic paths on curved surfaces, proper times are maximezed instead of minimized by geodesic paths through space-time.

Last edited: Dec 23, 2006