# A clock tower puzzle in a gravitational field

1. Aug 14, 2014

### rlshuler

In General Relativity spacetime is described by a metric. The most common one is the Schwarzschild metric, valid at radii greater than the surface radius of a mass. If we assume constant angular position so that dθ=dø=0, then this metric relates local (proper) coordinate time and distance dτ and ds to something called Schwarzschild coordinates dr and dt, which may be roughly taken to be coordinates as viewed by a distant observer on ephemeral time. I'll call these ephemeral coordinates.

If we stack meter sticks up on two sides of a planet to make towers of equal height, then all observers will agree on the number of meter sticks simply by counting them. According to a proper observer (traveling along the towers), this is also the combined "proper" height of the towers. For simplicity, assume each one drills down to the center of the planet, or just ignore the planet. Trust me it doesn't make any difference and just messes up the calculation.

The mathematician can calculate the proper height of the towers by solving for and integrating ds. This is easy because the towers are static, i.e. not changing in time, and we can set dt=0, leaving only one term in the metric. The proper distance ∫ds=S will be greater than the ephemeral coordinate distance ∫dr=R by some factor K. This factor K is just the average of the coefficient in the metric over r, as that coefficient is a function of r. Nothing too mysterious.

The ephemeral observer will notice that light rays take longer than would seem necessary to cross the gravitational field. If the time interval is T, then R/T < c. That's OK because this is a non-local measurement of lightspeed, not a proper lightspeed. It is called a "coordinate" velocity. The ephemeral observer notices that T is greater than the expected time by the factor K that was found in the proper distance computation. So we have K(R/T)=(KR)/T=S/T=c and everything is OK. Using ephemeral coordinates, slow light travels the distance R and this produces time dilation. Using proper time and lengths, normal light travels the greater distance S and this produces time dilation. Everyone is in agreement about the objective facts. Who cares if their coordinates are different, right?

Now let's place mirrors at the ends of each meter stick and make a light clock out of it. I assume everyone knows what a light clock is. Photons bounce between mirrors and a timing signal is derived from that. We can even put these clocks in series so that each issues one "tick" after the previous one until they have all ticked. Then if we want to keep ticking, it goes back the other way. Obviously the total time for a complete group of ticks is T in ephemeral coordinates. The tick period for each individual clock is T/S, i.e. divide the total time by the number of clocks or meter sticks. I have conveniently made the number of clocks equal to the number of meter sticks by making them 1 meter long.

Since the ephemeral observer's coordinates measure the total distance as R < S, but the total number of meter sticks (or clocks) is S, then the ephemeral observer concludes the clocks have shrunk and are now only (on average, as the amount varies along the path) S/K in length (since KR=S). This is gravitational length contraction, seen only in the ephemeral reference frame, occasionally discussed but not too often. It is just the ephemeral view of the effects of what is a longer path in the proper view. Again, nothing too mysterious. Just games with coordinates. The objective observables are so far consistent. Everyone agrees on the total number of clocks or sticks.

OK, so the coordinate average length of a light clock is 1/K, since there were S of them at 1 meter each in the proper units. In ephemeral coordinates each is 1/K meters long. (K us usually a number very close to one)

Recall from earlier that the coordinate velocity of light in the ephemeral frame was slightly slow, again by a factor of K. It is exactly c/K.

Now the ephemeral observer computes the ticking rate of each 1 meter light clock. I will call these little ticks t to distinguish them from big ticks through the whole distance T. It is just the coordinate length divided by the coordinate velocity of light, tick = (1/K)÷(c/K) = 1/c.

OK, so everything in the BIG picture checks out in any coordinate system. The total interval T=S/c=KR/c as it should be. The total BIG tick T is longer by K in ephemeral coordinates reflecting time dilation.

Uh oh, something is wrong. The average little tick t=1/c is completely immune to time dilation. An individual clock is ticking always at its nominal rate!

You can re-do this analysis focusing on the little clocks, and get them to stretch. I'm sure I'll get half a dozen answers telling me to do that. I understand. I've done it already. BUT, if you do this, then you can't fit S of them into the big distance. Only R=S/K of them. BUT SINCE THE METER STICKS (LIGHT CLOCKS) are the REFERENCE STANDARD for proper distance, i.e. their count is the proper distance, then you have to conclude S=R in violation of the metric.

Can anyone REALLY explain this? I mean not just dodge the question. This is not an oh I just thought of this today question. I've been trying to figure it out for years, and have posted on other forums without anyone really giving an answer (so far).

2. Aug 14, 2014

### pervect

Staff Emeritus
Not a good sign, you've probably gotten a lot of good explanations and rejected them :(. But to be motivated to answer I'll have to hope that it's different time (I wish us luck!).

I'll focus on what I see as the problem, it's a rather long post.

I would say that coordinates only measure distance without correction when you have a particular sort of metric, a diagonal metric with unit coefficients. In fact, the whole purpose of a metric is to change the observer-dependent notion of coordinates into the (mostly) observer-independent notions of distance.

An aside: distance is invariant here only for the class of static observers, but that's just the class of observers we are interested in.

So the ephemeral observer's coordinates don't measure distance when you just subtract the difference, because you need to take into account the metric.

We conclude that the height of the tower, R, the proper height, an observer independent quantity, is the number of meter sticks it contains. But this proper height is not the difference in coordinates between the top and bottom of the tower, because the coordinates don't directly measure distance.

There is a conflict between the idea that "distance is measured by prototype meter bars" and the idea that "distance is measured by the change in coordinates". All you need to do is resolve the conflict in favor of the former (i.e that distance is measured by the number of prototype meter bars), and regard a change in generalized coordinates as something that can be converted into distance (using the metric), but it is not a distance until you do this conversion.

3. Aug 15, 2014

### rlshuler

Ah... a much simpler formulation and possible explanation?

Suppose the coordinate distance in A's frame from A to B and back is R. A's radius is greater than B's. Then ∫ds=S=KR where K>1. So KR meter sticks can be laid from A to B and both parties will agree.

The coordinate velocity of light through each of A's coordinate meters is reduced by K in A's frame. This can be determined by dividing the coordinate distance R by the transit time in A's frame KR/c giving R/(KR/c)=c/K.

Time dilation for a proper 1 meter light clock is increased by K as viewed by A. This follows from time dilation alone without reference to length changes. But this is also the time dilation for A's coordinate meters, so A's coordinate meters must equal the proper meters.

Oh my ...

I think that the scale of local coordinates is arbitrary if one considers only one path segment, like a meter. The geodesic assumption removes this flexibility by requiring a scale choice that points the curvature tangent in the direction of motion derived from equivalence.

However, requiring that the whole equals the sum of the parts places also another constraint on the scale.

4. Aug 16, 2014

### rlshuler

John Macken on another thread has pointed out the reason for the disagreement in my algebra. The coordinate velocity of light is proportional to K squared, not K.

This seems to be related to the Shapiro delay. It should directly verify curvature, since it is not explainable by time dilation alone. I spent a couple of hours with Cliff Will's book on GR verification, and searched the web, but I cannot find detailed information on the Shapiro experiments. A number was thrown out in one source of 200 microseconds change in delay, but that must be a typo. My own calculations show a range of 90 to 280 nanoseconds, depending on what assumptions are used, which differs by 3 orders of magnitude. Anyone have any information on this?