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.(adsbygoogle = window.adsbygoogle || []).push({});

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).

Appreciate any help you can give. If no one can answer, what does that say?

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# A clock tower puzzle in a gravitational field

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