# Special relativity: local frames versus global frame in a loop scenario

1. May 29, 2010

### Cleonis

Take the following setup:
A series of pulses of radio signal is relayed around the world, along the equator. There is no "gap", it is a continuous loop along numerous relay stations build along the equator. The total number of pulses is fixed at 648,000 - I'll explain in a minute why that number. The spacing between the pulses is continuously monitored and if necessary adjusted, so that there is an even spacing.

To keep the numbers relatively simple I take a day of 86400 seconds (a solar day), I take the Earth circumference as 40,000,000 kilometer, and I take the speed of electromagnetic waves as 300,000 km/s

An electromagnetic wave travels around the equator in 0.1333333... seconds
During that 0.133333... second time interval the relay stations on the equator have traveled 61.7 meter, as they are co-rotating with the Earth. That 61.7 meter distance is 1/648,000th of the Earth's circumference.

Let two series of pulses be circumnavigating: one in the direction of the Earth's rotation, the other counter to it. If the Earth would be non-rotating then the two counterpropagating series will have the same spacing.
Since the Earth is rotating the co-propagating series of pulses will have a somewhat wider spacing 684,001/684,000 larger than in the non-rotating case. Similarly, the counter-propagating series will be spaced 683,999/684,000 smaller.

The above effect is an instance of the Sagnac effect. In particular, the above case is analogous to a Ring Laser Interferometer setup. The reason that the Sagnac effect occurs is the fact that electromagnetic waves propagate at a particular velocity. In a Sagnac setup the speed of light serves as a reference.

If you evaluate the loop as a whole then clearly the relay stations have a velocity relative to the reference created by the counterpropagating EM-waves. That's because the setup involves a closed loop.

Here is what I think is an interesting question:
If you evaluate just a section of the loop (in other words, if measurements are confined to a local frame) then what propagation speed of EM-waves relative to the relay stations will you find?

- If you confine measurements to a local frame, then the clocks in that frame must be synchronized using emissions from within that local frame only. Given that clock synchronization you will find a lightspeed of c in both directions.
- The global frame, evaluating the loop as a whole, does not have the same clock synchronization as the local frame. For the global frame a global time applies. If you use that global clock synchronization you find that light does not have velocity c relative to the relay stations.

This is why I called this thread 'Local frames versus global frame'.
If the setup involves a loop, and information flowing along the loop is evaluated, then global clock synchronization is different from clock synchronization along sub-sections.

2. May 29, 2010

### Passionflower

Can I ask you is there some kind of point you wish to make?

Remember that speed of light is locally always c.

3. May 29, 2010

### Cleonis

More generally, as long as the string of relay stations does not close a loop then the synchronization procedure that is used makes the speed of light come out as c over the entire length.

When the loop is closed, that is, when information travels in a loop topology, then the Einstein synchronization procedure does not apply.

Alternatively, you can disseminate time by travelling with clocks in opposite directions along the equator.

Generally, the weirdness of special relativity comes into focus when a loop is closed. The twin scenario is a loop scenario. At the instant that the siblings who have travelled along different worldlines exchange information again the relativistic effect becomes observable.

4. May 29, 2010

### starthaus

"Closing the loop" allows for:

Mathenatically: the two integrals calculating proper time to have the same limits.
Physically: the two twins get to be side by side when they split (so they can synchronize their clocks) and be again together at the end of the journey (such they can compare their clocks, exactly as in the Haefele-Keating experiment).