Detector Watch 2

This is a continuation of the previous thread.

It's time for a talk about time. :-)

Einstein probably said it all when saying "Time is that which is measured by clocks". Here one can get into alot of deep-ish water looking for a firmer definition - that is not a bad pursuit of itself, but it doesn't advance science much in practice. So Einstein tied the concept of time with it's measurement, pretty much like tying the premise of a 'rigid body' ruler to distance. I guess we all have an intuitive idea of time which is hard to capture precisely and consistently, so the science way is not to worry too much about that aspect, but to just create a working definition which covers our main sensibilities and work from there. So if we have a clock design that a priori ( in advance ) seems to be likely to produce a 'regular' signal then we go with that.

Any measurement of some clock's accuracy is only going to be made with reference to another clock, so like other fundamental quantities it is tied to some agreed standard. When I last looked a second was defined to be a certain number of oscillations of a light signal emitted during the transition of electrons from one atomic state to another ( Caesium I think ). There are a number of variants upon this technique - the short answer is that time ( as an international standard ) is based upon averaging from several clocks of that general type - safety in redundancy. That's not any better, or worse, than defining distance to be based upon a dead English king's stride ( Imperial Foot ) or a fraction of the circumference of the Earth ( the metre ).

However for general use, and LIGO in particular, a thingy called GPS time is used. As you probably know, the Global Positioning System relies on very accurate timing. A useful rule of thumb here is that light travels at about one foot per nanosecond - yes that's the stride of a dead English king in one billionth of a second. Nothing at all has been found to travel faster than that. So if you are, say, 60 feet from your neighbour then you are also 60 nanoseconds from them too. This introduces the tricky concept of when do things happen - when I record them or when my neighbour does? For everyday stuff who cares, eh?

Not so for many science purposes however. In general, separated observers ( people with recording instruments ) will disagree about whether events A and B ( stuff that happens ) occurred simultaneously, A before B, B before A, and how much by. Following the detail of such analyses leads into the formulation of the Special Theory of Relativity which is not the real path here. But I will talk of how one might in practice synchronise separated clocks.

A first attempt is to create two identical clocks - same design, materials etc - and tune them in to each other in some way so as they tick together. That is every advance of each clock's counter ( whatever it is ) is never different between the two when they are next to each other. Now leave one ( call it clock A ) at Site A and move the other ( called clock B ) to Site B. So they should be synchronised now, OK? Well not quite ..... as it leaves the possibility that some aspect of transport may have affected say clock B's function. I don't mean it got bumped in transit ( although that may have happened ), I refer to something more fundamental - the nature of the passage of time itself during the trip.

So to eliminate that possibility in our process, we try to signal between the clocks once they are set in their respective positions in order to then effect synchrony. Roughly speaking it goes like this:

- at A fire off a light signal to B: note down the moment on clock A that you did that ( say 12:00 PM exactly ).
- upon arrival at B: immediately respond with a signal back to A which also encodes the time showing on clock B when A's signal did arrive ( say 12:03 PM exactly ).
- when B's signal comes back to A: note the moment that occured ( say 12:04 PM exactly ) and discover whatever that encoded time was at B.

So then, at Site A, I'd have on my timesheet:

Event #1 - Signal left A: 12:00
Event #2 - Signal arrived at B: 12:03
Event #3 - Signal back at A: 12:04

NB. We can factor delays in response if you like, but it doesn't change the heart of the method. Also we can only record events at this stage using clocks placed at the position of those respective events - because we haven't yet achieved/decided how to compare/synchronise separated occurences yet!

So you will look over that sheet and deduce that our separated clocks are saying that it took 3 minutes for a signal to go one way but only 1 minute on the return leg. So inevitably we must conclude that either:

- there is some anisotropy ( directional dependence ) to our signalling

OR

- we need to adjust one or both clocks.

The solution preferred is to signal site B, again, and ask that clock B be wound back by one minute. This is because we conclude that the round trip time is 4 minutes ( as measured by clock A alone ), then by discarding anisotropy ( to date there is no evidence for it ), we assign 2 minutes per separate leg & equal each way. As clock B said 12:03, not 12:02, then that hints at the correct adjustment to it.

You can adjust this method for any other clock readings in similiar fashion. If that is then done we would consider, or define, that clocks A and B are synchronised. If you worry about some 'drift' in agreement over time, then repeat the above whenever you like and re-adjust, possibly fiddle with some 'rate' knob on the clocks etc... interpolate between readings .... and so on.

One characteristic of this approach is that if we have a third clock C, at a site C, which is twice as far from A as B is - and in the same direction ( ie. A, B, and C in an equally spaced line ) - then if we synchronise A and B, followed by B and C, then A and C must also be synchronised ( even if we never do the messenging! ). Thus this can give rise to a consistent, anisotropic, grid of clocks to work from. It will be consistent with theoretical aspects of time as used in Special/General Relativity. ( I'm talking about 'normal' space here, where a clock hasn't fallen into a black hole etc ).

Now you may be asking what the heck does this have to do with LIGO and GPS? In effect GPS is like the old 'speaking clock' on the phone systems - "at the third stroke it will be ...." - continuously broadcasting timing information via satellite and precision recievers. It plays the role of a clock saying 'I am here at position P and I say it is time X'. The GPS in turn relies, or calibrates, against the atomic clock standards mentioned above. Each LIGO installation distributes timing information within and is slaved to the GPS time channel. The data 'pipelines' regularly stamp the data 'frames' accordingly. This is absolutely crucial for any later deduction of the direction of a gravity wave's arrival at Earth.

I think the acceptable inaccuracy is no more than about 10 nanoseconds ( ten feet! ) across LIGO. The light flight time from Hanford to Livingstone is about 10 milliseconds ( ie. 10,000,000 feet or approximately 1800 miles ) on a direct line through the Earth ( which is curved after all ). I don't however know what angular resolution this yields for pinpointing sources in the celestial sky.

Cheers, Mike.

( edit ) Changed gif frame-rate from 14 to 8 ( per second ).