Secondly, Rincewind:

Some interesting challenges to relativity theory. Keep them coming, they’re getting me to really use my brain (for a change).

Re: the idea that displacement, not motion, causes time dilation – I think Michael’s example of circular motion dealt with that. (Nice one Michael – I wish I’d thought of that myself.)

Re: acceleration can’t be the key – hopefully you’ll find the following link as useful and interesting as I did. (The logic in this web page looks right to me, but I may be missing something.)

http://www.incentre.net/tcantine/TP.htmlRe: symmetry in relativity – there IS symmetry in relativity.

You need to get past the notion of ABSOLUTE time and ABSOLUTE space (which relativity says don’t exist), and accept that all we can do is measure the DIFFERENCE in lengths between points, and the DIFFERENCE in times between events.

Relativity is symmetrical in that it says that two observers stationary relative to each other (and ignoring the gravitational field too, to be pedantic) will measure the same rate of ticking of a clock as each other. Their clocks may have different starting times (9pm and 10pm), but so long as they’re stationary, their clocks will advance at the same rate.

I’ve a feeling (but I can’t prove it … until I’ve studied relativity more) that the symmetry in relativity would be destroyed if time slowed down when an astronaut is moving away from you and sped up when travelling towards you – such that their clocks read the same time again when they rejoin each other. ((Maybe an example of circular motion would help here too, Michael??))

Symmetry in physical laws can be expressed mathematically by saying that some quantity or other is conserved – i.e. that the quantity is “invariant” (doesn’t change).

You’re right that in normal (“Euclidean”) space - in the absence of motion or gravity - the distance from the origin is “invariant” under rotation. Mathematically, this is expressed as the invariance of the “Interval” I, where:

I^2 = x^2 + y^2 + z^2.

(That’s just Pythagoras’ rule for a hypoteneuse in 3 dimensions, by the way.)

In relativity, the invariant “Interval” I is calculated as:

I^2 = x^2 + y^2 + z^2 – (c.t)^2

(No, don’t ask me how they derive that.)

The fourth term above, the (c.t) term, is basically agreeing with what you said earlier - that a time difference of 1 second can be treated as equivalent to a length/displacement difference of 300,000 km along a space dimension. But this (c.t) term is NEGATIVE in the above equation, not positive – which (I think, maybe?) sort of explains the counter-intuitive nature of relativity. (For example, it means that a rotation of the axes - due to motion – leads to time differences dilating and lengths contracting, rather than length and time keeping their normal relations to each other.)

Maybe I’m not actually helping here (someone who doesn’t really understand what he’s talking about quoting mathematical formulae at you) – but you started it with your (0,0,5) coordinates! [:p]