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Author | Topic: The Twins Paradox and the speed of light | |||||||||||||||||||||||||||
cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
why is either watch affected by it's velocity? They're not. They are simply ticking as normal - but they are being taken on different length paths through space-time. The one that takes the shorter path naturally ticks less, and hence appears younger when the two watches get back together and are compared. The length of a path through space-time equals the time experienced along that path. But space-time is strange - the *longest* (space-time) distance between two points is a straight line!! Pick two points in space-time: say P1 is Time's Square 00:00:00 1st Jan 2000; and P2 is Time's Square 00:00:00 1st Jul 2008. So sitting still in Time's square in order to get from P1 to P2 is the LONGEST space-time path between these points. Any other path will be shorter! Repeatedly flying back and forth from JFK to Sydney to get from P1 to P2 will be slightly shorter than staying still, so your watch (and your heart) will tick slightly less on this journey - although almost immeasurably less. Travelling out to Alpha Centauri and back at just under the speed of light will just about get you from P1 to P2 and that path will be much much shorter than sitting still, and so your watch will tick considerably less seconds on this path - perhaps only a few days' worth!! So a watch left to sit still between P1 and P2 will tick away 8.5 years, and your watch on your space-trip may only tick away one week! Edited by cavediver, : No reason given.
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
How does flying back and forth make the space-time path shorter? Because the journey is not a striaght line through space-time, but is weaving about. Just like in everyday life: a wavy path between two points must be LONGER than the straight line; in the topsy turvy geometry of space-time, a wavy path must be SHORTER than the straight line.
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
Does that make any sense? Perfectly.
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
That makes sense. Yes, Randman's idea does seem to make sense and sort of reproduces the desired effect. Unfortunately, it has nothing to do with relativity and the twins "paradox". Multiplying 2 by 2 gives the correct answer 4, despite the question being "what do you get if you raise 2 to the power of 2?"... Edited by cavediver, : No reason given.
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
Thank you SG. That is simple and clear. What is more, we can essentially derive the entirety of Special Relativity just from our simple metric, ds^2 = dt^2 - dx^2 - dy^2 - dz^2. General Relativity then follows by allowing us to generalise the metric to ds^2 = A()dt^2 - B()dx^2 - C()dy^2 - D()dz^2 where A, B, C and D are functions of t,x,y,z. (there can also be cross-terms in general, dt.dx, etc, though we can diagonalise them away at a point). The Einstein Equation tells us what A, B, C, and D can be based upon the matter distribution of the space-time. Beautiful in its simplicity
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
Does this mean that if something travels faster than 300,000,000 meters in 1 second... Hmmm... now what travels 300,000,000 metres in 1 second Oh, yes of course! All things travel at the speed of light *ALL THE TIME* - it's just that mostly we travel in the temporal direction, so we don't notice anything other than time passing. What we call *speed*, is us rotating our speed-of-light velocity slightly out of the temporal direction and slightly into the spatial durections. The most *speed* we can ever achieve is what we get by rotating our speed-of-light velocity completely into the spatial directions... and surprise surprise, we see the greatest *speed* is the speed of light. Our velocity is ALWAYS the speed of light - so it is impossible to talk about going *faster*, just as it is impossible to talk about going *slower*
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
It was about how traveling in space can make the time distance shorter. Yes, I know.
That makes sense to me. I know, it does sort of make sense, but...
Is it accurate? Absolutely not, hence my warning. This is not how space-time works, and it is making a large confusion of time experienced with what we call coordinate time.
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
Time experienced is given by the ds term - it's an inifnitessimal so it needs integrating along your path through space-time, to given you the time experienced along that path. The dt is coordinate time, and is merely a way of puting a chart or map down on space-time in order to make calculations - rather like latitude and longitude on the Earth - they are merely calculating conventions to ease measurement on the 2d surface of the Earth. The 2d surface is real, as is the physical distance between two points, but the actual lat/long grid is purely arbitrary. The 4d of space-time is real, as is the time experienced along a particular path, but the dt, dx, dy, dz grid you use to decribe it is (fairly) arbitrary.
Edited by cavediver, : No reason given.
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
Now... to over-simplify it even further... Ts + Ss = Speed of light (300,000,000 m/s) Make it Ts2 + Ss2 = c2 and you'd be almost there (c is the speed of light) although it's a complete bastardisation of how we write these things
If I'm on track here, this also explains why faster-than-light travel is impossible... because it would either result in a negative temporal speed, or it would simply "break" the equation. And now we have the "squared" version of your equation, you can't even consider a negative Ts - the equation is simply sharing out your speed-of-light velocity between the spatial and temporal components... So we have shown that "faster than the speed of light" is a nonsense concept, can you start to see where time dilation comes from now? Edited by cavediver, : No reason given.
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
When I explained about how we are all "traveling" through time at the speed of light, someone asked "how do you know that". It is simply what relativity shows us, and relativity (in its Special form) is the most successfully tested theory we have ever discovered, so we tend to trust what it tells us To understand it at a deepr level, I'm afraid it is time for four-dimensional vectors and tensors - all good fun, but not the easiest mathematics to explain over a pint...
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
everything that exists is in motion, strangely even Quantum [Bosonic] Fields Of course the quantum fields are not in motion - what the hell does that even mean?? The fields are at a much more fundemental level than any concept of motion. It's the same as claiming that electrons have smell. You really do have absolutely no clue about any of this...
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
Hope this helps.
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
Yes, that's the idea. Though the units are a bit confused, as we don't measure distance through time in meters but in seconds
Note that I am glossing over a few bits with this picture. If I have some at some stage I may try to elucidate, but it does get a bit more confusing
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
Just remember that although SG is measuring the spacetime distance (we usually use the term interval as it is less confusing, applying equally well to space and time) in metres, he could just as well have measured it in seconds, and the for us beings with mass, spacetime distance is actually time experienced. We have three concepts here: coordinate space, coordinate time, and spacetime interval. The static twin moves through less (no) coordinate space, than the travelling twin. Both move the same amount of coordinate time (from *sometime then* to *sometime later*), and the traveller's spacetime path (the time he experiences) is shorter than the static twin's spacetime path. Clear as mud
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
The spacetime interval IS the experienced time - they are one and the same. And it will be shorter for all speeds - just reasonably measurable as the difference in speed approaches c.
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