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Author Topic:   Geometry of Spacetime
nwr
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Posts: 5592
From: Geneva, Illinois
Joined: 08-08-2005


(1)
Message 3 of 41 (701434)
06-18-2013 11:57 PM
Reply to: Message 1 by Iblis
06-18-2013 10:11 PM


a² + b² = c²

With the Pythagorean formula, we measure distance in two dimensions as . The equivalent for three dimensions is to measure distance as

Once we add in a time dimension, that becomes

This assumes that we have scaled things so that , the velocity of light, is 1.

That's pretty much the metric that is the basis for relativity.

Edited by nwr, : No reason given.


Fundamentalism - the anti-American, anti-Christian branch of American Christianity

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nwr
Member
Posts: 5592
From: Geneva, Illinois
Joined: 08-08-2005


Message 5 of 41 (701438)
06-19-2013 12:50 AM
Reply to: Message 4 by Coyote
06-19-2013 12:02 AM


Why do you + x, y, and z but –t? Why not +t?

It's connected to the fact that there's a minus in the wave equation:

Light (electromagnetic fields) satisfy the wave equation.

And how do you know?

I'll admit that my relativity is a bit rusty.

Fundamentalism - the anti-American, anti-Christian branch of American Christianity

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nwr
Member
Posts: 5592
From: Geneva, Illinois
Joined: 08-08-2005


Message 16 of 41 (701535)
06-20-2013 3:28 PM
Reply to: Message 15 by New Cat's Eye
06-20-2013 12:05 PM


The way I understand it, the only way for the time component of your path to be at a true 90 degrees would be if you were not moving.

And the way I see it, is that in space-time there is no such thing as "true 90 degrees" between a time-like direction and a space-like direction. It is all relative to the observers frame.

Fundamentalism - the anti-American, anti-Christian branch of American Christianity

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nwr
Member
Posts: 5592
From: Geneva, Illinois
Joined: 08-08-2005


Message 18 of 41 (701543)
06-20-2013 7:34 PM
Reply to: Message 17 by New Cat's Eye
06-20-2013 4:10 PM


But if you have a particle that is at rest (in whatever reference frame), then aren't the time-like and space-like directions at 90 degrees?

I'm not sure if that even makes sense.

Take ordinary 2-dimensional space. If we stretch out the x-axis, say rescale it so that what was one unit of length becomes 2 units, then angles change -- assuming that we don't also stretch out the y-axis. So, in some sense, the magnitude of angles is an artifact of how we measure them.

In the case of spatial directions, we normally require rotational symmetry. And if we require rotational symmetry, we cannot stretch out the x-axis without also stretching out the y-axis.

As far as I know, we cannot rotate things from a space-like direction to a time-like direction. So we don't have something like rotational symmetry to normalize our way of measuring. So I think that unavoidably leaves measurements of angles between space-like and time-like directions to be dependent on our arbitrarily chosen standards.


Fundamentalism - the anti-American, anti-Christian branch of American Christianity

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nwr
Member
Posts: 5592
From: Geneva, Illinois
Joined: 08-08-2005


Message 25 of 41 (701947)
06-28-2013 9:03 AM
Reply to: Message 24 by Iblis
06-28-2013 12:14 AM


Re: Anybody?
Some of this is basic calculus, and some of it comes from the mathematical model that Einstein used in special relativity.

Einstein's intuition told him that the velocity of light should be the same for all observers, and the Michelson-Morley experiment seemed to confirm that. But that was incompatible with the traditional Newtonian/Euclidean view of space, which saw time and distance as independent. So the problem was one of finding a new metric which connected time and space in such a way that the velocity of light could be the same for all observers.

It worked very well. It accurately predicted motion in particle accelerators. One could deduce which seemed to accurately account for the energy seen in radioactive materials, and which was confirmed by nuclear physics.

And yes, many people at that time thought it counter-intuitive. But science is a pragmatic enterprise, and it is hard to beat "it works very well."


Fundamentalism - the anti-American, anti-Christian branch of American Christianity

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