Originally posted by sonhouse
If I have that right, it is a 4th dimensional vector, right? BTW, what is s in the last equation?
Yes, I was talking about 4-vectors, but there is a complication. First of all the last equation. In relativity theory you have space-like and time-like vectors. You can express the ´length´ of a four vector as either a space-like quantity (ds) or as a time-like quantity (dT) when it is called proper time. The last formula just relates the two notations. There is no extra physical content.
The complication is to do with how vectors are defined. Vectors are defined in a
flat space tangent to the curved space you are in - you need a different tangent space at each point of the base space. If you think of a vector as being a
straight arrow then you cannot have a position vector on the surface of a sphere - it has to leave the surface because the sphere is curved and the arrow is straight.
You can define a velocity vector for arbitrarily large velocities because velocity is defined in the limit of very small displacements over very small times; and you can always get to a distance scale where the sphere appears flat locally - a map of your home town can afford to ignore the curvature of the earth, but if you want a map of the world you can´t.
If your base space is flat then you can define position vectors of arbitrary size as the base space is isomorphic with (*) with it´s own tangent space (where the vectors live). This is the case for special relativity. It is not the case for general relativity.
With that proviso the rule I gave is the rule for finding the length of a four-vector in special relativity. You need a more complicated rule in general relativity and there are no well defined position vectors which are not infinitesimal.
(*) Isomorphic is maths speak for identical with.
Explain this to me, pls
05 Mar '09 05:17
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