Originally posted by DeepThoughtIf I have that right, it is a 4th dimensional vector, right? BTW, what is s in the last equation?
Sort of, but not really, science fiction writers (except in very hard SciFi) are more concerned with it sounding good than being accurate. I have to say that quantum theory has nothing to do with this I don´t see why the posters above started talking about that.
In relativity theory observers are point-like entities who come equipped with imaginary c ...[text shortened]... about proper time, this is just the interval expressed as a time-like quanitity: c²dT² = - ds².
Originally posted by sonhouseYes, 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.
If I have that right, it is a 4th dimensional vector, right? BTW, what is s in the last equation?
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.