yeah, it's sort of the noob way of doing it but it was a point of reference. strictly speaking though, it's the only way of multiplying binomials. It's only the noob way because it doesn't work for polynomials or vectors with more than two terms.
okay, the importance of the metric tensor is that it tells us how to form the dot product and that allows us to measure the distance between points in space. if we have a point X = (1, 2) and Y = (3, 4) then to measure the square of the distance between them we just do:
(X - Y)^2 = (X - Y) * (X - Y) = (-2, -2) * (-2, -2) = 4 + 4 = 8
so the distance between them is sqrt(8). remember, this is with g11 = g22 = 1 and g12 = g21 = 0 in (eqn 1) above. Again, the importance of understanding that the formula for the distance between two points comes from the metric tensor is that we change it in relativity.
---------------------------------------------------
Really, a form relativity is already in newtonian mechanics. It's called Galilean relativity and has to do with changing from one frame of reference to another that is moving with constant velocity in relation to the first. First, we say that a frame of reference is an inertial frame if newtons first law holds in it. Essentially, this means that the frame isn't accellerating. So if you are in a car that's accelerating towards a street sign, that sign does not obey newtons first law because it is accelerating towards you even though there is no force acting on it. So we restrict ourselves to inertial frames. Even though it's not necessary, we will also restrict ourselves to frames of reference that aren't rotated with respect to each other so that all the coordinate axes are parallel. Otherwise, we have to bring matrices into it.
If an observer measures an event, E, with coordinates (x1, x2, x3) in one frame, lets call it O, and a second observer measures it in another frame that's at rest with respect to the first, let's call it O', then the we can determine the coordinates (x1', x2', x3') that the second observer assigns to E as follows:
We measure the distance from O to O' and get a vector D=(d1, d2, d3) so that the coordinates of O' in O are just D. Then the coordinates of E in the O' frame of reference are just (x1 - d1, x2 - d2, x3 - d3).
Does that make sense? We can get to the physics next post if so.
|
|
Bookmarks