Spacetime: Special Relativity
The development of the spacetime idea actually arose from the necessity of creating a self-consistent description of Maxwell's electrodynamics that worked for all observers no matter their relative motion. This led to the recognition by Lorentz that a special set of coordinate transformations allowed the FORM of Maxwell's equations to remain invariant. After Einstein published "Special Relativity" in 1905, it was Minkowski who took this one step further and showed that the Lorentz Transformation could be re-interpreted as a rotation within a 4-dimensional coordinate space that he dubbed "spacetime."
In an ordinary coordinate rotation, we can express the endpoints of a meter stick of length L in one coordinate system as P1 = (x1,y1) and P2 = (x2,y2). We can perform a rotation about the origin by an angle theta to get a second set of coordimnates for these points x' = Lcos(theta), y'= Lsin(theta). The result is that, in both coordinate systems X and X', the invariant length of the meter stick remains the same, L. However, the rotation changes the specific names we assign to the coordinates.
Special Relativity & Velocity
In special relativity, in the spacetime description, the rotation angle depends on the relative velocity between the two observers, Theta = V/C where C is the speed of light. When V is very small, Theta ~ 0 and so there is no significant difference between the two descriptions provided by the two observers, for electrodynamic phenomena covered by Maxwell's Equations. When V ~ C, however, the rotation angle is large, and in general the coordinate descriptions will be very different, leading to phenomena such as time dilation and length contraction.
Mathematically, we can write the Pythagorean Theorem in 2 dimensions as:
ds2 = Pxx dxdx + Pxy dxdy + Pyx dydx + Pyy dydy
where dx = (x2-x1) dy=(y2-y1)
For Euclidean geometry (flat plane) the coefficients are Pxx = 1, Pxy =- Pyx = 0 and Pyy=1.
In general, for any kind of geometry, Gauss defined a quantity gij as the Fundimental Metric Tensor that defines the geometric properties of a space, so that the generalized Pythagorean Theorem becomes:
ds2 = gij dxidxj
where i = 1,2,3 and j=1,2,3 and the coordinates for a point in 3-dimensional space are given by (x1, x2, x3). These coordinates can be the ordinary Cartesian X, Y and Z, or any other orthogonal space coordinate system ( e.g. spherical, cylindrical etc). For a Euclidean flat space we have g11 = 1, g22 = 1, g33=1 and all other terms are zero.
For spacetime, the analogous quantity to the 3-dimensional Metric Tensor is the 4-dimensional Minkowski Tensor ,nuv, which has the valuesnxx = 1 nyy= 1, nzz = 1, ntt = -1 and all other terms are zero. This gives us:
ds2 = -(cdt)2 + dx2
The interpretation for the metric tensor for Minkowski spacetime, a cornerstone for Special Relativity, is that it is a book-keeping tool to help us perform calculations in special relativity. There is nothing about it, and the nature of spacetime at this level, that demands a more detailed explanation. However, the advent of General Relativity by Albert Einstein in 1915 introduced a whole new way to regard spacetime.
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An artist's concept of twisted space-time around Earth. (Source: Spacetime Vortex - NASA.)