Special Relativity

.. warning: This is a rough work in progress!! Likely to be factual errors, poor grammar, etc.

.. note: Most of this content is based on a 2002 Caltech course taught by Kip Thorn [PH237]

As opposed to general relativity, special relativity takes place in a flat Minkowski space time: a 4-space with three spatial dimensions and one time dimension.

Index notation Variable Type
x^0 t Time
x^1 x Spatial
x^2 y Spatial
x^3 z Spatial

Separations

The separation (\Delta s)^2 between two events in space time, in a given Lorentzian/inertial frame, is defined as:

(\Delta s)^2 \equiv -(\Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2

or

(\Delta s)^2 \equiv -(\Delta x^0)^2 + \sum_{i,j} \delta_{ij} \Delta x^i \Delta x^j

where \delta_{ij} is the Kronecker delta (unity or 1 when i=j; zero otherwise), and the indices i and j are over the spatial dimensions 1,2,3 (corresponding to x,y,z). It can be shown that this separation is Lorentz-invariant; the scalar value of separation between two events does not depend on the inertial frame chosen.

Note the negative sign in front of the time dimension. The are three types of separations: space-like when (\Delta s)^2 > 0, null- or light-like when (\Delta s)^2 = 0, and time-like when (\Delta s)^2 < 0. When dealing with time-like separations, ignore the implication of an imaginary number. The difference in time \Delta \mathrm{T} is always real: (\Delta \mathrm{T})^2= -(\Delta s)^2.

References

[PH237]: Gravitational Waves (aka ph237), a course taught by Kip Thorne at Caltech in 2002. See http://elmer.tapir.caltech.edu/ph237/ for notes and lecture videos.