# 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 (*Δ**s*)^{2} between
two events in space time, in a given Lorentzian/inertial frame, is
defined as:

(*Δ**s*)^{2} ≡ − (*Δ**t*)^{2} + (*Δ**x*)^{2} + (*Δ**y*)^{2} + (*Δ**z*)^{2}

or

(*Δ**s*)^{2} ≡ − (*Δ**x*^{0})^{2} + ∑_{i, j}*δ*_{ij}*Δ**x*^{i}*Δ**x*^{j}

where *δ*_{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 (*Δ**s*)^{2} > 0,
**null-** or **light-like** when (*Δ**s*)^{2} = 0, and
**time-like** when (*Δ**s*)^{2} < 0.
When dealing with time-like separations, ignore the implication of an
imaginary number. The difference in time *Δ*T is always real: (*Δ*T)^{2} = − (*Δ**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.