Cosmology 5022

Relativity

Length Contraction

S S’

event #1 0 0 0 0

event #2 0 L t’ L’

t x t’ x’

S’S#1 #2

v

rod is stationary in S’ frame

Want to measure the length of the rod in S frame

In S frame the length of the rod is

In S’ event #2 takes place before event #1

LT applied to event #2

LT:

Simultaneous events

Time Dilation

S S’

tick #1 0 0 0 0

tick #2 dt x dt’ 0

t x t’ x’

S’S

v

clock is stationary in S’ frame

Want to measure dt, the time between ticks in S frame

In S frame the time between clock ticks is

LT applied to event #2

LT:

Time dilation has been verified experimentally

Basis and co-basis setsbasis set along coordinate axes

co-basis set

Parallel Transport

transport a vector without changing italong a closed loop

the transported vector is not the same as theoriginal one

the difference between the two depends on the path taken and the area enclosed

Newtonian Deviation Equation

Example:

particles approach each other in x1, x2 directions, separate along x3 direction

M

Note: in Eucledian space of Newtonian gravitycovariant = contravariant