1. Cosmology 5022
Relativity

2. S S’ Length Contraction S’ S v LT:
Want to measure the length of the rod in S frame S S’
LT: v
rod is stationary in S’ frame #1 #2
LT applied to event #2 S S’
t x t’ x’
event #
In S frame the length of the rod is
In S’ event #2 takes place before event #1
event # L t’ L’
Simultaneous events

3. S S’ Time Dilation S’ S v LT:
Want to measure dt, the time between ticks in S frame S S’
LT: v
clock is stationary in S’ frame
LT applied to event #2 S S’
t x t’ x’
tick #
In S frame the time between clock ticks is
tick # dt x dt’
Time dilation has been verified experimentally

4. Basis and co-basis sets
along coordinate axes
co-basis set

5. Parallel Transport transport a vector without changing it
along a closed loop
in general,
the transported vector
is not the same as the
original one
the difference between
the two depends on
the path taken and
the area enclosed
the curve (path) that parallel transports
its own tangent vector is a geodesic

6. Newtonian Deviation Equation
Note: in Eucledian space
of Newtonian gravity
covariant = contravariant
Example: M
assume
x1=x2=0
and x3=r
particles approach each
other in x1, x2 directions,
separate along x3 direction