ページ1:

Note: this is the formula sheet that will be given to you during the exam.
Time dilation:
1
At = YAT
where y=
Here, v is the relative velocity between frames, At is proper time measured in the rest frame of the clock.
Length contraction:
1=lo/Y
where lo is the proper length measured in the rest frame of the stick.
Doppler effect:
c+v
f'
=
if a source moves towards the observer.
v switches sign if the source moves away from the observer.
Spacetime interval: As²
=
(cAt) (Ax) (Ay)² - (Az)² invariant and either timelike or spacelike
Below, we assume frame S' moves to the right of frame S along x axis at velocity v. Also, the reverse
transformation can be found by flipping the sign of v while switching prime and non-prime.
Lorentz transformation : x' = y(x − vt); y' = y; z' = z; t' = y (t − 2 x)
Velocity addition:
=
Ux-v
vuxi
uy
Uz
-
; uz =
Y(1-vux)'
Y(1-x)
Momentum: Yumu
Energy: E =
Yumc²
Relation: E2 = p² c² + m² c4
4-velocity: V =
dx
dt
dx
= Yu =
dt
uxi
4-position: X = (ct, x, y, z)
dv
4-acceleration: A = r = Y₁a(½, 1,0,0), where the last part is only true for 1D motion along x-axis.
dt
4-momentum: P = mV = my₁ (c,u) = (-,p)
Lorentz transformation matrix:
[ct"
Vv
--% 0
v
0 10
200
0
Yv
Vv
0
0 y
1
Z
The invariant scalar product of any two 4-vectors:
A B AoBo A₁B1-A2B2 - A3 B3
Example:
As² = AX AX = (cAt)² - (Ax)² - (Ay)² - (Az)²