Special relativity

Postulate 1: the principle of relativity: the laws of physics are the same in all itertial frames

Postulate 2: The speed of light is the same in all inertial frames

Frames S: (x, t) and S': (x', t')

Most general relation:
x' = f(x, t)
t' = g(x, t)

1. Law of inertia: in inertial frame, particle travels at constant velocity. Maps straight lines to straight lines, which means:
x' = a1 x + a2 t
t' = a3 x + a4 t

2. S' has velocity v relative to S, therefore, x = v t maps to x' = 0. Also: when t = 0, x' = 0, therefore,
x' = gamma(v) (x - v t)

3. gamma(v) is even function:
x = gamma(v) (x' + v t')

4. speed of light:
x = c t maps to x' = c t':

c t' = gamma(v) (c - v) t
c t  = gamma(v) (c + v) t', therefore, gamma(v) = \sqrt{\frac{1}{1 - \frac{v^2}{c^2}}}


Lorentz transformations:
x' = gamma (x - v / c c t)
y' = y
z' = z
t' = gamma (c t - v / c x)


If c = 1:
x' = (x - v t) / sqrt(1 - v^2)
t' = (t - v x) / sqrt(1 - v^2)

In the low v limit, we get Galilean transformations


Clock in frame S', intervals T'.
Events occur at (ct', 0), then (ct' + c T', 0) and so on.
In the frame S: t = gamma (t' + v x' / c^2), clock at x' = 0, therefore, T = gamma T'. "The time runs slower in moving frame"

Twins paradox:
People A and B.
B jumps in a spaceship and flies to some planet at speed v, then turns around and returns after some time T and finds A dead since for A it was T/gamma.
However, we might consider it as: A flies away on some planet from B at speed v, then turns around and returns after time T and finds B dead since for B it was T / gamma.
Resolution: actually, no symmetry since someone has to change velocity from v to -v and accelerate (general relativity).

Length contraction:
TODO

Pole-barn paradox:
laddar of length 2L, barn of length L.

* if you run fast enough with the ladder, from the barn POV, the ladder contracts to the length 2L / gamma. Possible to fit.
* from the ladder POV, the barn contracts to the lenght L / gamma. Impossible to fit.
No paradox, does depend on the frame!


TODO Addition of velocities

Invariant interval: \Delta s^2 = c^2 \Delta t^2 - \Delta x^2

* \Delta s^2 > 0: timelike separated, within each others lightcones. Closer in space than in time.
* \Delta s^2 < 0: spacelike separated, outside each other's lightcones. Observers can disagree about the temporal ordering.
* \Delta s^2 = 0: lightlike separated.

Lorentz group:

Minkowski metric:

\eta

1  0  0  0
0 -1  0  0
0  0 -1  0
0  0  0 -1

inner product of 4-vectors:

<X, X> = X^T \eta X = X^i \eta_{ij} X^j

Lorentz transformation X' = \Lambda X
X'^i = \Lambda^i_j X^j

Lorentz transformation are those leaving inner product invariant, that is, <X', X'> = <X, X>

\Lambda^T \eta \Lambda = \eta

Both sides are symmetric 4x4 matrices, 10 constrains on coefficients of \Lambda, therefore, 16 - 10 = 6 independent solutions

# Solutions of form
1  0  0  0
0
0     R
0
R R^T = 1, R is space rotation matrix. Three independent matrices (rotations about the three spatial axis)

# Solutions of form
gamma         -gamma v / c  0  0
-gamma v / c  gamma         0  0
0             0             1  0
0             0             0  1
Three solutions, for x, y and z axis.

Set of all matrices is Lorentz group O(1, 3).

det \Lambda^2 = 1
* subgroup SO(3): spatial rotations
* subgroup det \Lambda = 1: proper Lorentz group SO(1, 3)
* subgroup det \Lambda = -1


Proper time: \Delta \tau = \Delta s / c


4-velocity: derivative w.r.t. to infinitesimal proper time

Action principle: minimal proper time along the trajectory


https://en.wikipedia.org/wiki/Four-vector

Time dilation: moving clocks are observed to be running slower
Two observers still can measure time between two intervals to be equal

Nice formal treatment of relativistic Doppler effect https://en.wikipedia.org/wiki/Relativistic_Doppler_effect#Systematic_derivation_for_inertial_observers

Four-velocity U = dx / dtau: tangent four-vector to worldline, of magnitude 1

In the object's O rest frame: U = (1, 0, 0, 0)

t = gamma tau

O' moving at velocity v from O.

Applying Lorentz transformations: U' = (gamma, -v gamma, 0, 0)

Derivation of velocity addition:

A.     B.->u(relative to A)     C.->v(relative to B)

* in C's frame: C's 4-velocity is U_C = (1, 0)
* in B's frame: C's 4-velocity is U_B = LT(v) U_C = (gamma_v, -v gamma_v)
* in B's frame: C's 4-velocity is LT(u) U_B = TODO

https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations

Hyperbolic rotations of coordinates https://en.wikipedia.org/wiki/Lorentz_transformation#Hyperbolic_rotation_of_coordinates


	twins paradox
Acceleration

• t'² = t² - x²
  
• The gravity on the poles in a bit larger than the gravity on the equator due to the centrifugal force.
  
• Galilean group of transformations:
  
  1. Translation x' = x + a
     
  2. Rotation x' = Rx, R R^T = 1
     
  3. Boost: x' = x + vt
     
• t' = t + t₀
  
• map intertial frames to intertial
  
• dx²/dt² = 0, then, for each transformation, dx'²/dt² = 0
  
• the principle of relativity: the Newton's laws are the same in all itertial frames
  
• The equation of motion is second order
  
• Potential V(x) is defined by: F(x) = -dV(x)/dx
  
• Energy E = 1/2 m v² + V(x). It is conserved, E' = 0 for any trajectory that obeys the equation of motion
  
• dynrel, p.20, potential!
  
• Energy is conserved iff there exists V such that F = - grad V.
  
• Central forces: angular momentum is conserved. L = m x × x'. dL/dt = mx × x'' = x × F.
  

only preferred local frames of reference are forbidden, you can still have preferred global frames of reference. For example, big bang frame of reference, where the CMB appears still?

very good intuitive explanation! Basically, since charges in wire (protons/electrons) are moving relative to each other, they are slightly contracted so in other frames of reference it creates a force

• Box 20.1
  
• 224 the cosmological constant
  

A's frame: (1, 0, 0)
O's frame: gamma (1, 0, 0.9)
B's frame: (gamma^2, 0.9 gamma^2, 0.9 gamma)

A's frame: U_A = (1, 0, 0)
Boost at the Y direction: u
LT(u) =
{
	gamma_u   , 0, -gamma_u u
	0         , 1, 0
	-gamma_u u, 0, gamma_u
}
O's frame: U_O = LT(u) U_A = (gamma_u, 0, -gamma_u u)
Boost at the X direction: v
LT(v) =
{
	gamma_v   , -gamma_v v, 0
	-gamma_v v, gamma_v   , 0
	0         , 0         , 1
}
B's frame: U_B = LT(v) U_O = (gamma_u gamma_v, gamma_u * -gamma_v v, -gamma_u u)

U_B = LT(w) U_A (1, 0, 0)

gamma_w = gamma_u gamma_v
-gamma_w w_x = -gamma_u gamma_v v
-gamma_w w_y = -gamma_u u

w_x = v
w_y = u / gamma_v