relativity

# Special relativity

## ¶[2015-03-16] old notes 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


## ¶[2015-03-16] some special relativity notes relativity

• 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 RT = 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.

## ¶[2021-01-20](1) Could A Spaceship Wrap Around The Universe & Destroy Itself? - YouTuberelativity

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?

## ¶[2019-08-25] How Special Relativity Makes Magnets Work - YouTube https://www.youtube.com/watch?v=1TKSfAkWWN0relativity

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

## ¶TODO GR workbook? relativitystudy

CREATED: [2017-05-15]
• Box 20.1
• 224 the cosmological constant

## ¶[D][2015-01-12] perpendicular velocity addition in special relatility relativity

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

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