Table of Contents
[20150316]
old notes [[related]] [[physics]]
[20200826]
Velocity Raptor  TestTubeGames [[relativity]] [[game]][20150316]
some special relativity notes [[relativity]][20210120]
(1) Could A Spaceship Wrap Around The Universe & Destroy Itself?  YouTube [[relativity]][20190825]
How Special Relativity Makes Magnets Work  YouTube https://www.youtube.com/watch?v=1TKSfAkWWN0 [[relativity]][20190906]
Например, космический корабль, который движется с ускорением свободного падения g, пройдет расстояние 13 миллиардов световых лет (долетит до края наблюдаемой Вселенной!) менее чем за сто лет, если считать время в собственной системе отсчета. [[GR workbook?]] [[study]]
[20150112]
perpendicular velocity addition in special relatility
[20150316]
old notes
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
Polebarn 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
* \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 4vectors:
<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
4velocity: derivative w.r.t. to infinitesimal proper time
Action principle: minimal proper time along the trajectory
https://en.wikipedia.org/wiki/Fourvector <button class="pullurl" value="https://en.wikipedia.org/wiki/Fourvector">pull</button>
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 <button class="pullurl" value="https://en.wikipedia.org/wiki/Relativistic_Doppler_effect#Systematic_derivation_for_inertial_observers">pull</button>
Fourvelocity U = dx / dtau: tangent fourvector 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 4velocity is U_C = (1, 0)
* in B's frame: C's 4velocity is U_B = LT(v) U_C = (gamma_v, v gamma_v)
* in B's frame: C's 4velocity is LT(u) U_B = TODO
https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations <button class="pullurl" value="https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations">pull</button>
Hyperbolic rotations of coordinates https://en.wikipedia.org/wiki/Lorentz_transformation#Hyperbolic_rotation_of_coordinates <button class="pullurl" value="https://en.wikipedia.org/wiki/Lorentz_transformation#Hyperbolic_rotation_of_coordinates">pull</button>
twins paradox
Acceleration
related [[physics]]
[20200826]
Velocity Raptor  TestTubeGames [[relativity]] [[game]]
[20150316]
some special relativity notes [[relativity]]

t'^{2} = t^{2}  x^{2}

The gravity on the poles in a bit larger than the gravity on the equator due to the centrifugal force.

Galilean group of transformations:
 Translation x' = x + a
 Rotation x' = Rx, R R^{T} = 1
 Boost: x' = x + vt

t' = t + t_{0}

map intertial frames to intertial

dx^{2}/dt^{2} = 0, then, for each transformation, dx'^{2}/dt^{2} = 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 v^{2} + 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.
[20210120]
(1) Could A Spaceship Wrap Around The Universe & Destroy Itself?  YouTube [[relativity]]
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?
[20190825]
How Special Relativity Makes Magnets Work  YouTube https://www.youtube.com/watch?v=1TKSfAkWWN0 [[relativity]]
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
[20190906]
Например, космический корабль, который движется с ускорением свободного падения g, пройдет расстояние 13 миллиардов световых лет (долетит до края наблюдаемой Вселенной!) менее чем за сто лет, если считать время в собственной системе отсчета.
GR workbook? [[study]]
 Box 20.1
 224 the cosmological constant
[20150112]
perpendicular velocity addition in special relatility
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
 public document at doc.anagora.org/relativity
 video call at meet.jit.si/relativity
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