# Table of Contents - [`[2015-03-16]` old notes](#ldnts) - [related](#rltd) [[physics]] - [`[2020-08-26]` Velocity Raptor | TestTubeGames](#ststtbgmscmvlctyrptrhtmlvlctyrptrtsttbgms) [[relativity]] [[game]] - [`[2015-03-16]` some special relativity notes](#smspclrltvtynts) [[relativity]] - [`[2021-01-20]` (1) Could A Spaceship Wrap Around The Universe & Destroy Itself? - YouTube](#swwwytbcmwtchvmfjlkbycldspwrprndthnvrsdstrytslfytb) [[relativity]] - [`[2019-08-25]` How Special Relativity Makes Magnets Work - YouTube https://www.youtube.com/watch?v=1TKSfAkWWN0](#hwspclrltvtymksmgntswrkytbswwwytbcmwtchvtksfkwwn) [[relativity]] - [`[2019-09-06]` 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.](#напримеркосмическийкораблсобственнойсистемеотсчета) - [GR workbook?](#grwrkbk) [[study]] - [`[2015-01-12]` perpendicular velocity addition in special relatility](#prpndclrvlctyddtnnspclrltlty) # `[2015-03-16]` 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 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^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, = \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 # related [[physics]] # `[2020-08-26]` [Velocity Raptor | TestTubeGames](https://testtubegames.com/velocityraptor.html) [[relativity]] [[game]] # `[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? - YouTube](https://www.youtube.com/watch?v=6MfJ59lkABY) [[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? # `[2019-08-25]` How Special Relativity Makes Magnets Work - YouTube [[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 # `[2019-09-06]` 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 # `[2015-01-12]` 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