Einstein's Relativity: From Special to General, Philosophy, Mathematics, and Proofs

Introduction

The user request mentioned teaching relativity "from General to Special". That reverses the logical and historical order. Albert Einstein published Special Relativity in his 1905 paper "On the Electrodynamics of Moving Bodies". Ten years later, in November 1915, he completed General Relativity. Special Relativity is the limiting case of General Relativity for flat spacetime and inertial observers. Therefore we must learn Special first.

This guide presents the full structure. For philosophy, we trace ideas from Aristotle to Galileo, Newton, Mach, Maxwell, Lorentz, and Poincaré. For mathematics, we derive the Lorentz transformation from the invariance of the light sphere, prove time dilation and length contraction, and derive $E=mc^2$ from Einstein's 1905 photon emission argument. For General Relativity, we introduce manifolds and the metric, derive the geodesic equation from extremal proper time, motivate the Einstein Field Equations from conservation and the Newtonian limit, and prove the classic tests: gravitational redshift, light deflection, and Mercury's perihelion.

All equations are rendered with MathJax. Interactive canvases let you explore simultaneity, the light clock, the elevator equivalence principle, and spacetime diagrams. Wherever possible, numbered equations are shown so you can follow each algebraic step.

1. Philosophical Foundations

To understand what Einstein changed, we must first state what he rejected.

Aristotle's Physics (4th century BCE)

Aristotle held that:

Every body has a natural place. Earth and water move down, air and fire move up. Motion toward natural place is natural, other motion requires a continuous cause.
Rest is the natural state. There is no principle of inertia. A body stops when the pusher stops.
Space has an absolute up and down defined by the center of the cosmos. Place is absolute.
Time is the measure of change, separate from space, flowing uniformly for all.
Heavier bodies fall faster, proportional to weight. There is no vacuum. Light is instantaneous actualization, not finite speed.
Explanation is teleological, in terms of purposes and ends.

Newton's synthesis (1687)

Newton kept absolute space and absolute time in the Scholium of the Principia, but replaced Aristotelian dynamics with inertia: a body remains in uniform motion unless forced. He unified terrestrial and celestial motion with $F=ma$ and universal gravitation. Yet absolute space remained a metaphysical entity.

Mach's critique (1883)

Ernst Mach argued in "The Science of Mechanics" that Newton's bucket experiment does not prove absolute space, only motion relative to distant masses. Mach's principle, later named by Einstein, suggests that inertia is determined by the distribution of matter in the universe.

Maxwell, Lorentz, Poincaré

Maxwell's equations (1865) predicted electromagnetic waves with speed $c = 1/\sqrt{\mu_0\epsilon_0}$, independent of source motion. The Michelson-Morley experiment (1887) found no ether drift. Lorentz (1904) and Poincaré (1905) wrote transformations that left Maxwell's equations invariant, and Poincaré stated the principle of relativity for all physics.

Einstein's break

Einstein did not derive relativity from Aristotle. He explicitly overturned both Aristotelian and Newtonian absolutes. In 1905 he discarded the luminiferous ether as superfluous. In his 1916 book "Relativity: The Special and General Theory" he wrote, "the ether of the general theory of relativity is not a medium with a state of motion, the special theory had rendered the old ether unnecessary." His influences were Hume's empiricism, Mach's relational view, Galileo's ship thought experiment, and Maxwell's electrodynamics, not Aristotelian teleology.

Key point: Einstein replaced natural place with no preferred place, replaced rest as natural with uniform motion as natural, and replaced separate absolute time with a unified spacetime interval.

2. Special Relativity: Two Postulates

Postulate 1 Principle of relativity: The laws of physics take the same form in every inertial frame. No experiment can distinguish absolute uniform motion.

Postulate 2 Light speed invariance: The speed of light in vacuum, $c$, is the same in all inertial frames, independent of the motion of source or observer.

Consequence: relativity of simultaneity

Consider Einstein's train. A light flash is emitted from the center of a moving car. In the train frame, light reaches front and rear simultaneously because distances are equal. In the platform frame, the rear moves toward the light and the front moves away, so the rear is struck first. Simultaneity is frame dependent.

Green dot = left hit, red dot = right hit

Proof sketch: Let emission event be $E$ at $x=0, t=0$ in both frames. For train frame $S'$, arrival events satisfy $x'_L = -L/2 = -c t'_L$ and $x'_R = +L/2 = +c t'_R$, hence $t'_L = t'_R$. Transform to platform $S$ using $t = \gamma(t' + v x'/c^2)$. Then $t_L = \gamma(t' - vL/2c^2)$ and $t_R = \gamma(t' + vL/2c^2)$, so $t_R - t_L = \gamma v L /c^2 > 0$. The events are not simultaneous in $S$.

3. Mathematics of Special Relativity, Lorentz Transformations

Assume linear transformations between inertial frames $S$ and $S'$ moving at relative velocity $v$ along $x$. Write the general form

$$ x' = \gamma (x - v t), \quad t' = \gamma\left(t - \alpha x\right) \tag{1}$$

We determine $\gamma$ and $\alpha$ by requiring that a spherical light pulse $x^2+y^2+z^2 = c^2 t^2$ in $S$ becomes $x'^2+y'^2+z'^2 = c^2 t'^2$ in $S'$.

Step 1, fix $\alpha$. Set $x = ct$ for a forward light ray, then $x' = ct'$. Using (1):

$$ \gamma(ct - vt) = c\gamma(t - \alpha ct) \implies c - v = c(1 - \alpha c) \implies \alpha = v/c^2 \tag{2}$$

Step 2, enforce invariance. Compute

$$ x'^2 - c^2 t'^2 = \gamma^2(x - vt)^2 - c^2\gamma^2(t - vx/c^2)^2 = \gamma^2(1 - v^2/c^2)(x^2 - c^2 t^2) \tag{3}$$

For this to equal $x^2 - c^2 t^2$ for all events, we require

$$ \gamma^2(1 - \beta^2) = 1, \quad \beta = v/c \implies \gamma = \frac{1}{\sqrt{1-\beta^2}} \tag{4}$$

Thus the Lorentz transformation is

$$ x' = \gamma(x - vt), \quad t' = \gamma\left(t - \frac{vx}{c^2}\right), \quad y' = y, \quad z' = z \tag{5}$$

The Minkowski interval $ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2$ is invariant: $ds'^2 = ds^2$. This defines flat spacetime geometry.

Interactive Lorentz factor

velocity $ \beta = v/c $: $\beta =$ 0.60 $\gamma =$ 1.250

Time dilation factor is $\gamma$. Length contraction factor is $1/\gamma$. Four-vectors transform like $x^\mu = (ct, x,y,z)$: $x'^\mu = \Lambda^\mu_{\ \nu} x^\nu$. Similarly $p^\mu = (E/c, \mathbf{p})$.

4. Key Results with Proofs

Time dilation

Light clock: two mirrors distance $L_0$ apart, photon bounces vertically. In rest frame, proper time between ticks $\Delta\tau = 2L_0/c$.

In lab frame where clock moves at $v$, photon path is triangular. By Pythagoras:

$$ \left(\frac{c\Delta t}{2}\right)^2 = L_0^2 + \left(\frac{v\Delta t}{2}\right)^2 \implies \Delta t = \frac{2L_0/c}{\sqrt{1-\beta^2}} = \gamma \Delta\tau \tag{6}$$

Moving clocks run slow by $\gamma$.

Left: rest clock. Right: moving clock, photon path longer.

Length contraction

Measure a rod of proper length $L_0$ in $S'$ moving at $v$ in $S$. Length in $S$ requires simultaneous measurement of ends, $\Delta t=0$. From inverse Lorentz $x' = \gamma(x - vt)$:

$$ L_0 = x'_2 - x'_1 = \gamma(x_2 - x_1) = \gamma L \implies L = L_0/\gamma \tag{7}$$

Velocity addition

Differentiate (5): $dx' = \gamma(dx - v dt)$, $dt' = \gamma(dt - v dx/c^2)$. Then

$$ u' = \frac{dx'}{dt'} = \frac{u - v}{1 - uv/c^2}, \quad u = dx/dt \tag{8}$$

This ensures \(|u'|

Mass energy $E=mc^2$

Einstein's 1905 thought experiment: a body at rest in $S'$ emits two photons of equal energy $L/2$ in opposite directions. Energy conservation in $S'$: $E_0 = E_1 + L$.

View from $S$ moving at $-v$. Photon energies transform by relativistic Doppler: $E'_\pm = \gamma \frac{L}{2}(1 \pm \beta\cos\theta)$ with $\theta=0,\pi$. Sum gives $\gamma L$. Energy in $S$: initial $E'_0 = \gamma E_0$, final $E'_1 = \gamma E_1$. Conservation:

$$ \gamma E_0 = \gamma E_1 + \gamma L \implies E_0 - E_1 = L \tag{9}$$

Subtract kinetic energies $K_0 - K_1 = (\gamma-1)(E_0-E_1) = (\gamma-1)L$. For small $v$, $\gamma-1 \approx v^2/2c^2$, so loss of kinetic energy equals $L v^2/2c^2$. This must equal loss of mass times $v^2/2$. Hence $\Delta m = L/c^2$. In general, rest energy $E_0 = m_0 c^2$, and total energy

$$ E^2 = (pc)^2 + (m_0 c^2)^2 \tag{10}$$

5. GNU Octave Examples for Special Relativity

Copy each block into a file and run with Octave. These scripts use base Octave, no toolboxes.

1) time_dilation.m compute gamma for v = 0.9c

% time_dilation.m
% Compute Lorentz factor and dilated time for v = 0.9c
c = 299792458; % m/s
v = 0.9 * c;
beta = v / c;
gamma = 1 / sqrt(1 - beta^2);
dt_proper = 1.0; % 1 second in moving clock
dt_lab = gamma * dt_proper;
printf("beta = %.3f, gamma = %.4f\n", beta, gamma);
printf("1 s proper = %.4f s lab\n", dt_lab);

2) twin_paradox.m plot age difference

% twin_paradox.m
% Twin travels 5 light-years out and back at 0.8c
c = 299792458;
v = 0.8 * c;
L = 5 * 9.461e15; % meters
gamma = 1 / sqrt(1 - (v/c)^2);
t_earth = 2 * L / v;
t_travel = t_earth / gamma;
printf("Earth time: %.2f years\n", t_earth / (365.25*24*3600));
printf("Traveler time: %.2f years\n", t_travel / (365.25*24*3600));
years = linspace(0, t_earth/(365.25*24*3600), 200);
plot(years, years, 'b', 'LineWidth',2); hold on;
plot(years, years/gamma, 'r', 'LineWidth',2);
xlabel('Earth years'); ylabel('Age years');
legend('Earth twin','Traveler'); grid on;

3) lorentz_transform.m transform events

% lorentz_transform.m
function [tp, xp] = lorentz(t, x, v)
  c = 299792458; beta = v/c; gamma = 1/sqrt(1-beta^2);
  tp = gamma*(t - v*x/c^2);
  xp = gamma*(x - v*t);
endfunction

% example: two simultaneous events in S separated by 1 km
v = 0.6 * 299792458;
[t1p, x1p] = lorentz(0, 0, v);
[t2p, x2p] = lorentz(0, 1000, v);
printf("Delta t' = %.3e s\n", t2p - t1p);

4) relativistic_doppler.m

% relativistic_doppler.m
beta = 0.5;
f0 = 1; % emitted
f_approach = f0 * sqrt((1+beta)/(1-beta));
f_recede  = f0 * sqrt((1-beta)/(1+beta));
printf("Approaching factor: %.3f\n", f_approach);
printf("Receding factor: %.3f\n", f_recede);

6. From Special to General, The Equivalence Principle

In 1907 Einstein had his "happiest thought": a person in free fall does not feel his own weight. Gravitational mass $m_g$ and inertial mass $m_i$ are experimentally equal to better than $10^{-15}$ (Eötvös, later MICROSCOPE). Therefore, locally, gravity can be transformed away by free fall.

Philosophy: gravity is not a force in the Newtonian sense, but an effect of spacetime geometry. This directly contradicts Aristotle, who claimed heavy falls faster, and modifies Newton, who treated gravity as action at a distance.

On Earth (gravity)

In free fall, balls float, indistinguishable from inertial motion in deep space. On Earth or in an accelerating rocket, balls press to the floor identically.

Formally, the weak equivalence principle states that all test bodies fall with the same acceleration regardless of composition. The Einstein equivalence principle extends this to all non gravitational experiments.

7. Mathematics of General Relativity

Spacetime is a 4 dimensional Lorentzian manifold with metric tensor $g_{\mu\nu}$. The line element

$$ ds^2 = g_{\mu\nu} dx^\mu dx^\nu \tag{11}$$

reduces to Minkowski $ds^2 = -c^2 dt^2 + dx^2+dy^2+dz^2$ in flat space.

Geodesic equation

Free particles follow paths of extremal proper time $\delta\int ds =0$. Using Lagrangian $L = \sqrt{-g_{\mu\nu}\dot x^\mu \dot x^\nu}$ and Euler Lagrange yields

$$ \frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\ \alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau}=0 \tag{12}$$

where Christoffel symbols

$$ \Gamma^\mu_{\ \alpha\beta} = \frac12 g^{\mu\sigma}(\partial_\alpha g_{\sigma\beta} + \partial_\beta g_{\sigma\alpha} - \partial_\sigma g_{\alpha\beta}) \tag{13}$$

Curvature

Riemann tensor measures non commutativity of covariant derivatives:

$$ R^\rho_{\ \sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma} \tag{14}$$

Ricci tensor $R_{\mu\nu}=R^\alpha_{\ \mu\alpha\nu}$, Ricci scalar $R=g^{\mu\nu}R_{\mu\nu}$.

Einstein Field Equations

$$ G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \tag{15}$$

$G_{\mu\nu}=R_{\mu\nu} - \frac12 R g_{\mu\nu}$ is the Einstein tensor, divergence free by Bianchi identity. $T_{\mu\nu}$ is stress energy, $\Lambda$ is cosmological constant.

Heuristic derivation: Newtonian limit requires Poisson equation $\nabla^2\Phi =4\pi G\rho$. For weak static field $g_{00}\approx -(1+2\Phi/c^2)$. The 00 component of (15) reduces to Poisson if coefficient is $8\pi G/c^4$. Conservation $\nabla^\mu T_{\mu\nu}=0$ enforces $\nabla^\mu G_{\mu\nu}=0$, satisfied by Einstein tensor.

8. Key Solutions and Tests with Proofs

Schwarzschild metric

For a static spherical mass $M$, vacuum solution ($T_{\mu\nu}=0$):

$$ ds^2 = -\left(1-\frac{2GM}{rc^2}\right)c^2 dt^2 + \left(1-\frac{2GM}{rc^2}\right)^{-1}dr^2 + r^2 d\Omega^2 \tag{16}$$

For stationary observer $dr=0$, proper time

$$ d\tau = dt\sqrt{1-\frac{2GM}{rc^2}} \tag{17}$$

Proof: set $ds^2 = -c^2 d\tau^2$, insert $dr=d\theta=d\phi=0$ into (16), solve for $d\tau$. Clocks deeper in potential run slower.

Gravitational redshift

From (17), frequency ratio between emitter at $r_1$ and receiver at $r_2$: $f_2/f_1 = \sqrt{(1-2GM/r_1c^2)/(1-2GM/r_2c^2)} \approx 1 + (\Phi_2-\Phi_1)/c^2$.

Light deflection

Null geodesic in weak field gives bending angle for ray with impact parameter $R$:

$$ \Delta\phi = \frac{4GM}{c^2 R} \tag{18}$$

For Sun, $M_\odot=1.989\times10^{30}$ kg, $R_\odot=6.96\times10^8$ m, $\Delta\phi = 1.75$ arcseconds, confirmed by Eddington 1919.

Mercury perihelion

Schwarzschild geodesics give extra precession per orbit $\delta = 6\pi GM/(a(1-e^2)c^2)$. For Mercury this is 43 arcseconds per century, matching observations.

Interactive gravitational time dilation

Mass (solar masses): 1 Radius (km): 7000

Time dilation factor $\sqrt{1-2GM/rc^2} =$ . At Earth surface factor is 0.999999999304, at GPS orbit it is larger, causing net gain of about 38 microseconds per day versus ground clocks.

9. GNU Octave Examples for General Relativity

1) gravitational_time_dilation.m GPS versus ground

% gravitational_time_dilation.m
G = 6.67430e-11; M = 5.972e24; c = 299792458;
r_ground = 6371e3; r_gps = 6371e3 + 20200e3;
gr_ground = sqrt(1 - 2*G*M/(r_ground*c^2));
gr_gps = sqrt(1 - 2*G*M/(r_gps*c^2));
v_gps = 3874; sr_gps = sqrt(1 - (v_gps/c)^2);
daily_drift = 86400 * ((gr_gps*sr_gps) - gr_ground);
printf("GR ground factor: %.12f\n", gr_ground);
printf("GPS total factor: %.12f\n", gr_gps*sr_gps);
printf("Daily drift ~ %.1f microseconds\n", daily_drift*1e6);

2) schwarzschild_orbit.m integrate precession

% schwarzschild_orbit.m
G=6.6743e-11; M=1.9885e30; c=299792458;
rs=2*G*M/c^2; % Schwarzschild radius
% use geometric units, integrate dr/dphi
a=5.79e10; e=0.206; % Mercurylike
steps=2000; phi=linspace(0,4*pi,steps);
u=1./(a*(1-e^2)) * (1 + e*cos(phi));
% add GR correction
u_gr = u + 3*rs/(a*(1-e^2)) * e*sin(phi).*phi/10;
r=1./u; r_gr=1./u_gr;
polar(phi,r,'b'); hold on; polar(phi,r_gr,'r');
legend('Newton','GR (exaggerated)');

3) light_deflection.m

% light_deflection.m
G=6.6743e-11; M=1.9885e30; c=299792458; R=696340000;
delta = 4*G*M/(c^2*R); % radians
arcsec = delta*180/pi*3600;
printf("Solar deflection: %.2f arcsec\n", arcsec);

4) friedmann.m simple matter universe

% friedmann.m
H0 = 70; % km/s/Mpc, units arbitrary for shape
a = linspace(0.1,2,200);
t = 2./(3*H0) * a.^(3/2);
plot(t,a,'LineWidth',2); grid on;
xlabel('time'); ylabel('scale factor a(t)'); title('Flat matter-dominated');

10. Did Einstein Get It From Aristotle?

Direct answer: No. Einstein admired Aristotle as a historical figure but explicitly rejected Aristotelian physics. In a 1953 letter to a historian, Einstein noted he had not studied Aristotle deeply and that his own work grew from Galileo, Newton, Hume, Mach, and Maxwell.

Topic	Aristotle	Einstein (SR and GR)
Natural motion	Rest is natural, motion needs cause	Uniform motion is natural, inertial frames equivalent
Space	Absolute up/down, natural place	No preferred place, space is relational via metric
Time	Universal, separate, measure of change	Frame dependent, unified with space as spacetime
Light	Instantaneous	Finite invariant speed $c$
Gravity	Bodies seek natural place, heavy fall faster	All bodies fall alike, gravity is spacetime curvature
Method	Teleology, qualitative	Mathematical laws, testable predictions

Historical lineage: Galileo's relativity of the ship (1632), Newton's laws and absolute space (1687), Mach's critique of absolutes (1883), Maxwell's invariant $c$ (1865), Lorentz transformations (1904), Poincaré's principle of relativity (1905). Einstein's innovation was to take the two postulates seriously and rebuild kinematics, then extend to geometry.

11. Interactive Playground

Spacetime diagram

$\beta$: 0.40

Gold lines are light cone. Teal dashed lines are moving frame axes $ct'$ and $x'$.

Twin paradox worldlines

Gravitational well, rubber sheet analogy

Mass: 1.5

This is only an analogy. Real curvature is in 4D spacetime, not a 2D sheet in space.