E = mc² Gμν + Λgμν = 8πG/c⁴ · Tμν

Einstein's Theory of Relativity

Special Relativity 1905 · General Relativity 1915 · Albert Einstein

I. Historical & Physical Motivation

By 1900, two pillars of physics stood in stark contradiction. Newtonian mechanics assumed absolute time and absolute space — a fixed cosmic stage. Maxwell's electromagnetism, completed in 1865, embedded the speed of light \(c = 1/\sqrt{\mu_0\varepsilon_0} \approx 2.998\times10^8\) m/s as an absolute constant. But constant relative to what?

The Michelson-Morley experiment (1887) found no evidence of the luminiferous aether — light's speed was identical in all directions, regardless of Earth's motion. Lorentz and Poincaré proposed ad-hoc transformation rules that preserved Maxwell's equations. Einstein, in 1905, made the radical move: start from the constancy of \(c\) as a postulate and derive everything else.

YearEventSignificance
1864Maxwell's equationsLight speed \(c\) predicted from electromagnetism
1887Michelson-MorleyNo aether drift detected — \(c\) is frame-independent
1905Einstein's SR paperZur Elektrodynamik bewegter Körper
1905Mass-energy paper\(E = mc^2\) — energy and mass are equivalent
1907Equivalence principleGravity and acceleration are locally indistinguishable
1915General RelativityGravity = curvature of spacetime
1916Gravitational wavesPredicted; detected by LIGO 2015 (100 years later)
1919Eddington eclipseLight bending confirmed GR; doubled Newtonian prediction

II. The Two Postulates of Special Relativity

Postulate I — Principle of Relativity

The laws of physics are identical in all inertial (non-accelerating) reference frames. No experiment can distinguish between being at rest and moving with constant velocity.

$$\text{Physics: Frame } S \longleftrightarrow \text{Frame } S' \text{ (moving at constant }v\text{)}$$
Postulate II — Constancy of the Speed of Light

The speed of light in vacuum \(c = 299{,}792{,}458\) m/s is the same in all inertial frames, regardless of the motion of the source or observer.

$$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} = \text{const in all frames}$$

These two postulates together force us to abandon absolute simultaneity, and lead inexorably to the Lorentz transformation, time dilation, length contraction, and the equivalence of mass and energy.

III. Spacetime & the Minkowski Metric

Hermann Minkowski (1908) gave Einstein's kinematics its geometric home: spacetime — a four-dimensional manifold where space and time are unified. An event is a point \((ct, x, y, z)\) in spacetime.

The Spacetime Interval

The fundamental invariant — the same in every inertial frame — is the spacetime interval:

$$\boxed{s^2 = -c^2\Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2}$$
Theorem — Invariance of the Interval

If \(s^2 = 0\) in frame \(S\) (a light signal), then \(s'^2 = 0\) in any frame \(S'\). By Postulate II, the Lorentz transformation must preserve \(s^2\).

Minkowski Metric Tensor

With signature \((-,+,+,+)\) the Minkowski metric is:

$$\eta_{\mu\nu} = \begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$$

The line element: \(\d s^2 = \eta_{\mu\nu}\,\d x^\mu\,\d x^\nu\). Greek indices \(\mu,\nu \in \{0,1,2,3\}\); \(x^0=ct\), \(x^1=x\), \(x^2=y\), \(x^3=z\).

Causal Structure — The Light Cone

IntervalNamePhysical Meaning
\(s^2 < 0\)TimelikeCausally connected; material particles travel this way
\(s^2 = 0\)Lightlike (null)Light travels on null geodesics
\(s^2 > 0\)SpacelikeNo causal connection; simultaneous in some frame

Proper Time

For a massive particle, the proper time \(\tau\) measured by its own clock satisfies:

$$\d\tau^2 = -\frac{\d s^2}{c^2} = \d t^2 - \frac{\d x^2+\d y^2+\d z^2}{c^2} = \d t^2\left(1-\frac{v^2}{c^2}\right)$$ $$\boxed{\d\tau = \frac{\d t}{\gamma},\quad \gamma = \frac{1}{\sqrt{1-v^2/c^2}} \ge 1}$$

IV. The Lorentz Transformation — Full Derivation

We derive the transformation between frame \(S\) and frame \(S'\) moving at velocity \(v\) along the \(x\)-axis, starting from the two postulates alone.

Derivation from First Principles

Step 1 — Linearity. Homogeneity of spacetime demands the transformation be linear: \(x' = ax + bt\), \(t' = cx + dt\) for constants \(a,b,c,d\).

Step 2 — Origin of \(S'\). At \(x'=0\), \(x = vt\), so \(b = -av\).

Step 3 — Light speed invariance. A light flash at origin at \(t=0\): \(x=ct\). In \(S'\): \(x'=ct'\). So:

$$ct' = c\cdot\frac{cx+bt}{c^2x/c+d} \implies \text{ratio must give } \frac{x'}{t'} = c$$

Step 4 — Symmetry. The inverse transformation must have the same form with \(v\to -v\). This fixes \(d = a\).

Step 5 — Determine \(a\). Consider an event at \(x=0\), measured in \(S'\): \(t' = at - 0 = at\). And from the inverse: \(x = a(x'+vt')\), giving \(0 = a(x'-(-v)t')\) consistently. Apply \(s^2\) invariance to a light signal: \(c^2t^2=x^2 \implies c^2t'^2=x'^2\):

$$(ct')^2 - x'^2 = (ct)^2-x^2 \implies a^2\left(1-\frac{v^2}{c^2}\right) = 1 \implies a = \gamma = \frac{1}{\sqrt{1-\beta^2}}$$

where \(\beta = v/c\).

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Lorentz Transformation (boost along x) $$\boxed{\begin{pmatrix}ct'\\x'\\y'\\z'\end{pmatrix} = \begin{pmatrix}\gamma&-\gamma\beta&0&0\\-\gamma\beta&\gamma&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}\begin{pmatrix}ct\\x\\y\\z\end{pmatrix}}$$

Explicitly: \(\quad ct' = \gamma(ct - \beta x),\quad x' = \gamma(x - vt),\quad y'=y,\quad z'=z\)

The Lorentz Factor \(\gamma\)

\(\beta = v/c\)\(\gamma\)Effect
0.101.0050.5% time dilation — negligible
0.501.15515.5% — measurable
0.862.000Clocks run at half speed
0.997.089Cosmic rays reach ground
0.999970.71LHC protons
0.9999999917453Highest-energy cosmic ray

V. Time Dilation

Theorem — Time Dilation

A clock at rest in \(S'\) ticks with proper time \(\Delta\tau\). An observer in \(S\) measures:

$$\boxed{\Delta t = \gamma\,\Delta\tau \ge \Delta\tau}$$

Moving clocks run slow. \(\gamma \ge 1\) always, with equality only at rest.

Proof

Clock at rest in \(S'\): two ticks at \((t_1', x'=0)\) and \((t_2', x'=0)\), so \(\Delta t' = \Delta\tau\). Apply inverse Lorentz: \(t = \gamma(t' + \beta x'/c)\). At \(x'=0\):

$$\Delta t = \gamma\,\Delta t' = \gamma\,\Delta\tau$$
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Experimental Confirmation

VI. Length Contraction

Theorem — Lorentz–Fitzgerald Contraction

A rod of rest length \(L_0\) (measured in its rest frame) has length in frame \(S\):

$$\boxed{L = \frac{L_0}{\gamma} \le L_0}$$

Contraction occurs only along the direction of motion.

Proof

Rod at rest in \(S'\), endpoints at \(x'=0\) and \(x'=L_0\). To measure length in \(S\), record both endpoints simultaneously in \(S\) at time \(t\).

From Lorentz: \(x' = \gamma(x - vt)\). At simultaneous time \(t\) in \(S\):

$$L_0 = x_2' - x_1' = \gamma(x_2 - vt) - \gamma(x_1 - vt) = \gamma(x_2 - x_1) = \gamma L$$ $$\implies L = L_0/\gamma$$
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Key point: Length contraction and time dilation are perspectival — they describe how measurements differ between frames. The spacetime interval \(s^2\) is the objective, frame-independent quantity.

VII. Relativity of Simultaneity

Two events simultaneous in \(S\) (\(\Delta t = 0\)) are generally not simultaneous in \(S'\):

$$\Delta t' = \gamma\left(\Delta t - \frac{v\,\Delta x}{c^2}\right)$$

For \(\Delta t = 0\): \(\Delta t' = -\gamma v\,\Delta x / c^2\). Events separated in space and simultaneous in one frame are time-ordered differently in others — there is no universal "now."

Ladder Paradox (Resolved)

A ladder of rest length \(L_0 > D\) (garage door width) can fit inside the garage when moving, because from the garage frame the ladder is Lorentz-contracted to \(L_0/\gamma < D\). From the ladder frame, the garage is contracted. The paradox resolves because the front and back door events are not simultaneous in the ladder frame — no contradiction. Causality is always preserved.

VIII. Relativistic Velocity Addition

Galilean addition \(u = u' + v\) would allow exceeding \(c\). The relativistic formula, derived directly from the Lorentz transformation, is:

Einstein Velocity Addition $$\boxed{u_x = \frac{u_x' + v}{1 + u_x' v/c^2},\qquad u_y = \frac{u_y'}{\gamma(1 + u_x' v/c^2)},\qquad u_z = \frac{u_z'}{\gamma(1 + u_x' v/c^2)}}$$
Derivation

Differentiate the Lorentz transformation: \(x = \gamma(x'+vt')\), \(t = \gamma(t'+vx'/c^2)\):

$$u_x = \frac{\d x}{\d t} = \frac{\gamma(\d x'+v\,\d t')}{\gamma(\d t'+v\,\d x'/c^2)} = \frac{u_x'+v}{1+u_x'v/c^2}$$
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Limiting case: \(u_x'=c\) (light in \(S'\)) gives \(u_x = (c+v)/(1+v/c) = c\). Light speed is invariant. ✓

IX. Four-Vectors & Covariant Notation

Four-vectors are objects that transform under Lorentz boosts. A contravariant four-vector \(A^\mu\) has upper index; a covariant \(A_\mu = \eta_{\mu\nu}A^\nu\) has lower index. The Lorentz-invariant inner product is:

$$A \cdot B = A^\mu B_\mu = \eta_{\mu\nu}A^\mu B^\nu = -A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3$$

Key Four-Vectors

NameSymbolComponentsInvariant
Position\(x^\mu\)\((ct,\,x,\,y,\,z)\)\(s^2 = x^\mu x_\mu\)
Velocity\(u^\mu = \d x^\mu/\d\tau\)\(\gamma c(\,1,\,\bv{v}/c)\)\(u^\mu u_\mu = -c^2\)
Momentum\(p^\mu = m u^\mu\)\((E/c,\,\bv{p})\)\(p^\mu p_\mu = -m^2c^2\)
Wave\(k^\mu\)\((\omega/c,\,\bv{k})\)\(k^\mu k_\mu = 0\) (light)
Current\(J^\mu\)\((c\rho,\,\bv{J})\)Continuity: \(\partial_\mu J^\mu=0\)

Electromagnetic Four-Tensor

Maxwell's equations unify into two tensor equations using the field-strength tensor \(F^{\mu\nu}\) and the dual \(\tilde{F}^{\mu\nu}\):

$$\partial_\nu F^{\mu\nu} = \mu_0 J^\mu, \qquad \partial_\nu \tilde{F}^{\mu\nu} = 0$$

This is the complete, Lorentz-covariant form of Maxwell's 4 equations.

X. Relativistic Momentum & Energy

Relativistic Momentum $$\bv{p} = \gamma m\bv{v} = \frac{m\bv{v}}{\sqrt{1-v^2/c^2}}$$

As \(v\to c\), \(\bv{p}\to\infty\) — no finite force can accelerate a massive body to \(c\). The relativistic second law becomes:

$$\bv{F} = \frac{\d\bv{p}}{\d t} = \frac{\d(\gamma m\bv{v})}{\d t}$$

Energy-Momentum Relation

The energy of a free particle is \(E = \gamma mc^2\). Together with momentum:

$$\boxed{E^2 = (pc)^2 + (mc^2)^2}$$

This is the mass-shell condition — a Lorentz-invariant hyperbola in \((E,p)\) space. For massless particles (\(m=0\), e.g. photons): \(E = pc\).

Kinetic Energy

$$T = E - mc^2 = (\gamma-1)mc^2$$

For \(v\ll c\): \(\gamma \approx 1+\frac{v^2}{2c^2}\), so \(T\approx \frac{1}{2}mv^2\). ✓ Newtonian limit recovered.

XI. E = mc² — Full Derivation

Einstein's Mass-Energy Equivalence (1905) $$\boxed{E_0 = mc^2}$$

The rest energy of a body is proportional to its mass — mass and energy are the same physical quantity in different units.

Derivation via Photon Emission

Setup: A body at rest in frame \(S\) emits two photons in opposite directions, each with energy \(L/2\). By symmetry, the body remains at rest in \(S\).

In frame \(S'\) moving at velocity \(v\): Using the relativistic Doppler formula \(E' = \gamma E(1\mp\beta)\) for the two photons (forward/backward):

$$E_\text{fwd}' = \frac{L}{2}\gamma(1+\beta),\qquad E_\text{back}' = \frac{L}{2}\gamma(1-\beta)$$ $$E_1' + E_2' = L\gamma$$

Energy conservation in \(S'\): Before emission, the body has kinetic energy \(T_i' = E_i' - M c^2\). After: \(T_f' = E_f' - M'c^2\). Conservation:

$$T_i' - T_f' = L\gamma - L = L(\gamma-1) \approx \frac{1}{2}\frac{L}{c^2}v^2$$

Newtonian limit: The body loses kinetic energy \(\frac{1}{2}\Delta m \cdot v^2\). Comparing: \(\Delta m = L/c^2\).

Conclusion: Emitting energy \(L\) reduces mass by \(\Delta m = L/c^2\), i.e., \(\Delta E = \Delta m \cdot c^2\). Integrating from rest: \(E_0 = mc^2\).

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Nuclear consequences: In the fission of one ²³⁵U nucleus, mass defect \(\Delta m \approx 0.19\) u = \(3.15\times10^{-28}\) kg, yielding \(E = \Delta m c^2 \approx 2.8\times10^{-11}\) J = 177 MeV per nucleus. One kilogram of ²³⁵U releases \(\sim 8\times10^{13}\) J — equivalent to 20,000 tons of TNT.

XII. Relativistic Doppler Effect

For a source moving at velocity \(v\) toward/away from observer at angle \(\theta\) to line of sight:

$$f_\text{obs} = \frac{f_0}{\gamma(1 - \beta\cos\theta)}$$

Longitudinal cases:

$$f_\text{approach} = f_0\sqrt{\frac{1+\beta}{1-\beta}},\qquad f_\text{recede} = f_0\sqrt{\frac{1-\beta}{1+\beta}}$$

Transverse Doppler (\(\theta=90°\)): \(f_\text{obs} = f_0/\gamma < f_0\) — purely relativistic, no classical analog. A transversely moving source is redshifted due to time dilation alone. Confirmed by Ives–Stilwell experiment (1938).

XIII. The Twin Paradox

Alice stays on Earth; Bob travels at \(\beta = 0.8\) (\(\gamma = 5/3\)) to a star 4 light-years away and returns. Earth time elapsed: \(T = 2\times4/0.8 = 10\) yr. Bob's proper time: \(\tau = T/\gamma = 6\) yr.

Resolution

This is not a paradox — the situations are asymmetric. Bob must accelerate to reverse direction; he changes inertial frames. Alice remains in a single inertial frame throughout. The calculation using proper time along each worldline gives an unambiguous answer: Bob ages 6 years, Alice ages 10. The spacetime interval along Alice's worldline is longer (more proper time elapses).

Quantitatively, proper time is the length of a worldline in spacetime:

$$\tau = \int\d\tau = \int\sqrt{1-v(t)^2/c^2}\,\d t$$

The straight worldline (Alice) maximises proper time — the clock postulate / twin effect.

XIV. The Equivalence Principle — Path to GR

Weak Equivalence Principle (WEP)

Gravitational and inertial masses are equal: \(m_\text{grav} = m_\text{inert}\). All bodies fall with the same acceleration in a gravitational field, regardless of composition. Tested to \(10^{-15}\) precision by MICROSCOPE (2022).

Einstein Equivalence Principle (EEP)

In a freely falling reference frame, all non-gravitational physics reduces to special relativity. Gravity is locally indistinguishable from acceleration. A person in a sealed box cannot distinguish uniform gravity from being accelerated.

Consequences of EEP

Gravitational time dilation: A photon climbing height \(h\) in gravity \(g\) loses energy, hence frequency: \(\Delta f/f = -gh/c^2 = -\Delta\Phi/c^2\). Clocks higher in a gravitational field run faster.

$$\frac{\Delta f}{f} = \frac{\Delta\Phi}{c^2},\quad \frac{\d\tau_1}{\d\tau_2} = \sqrt{\frac{1-2\Phi_1/c^2}{1-2\Phi_2/c^2}}$$

Gravitational light bending: If gravity = acceleration, and acceleration bends light paths (in the accelerated frame), then gravity must bend light. EEP predicts \(\delta\theta = 2GM/(Rc^2)\) — exactly half the GR result; the second factor of 2 comes from spacetime curvature.

XV. Tensor Mathematics

Einstein Summation Convention

Repeated indices (one up, one down) are summed: \(A^\mu B_\mu \equiv \sum_{\mu=0}^3 A^\mu B_\mu\).

Tensors — Definition

Rank-(p,q) Tensor

A tensor \(T^{\mu_1\cdots\mu_p}{}_{\nu_1\cdots\nu_q}\) transforms under coordinate change \(x^\mu \to x^{\mu'}\) as:

$$T^{\mu_1'\cdots\mu_p'}{}_{\nu_1'\cdots\nu_q'} = \frac{\partial x^{\mu_1'}}{\partial x^{\mu_1}}\cdots\frac{\partial x^{\nu_q}}{\partial x^{\nu_q'}} T^{\mu_1\cdots\mu_p}{}_{\nu_1\cdots\nu_q}$$

Covariant Derivative

In curved spacetime, partial derivatives do not transform as tensors. The covariant derivative \(\nabla_\mu\) does:

$$\nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu_{\mu\lambda}V^\lambda$$ $$\nabla_\mu V_\nu = \partial_\mu V_\nu - \Gamma^\lambda_{\mu\nu}V_\lambda$$

where \(\Gamma^\lambda_{\mu\nu}\) are the Christoffel symbols.

Metric Compatibility & Torsion-Free Conditions

In GR we use the Levi-Civita connection, which satisfies:

$$\nabla_\rho g_{\mu\nu} = 0 \quad\text{(metric compatible)}, \qquad \Gamma^\lambda_{\mu\nu} = \Gamma^\lambda_{\nu\mu}\quad\text{(torsion-free)}$$

XVI. The Metric Tensor

The metric tensor \(g_{\mu\nu}\) encodes the geometry of spacetime — distances, angles, volumes:

$$\d s^2 = g_{\mu\nu}\,\d x^\mu\,\d x^\nu$$

The inverse metric \(g^{\mu\nu}\) satisfies \(g^{\mu\lambda}g_{\lambda\nu} = \delta^\mu{}_\nu\). The metric raises and lowers indices:

$$V_\mu = g_{\mu\nu}V^\nu, \qquad V^\mu = g^{\mu\nu}V_\nu$$

Important Metrics

SpacetimeMetric \(\d s^2\)
Flat (Minkowski)\(-c^2\d t^2+\d x^2+\d y^2+\d z^2\)
Schwarzschild\(-\left(1-\frac{r_s}{r}\right)c^2\d t^2+\frac{\d r^2}{1-r_s/r}+r^2\d\Omega^2\)
Kerr (rotating BH)(complicated — involves spin \(a=J/Mc\))
FLRW (cosmology)\(-c^2\d t^2+a(t)^2\left[\frac{\d r^2}{1-kr^2}+r^2\d\Omega^2\right]\)
de Sitter (\(\Lambda>0\))\(-\left(1-\frac{\Lambda r^2}{3}\right)c^2\d t^2+\cdots\)

where \(\d\Omega^2 = \d\theta^2+\sin^2\theta\,\d\phi^2\) is the unit 2-sphere metric.

XVII. Christoffel Symbols

The Christoffel symbols (connection coefficients) measure how basis vectors change across spacetime. For the Levi-Civita connection:

Christoffel Symbol of the Second Kind $$\boxed{\Gamma^\lambda{}_{\mu\nu} = \frac{1}{2}g^{\lambda\rho}\!\left(\partial_\mu g_{\nu\rho}+\partial_\nu g_{\mu\rho}-\partial_\rho g_{\mu\nu}\right)}$$

They are symmetric in the lower indices: \(\Gamma^\lambda_{\mu\nu} = \Gamma^\lambda_{\nu\mu}\). In flat spacetime (Cartesian coordinates), all \(\Gamma^\lambda_{\mu\nu} = 0\). In curved spacetime or curvilinear coordinates, they are non-zero.

Example: Schwarzschild Christoffel Symbols (non-zero)

For the Schwarzschild metric with \(f(r)=1-r_s/r\):

$$\Gamma^t_{tr} = \frac{r_s}{2r(r-r_s)},\quad \Gamma^r_{tt} = \frac{c^2 r_s(r-r_s)}{2r^3},\quad \Gamma^r_{rr} = -\frac{r_s}{2r(r-r_s)}$$ $$\Gamma^r_{\theta\theta} = -(r-r_s),\quad \Gamma^r_{\phi\phi} = -(r-r_s)\sin^2\theta$$ $$\Gamma^\theta_{\phi\phi} = -\sin\theta\cos\theta,\quad \Gamma^\phi_{r\phi} = \Gamma^\phi_{\phi r} = \frac{1}{r},\quad \Gamma^\phi_{\theta\phi} = \cot\theta$$

XVIII. The Geodesic Equation

In GR, freely falling particles follow geodesics — the straightest possible paths in curved spacetime. "Gravity is not a force; it is curvature of spacetime that particles follow naturally."

Geodesic Equation $$\boxed{\frac{\d^2 x^\lambda}{\d\tau^2} + \Gamma^\lambda{}_{\mu\nu}\frac{\d x^\mu}{\d\tau}\frac{\d x^\nu}{\d\tau} = 0}$$
Derivation via Variational Principle

The geodesic is the curve that extremises proper time \(\tau = \int\sqrt{-g_{\mu\nu}\dot x^\mu\dot x^\nu}\,\d\lambda\). Taking the Euler-Lagrange equations for the Lagrangian \(\mathcal{L} = g_{\mu\nu}\dot x^\mu\dot x^\nu\):

$$\frac{\d}{\d\tau}\frac{\partial\mathcal{L}}{\partial \dot x^\lambda} - \frac{\partial\mathcal{L}}{\partial x^\lambda} = 0$$ $$2g_{\lambda\nu}\ddot x^\nu + 2(\partial_\mu g_{\lambda\nu})\dot x^\mu\dot x^\nu - \partial_\lambda g_{\mu\nu}\dot x^\mu\dot x^\nu = 0$$

Multiply by \(g^{\rho\lambda}/2\), use symmetry of \(\dot x^\mu\dot x^\nu\) to combine derivative terms:

$$\ddot x^\rho + \frac{1}{2}g^{\rho\lambda}(\partial_\mu g_{\lambda\nu}+\partial_\nu g_{\lambda\mu}-\partial_\lambda g_{\mu\nu})\dot x^\mu\dot x^\nu = 0$$ $$\ddot x^\rho + \Gamma^\rho_{\mu\nu}\dot x^\mu\dot x^\nu = 0 \qquad \checkmark$$
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Newtonian Limit

In the weak-field, slow-motion limit: \(g_{00} \approx -(1+2\Phi/c^2)\), all other \(\Gamma\approx 0\) except \(\Gamma^i_{00}\approx -\partial_i\Phi/c^2\). The geodesic equation becomes:

$$\ddot{\bv{r}} = -\nabla\Phi$$

Newtonian gravity recovered! Einstein's gravity contains Newton's as a special case.

XIX. The Riemann Curvature Tensor

The Riemann tensor measures the failure of parallel transport around a closed loop — i.e., intrinsic curvature:

Riemann Tensor $$\boxed{R^\rho{}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}}$$

Symmetries of the Riemann Tensor

$$R_{\rho\sigma\mu\nu} = -R_{\sigma\rho\mu\nu} = -R_{\rho\sigma\nu\mu} = R_{\mu\nu\rho\sigma}$$

In 4D: 20 independent components (from 256 = 4⁴).

Bianchi Identity

$$\nabla_\lambda R_{\rho\sigma\mu\nu} + \nabla_\rho R_{\sigma\lambda\mu\nu} + \nabla_\sigma R_{\lambda\rho\mu\nu} = 0$$

Contracting twice with \(g^{\mu\rho}g^{\nu\sigma}\) yields the contracted Bianchi identity: \(\nabla_\mu G^{\mu\nu} = 0\) — essential for energy-momentum conservation in GR.

Ricci Tensor & Scalar

$$R_{\mu\nu} = R^\lambda{}_{\mu\lambda\nu} \quad\text{(Ricci tensor, contraction of Riemann)}$$ $$R = g^{\mu\nu}R_{\mu\nu} \quad\text{(Ricci scalar)}$$

Weyl Tensor

$$C_{\rho\sigma\mu\nu} = R_{\rho\sigma\mu\nu} - \text{(terms involving }R_{\mu\nu}\text{ and }R)$$

The Weyl tensor encodes "free" gravitational degrees of freedom — tidal forces and gravitational waves — and vanishes in conformally flat spacetimes.

XX. Einstein's Field Equations

Einstein Field Equations (1915) $$\boxed{G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}\,T_{\mu\nu}}$$

where \(G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R\) is the Einstein tensor, \(\Lambda\) is the cosmological constant, and \(T_{\mu\nu}\) is the stress-energy tensor.

This is 10 coupled, nonlinear PDEs (symmetric rank-2 tensor in 4D). The left side encodes spacetime geometry; the right side encodes matter and energy content. Wheeler's summary: "Spacetime tells matter how to move; matter tells spacetime how to curve."

Stress-Energy Tensor

$$T^{\mu\nu} = \begin{pmatrix} \rho c^2 & \text{momentum flux} \\ \text{momentum density} & \text{stress tensor}\end{pmatrix}$$

For a perfect fluid with energy density \(\rho\), pressure \(P\), and four-velocity \(u^\mu\):

$$T^{\mu\nu} = \left(\rho + \frac{P}{c^2}\right)u^\mu u^\nu + P g^{\mu\nu}$$

Linearised GR — Gravitational Waves

For a weak perturbation \(g_{\mu\nu} = \eta_{\mu\nu}+h_{\mu\nu}\), \(|h_{\mu\nu}|\ll 1\), in Lorenz gauge \(\partial^\nu\bar h_{\mu\nu}=0\):

$$\Box\,\bar{h}_{\mu\nu} = -\frac{16\pi G}{c^4}T_{\mu\nu}$$

This is a wave equation! Gravitational waves propagate at speed \(c\).

Motivation — from Bianchi Identity

The Einstein tensor \(G^{\mu\nu} = R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R\) satisfies \(\nabla_\mu G^{\mu\nu}=0\) (contracted Bianchi identity). The field equation \(G^{\mu\nu} = \kappa T^{\mu\nu}\) then implies \(\nabla_\mu T^{\mu\nu}=0\) — energy-momentum conservation is built in.

The constant \(\kappa = 8\pi G/c^4\) is fixed by demanding the Newtonian limit \(\nabla^2\Phi = 4\pi G\rho\) be reproduced.

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XXI. The Schwarzschild Solution

Karl Schwarzschild (1916, on the Russian front) found the first exact solution to Einstein's equations — the vacuum solution (\(T_{\mu\nu}=0\)) exterior to a spherically symmetric, non-rotating mass \(M\):

Schwarzschild Metric $$\boxed{\d s^2 = -\!\left(1-\frac{r_s}{r}\right)c^2\d t^2 + \frac{\d r^2}{1-r_s/r} + r^2\!\left(\d\theta^2+\sin^2\!\theta\,\d\phi^2\right)}$$

Schwarzschild radius: \(\quad r_s = \dfrac{2GM}{c^2}\)

ObjectMass \(M\)\(r_s\)Actual radius
Earth\(5.97\times10^{24}\) kg8.87 mm6371 km
Sun\(1.99\times10^{30}\) kg2.95 km696,000 km
Sgr A*\(4.1\times10^6\,M_\odot\)12.1 M kmCompact — BH
M87*\(6.5\times10^9\,M_\odot\)19.2 GmImaged in 2019

Orbital Mechanics in Schwarzschild

For equatorial orbits \((\theta=\pi/2)\), there are two conserved quantities:

$$\tilde{E} = \left(1-\frac{r_s}{r}\right)\frac{c^2\,\d t}{\d\tau},\quad \tilde{L} = r^2\frac{\d\phi}{\d\tau}$$

The radial equation (from \(g_{\mu\nu}\dot x^\mu\dot x^\nu = -c^2\)):

$$\left(\frac{\d r}{\d\tau}\right)^2 = \tilde{E}^2 - c^2\left(1-\frac{r_s}{r}\right)\left(1+\frac{\tilde{L}^2}{r^2 c^2}\right)$$

Mercury's Perihelion Advance

The GR correction to the orbit equation gives a precessing ellipse. Per orbit:

$$\Delta\phi = \frac{6\pi G M}{a(1-e^2)c^2} = \frac{3\pi r_s}{a(1-e^2)}$$

For Mercury: \(\Delta\phi = 43.0''/\text{century}\). Observed: \(43.1''/\text{century}\). Agreement is exact. ✓

Gravitational Redshift

A photon emitted at \(r_1\) and received at \(r_2 > r_1\):

$$\frac{\lambda_2}{\lambda_1} = \frac{f_1}{f_2} = \sqrt{\frac{1-r_s/r_1}{1-r_s/r_2}} > 1$$

Light climbing out of a gravitational well is redshifted. Confirmed to \(10^{-4}\) precision by Pound-Rebka (1959) and to \(2\times10^{-5}\) by the Galileo satellite (2021).

XXII. Black Holes

When a mass is compressed inside its Schwarzschild radius, \(r_s = 2GM/c^2\), a black hole forms. The surface \(r=r_s\) is the event horizon — a null surface from which nothing (not even light) can escape.

Properties of the Event Horizon

Hawking Radiation

Quantum field theory in curved spacetime (Hawking, 1974) shows black holes emit thermal radiation at temperature:

$$T_H = \frac{\hbar c^3}{8\pi G M k_B} = \frac{6.17\times10^{-8}}{M/M_\odot}\ \text{K}$$

Stellar-mass BHs: \(T_H \sim 10^{-8}\) K — unobservably cold. Primordial micro-BHs of mass \(\sim10^{11}\) kg would be evaporating today.

Penrose Diagram

The maximally extended Schwarzschild solution (Kruskal-Szekeres coordinates) has four regions: our exterior universe (I), the interior/future singularity (II), a white hole (III, time-reverse of BH), and a second exterior universe (IV) — connected only through the Einstein-Rosen bridge (wormhole).

XXIII. Experimental Tests of GR

TestPredictionResult
Mercury perihelion advance43.0″/century43.1″/century ✓
Deflection of light (Sun)1.7505″1.75″ ± 0.09″ ✓
Gravitational redshift\(\Delta f/f = \Delta\Phi/c^2\)Confirmed \(10^{-4}\) (Pound-Rebka) ✓
Shapiro time delayRadar signal delay near SunCassini: \(10^{-5}\) agreement ✓
Gravitational wavesInspiral chirp from mergersLIGO 2015: GW150914 ✓
Black hole shadowPhoton ring \(r = 3r_s/2\)EHT M87* 2019, Sgr A* 2022 ✓
Frame dragging (Kerr)Lense-Thirring precessionGravity Probe B ✓
Strong-field GR (BNS)Inspiral waveformGW170817 ✓

XXIV. Gravitational Waves

GWs are ripples in spacetime curvature — perturbations \(h_{\mu\nu}\) of the metric propagating at \(c\). They carry energy and angular momentum away from accelerating masses.

Transverse-Traceless Gauge

In the TT gauge, propagating along \(z\), only two polarization modes:

$$h_{\mu\nu}^{TT} = \begin{pmatrix}0&0&0&0\\0&h_+&h_\times&0\\0&h_\times&-h_+&0\\0&0&0&0\end{pmatrix}e^{i(kz-\omega t)}$$

\(h_+\) (plus polarization) stretches \(x\) and squeezes \(y\); \(h_\times\) does so at 45°.

Quadrupole Formula

$$h_{ij}^{TT} = \frac{2G}{c^4 r}\ddot{I}_{ij}^{TT}(t-r/c)$$

where \(I_{ij} = \int\rho x_i x_j\,\d^3x\) is the mass quadrupole moment. Power radiated:

$$P = \frac{G}{5c^5}\langle\dddot{I}_{ij}\dddot{I}^{ij}\rangle = -\frac{\d E}{\d t}$$

GW150914 — First Detection

LIGO detected the merger of two \(\sim 30\,M_\odot\) black holes at \(d\approx 410\) Mpc. At merger:

XXV. Cosmology — FLRW Metric & Friedmann Equations

The large-scale universe is homogeneous and isotropic (Cosmological Principle). The metric is the Friedmann–Lemaître–Robertson–Walker (FLRW) form:

$$\d s^2 = -c^2\d t^2 + a(t)^2\left[\frac{\d r^2}{1-kr^2}+r^2\d\Omega^2\right]$$

where \(a(t)\) is the scale factor (\(a=1\) today), and \(k = 0,\pm1\) for flat, closed, open universe.

Friedmann Equations

Inserting FLRW into Einstein's equations with a perfect fluid:

$$\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} \quad\text{(Friedmann I)}$$ $$\frac{\ddot a}{a} = -\frac{4\pi G}{3}\left(\rho+\frac{3P}{c^2}\right)+\frac{\Lambda c^2}{3} \quad\text{(Friedmann II)}$$

The Hubble constant \(H_0 = \dot a/a|_0 \approx 67.4\text{ km/s/Mpc}\) (Planck 2018). The universe is currently accelerating (\(\ddot a > 0\)) driven by dark energy (\(\Lambda > 0\)).

Cosmological Parameters

ComponentDensity parameter \(\Omega\)Equation of state \(w=P/\rho c^2\)
Matter (baryonic)0.0490
Dark matter0.2650
Dark energy (\(\Lambda\))0.685-1
Radiation\(9\times10^{-5}\)1/3
Total \(\Omega\)≈ 1.000(flat universe)

XXVI. GNU Octave Examples

1 — Lorentz Transformations & Spacetime Diagrams

Lorentz Boost — Spacetime DiagramsGNU Octave
%% Lorentz transformation matrix and spacetime diagram
%% Shows how the (ct,x) coordinate grid transforms under a boost

clear; clc;
c = 1;   % natural units c=1

%% ─── Lorentz boost matrix for velocity v/c = beta ────────────
function L = lorentz_boost(beta)
  gamma = 1/sqrt(1-beta^2);
  L = [gamma, -gamma*beta; -gamma*beta, gamma];
end

%% ─── Transform a set of spacetime events ─────────────────────
betas = [0, 0.3, 0.6, 0.8, 0.95];

% Events in frame S: [ct; x]
events = [0 1 2 3 4; ...
          0 1 0 2 1];

printf('%-8s %-12s %-12s %-12s %-12s %-12s\n', 'Event', 'beta=0', ...
       'beta=0.3', 'beta=0.6', 'beta=0.8', 'beta=0.95');

for ev = 1:size(events,2)
  printf('(%d,%d)   ', events(2,ev), events(1,ev));
  for ib = 1:length(betas)
    ep = lorentz_boost(betas(ib)) * events(:,ev);
    printf('(%.2f,%.2f)   ', ep(2), ep(1));
  end
  printf('\n');
end

%% ─── Spacetime diagram ────────────────────────────────────────
figure(1); clf;
subplot(1,2,1); hold on; grid on; axis equal;
title('Spacetime Diagram — Frame S');
xlabel('x (light-seconds)'); ylabel('ct (light-seconds)');

% Light cone
t_arr = linspace(-3,3,100);
plot(t_arr, t_arr, 'y--', t_arr, -t_arr, 'y--', 'LineWidth', 1.5);

% World lines of objects at rest at x=0,1,2
for x0 = [0 1 2]
  plot([x0 x0],[-3 3],'b-','LineWidth',1.2);
end
% Surfaces of simultaneity at t=0,1,2
for t0 = [0 1 2]
  plot([-3 3],[t0 t0],'b:','LineWidth',0.8);
end

% Boosted frame S' (beta=0.6)
beta = 0.6; gamma = 1/sqrt(1-beta^2);
% ct' axis: x = beta*ct (worldline of origin of S')
plot(beta*t_arr, t_arr, 'r-', 'LineWidth', 2);
% x' axis: ct = beta*x (surface ct'=0)
plot(t_arr, beta*t_arr, 'r-', 'LineWidth', 2);
legend({'Light cone','','World lines (S)','','', ...
        'Simultaneity (S)','','',"ct' axis (S')","x' axis (S')"});
axis([-3 3 -3 3]);

%% ─── Invariant interval verification ─────────────────────────
subplot(1,2,2); hold on;
title('Spacetime Interval Invariance');
xlabel('\beta'); ylabel('s² = -\Deltat² + \Deltax²');

% Event A at (ct=2,x=1), Event B at (ct=3,x=2)
A = [2;1]; B = [3;2];
s2_S = -(B(1)-A(1))^2 + (B(2)-A(2))^2;
bv = linspace(0, 0.99, 200);
s2_primes = zeros(1,length(bv));

for k = 1:length(bv)
  L = lorentz_boost(bv(k));
  Ap = L*A; Bp = L*B;
  s2_primes(k) = -(Bp(1)-Ap(1))^2 + (Bp(2)-Ap(2))^2;
end

plot(bv, s2_primes, 'b-', 'LineWidth', 2);
plot(bv, s2_S*ones(1,length(bv)), 'r--', 'LineWidth', 1.5);
legend({"s²(β) in boosted frame", 's² in S (constant)'});
grid on;
printf('Interval s² = %.4f (should be same in all frames)\n', s2_S);
printf('Max deviation: %.2e\n', max(abs(s2_primes - s2_S)));

2 — Relativistic Dynamics: Particle Acceleration

Relativistic vs Classical DynamicsGNU Octave
%% Compare classical and relativistic dynamics under constant force
%% Shows how relativistic momentum prevents v → c
%% Also demonstrates E² = (pc)² + (mc²)²

clear; clc;
c = 2.998e8;  m = 9.109e-31;  % electron
F = 1e-17;                        % constant force (N)
dt = 1e-10;  T = 5e-7;           % time step, total time
N = round(T/dt);

%% Classical and relativistic integration
v_cl = zeros(1,N); v_rel = zeros(1,N);
p_cl = zeros(1,N); p_rel= zeros(1,N);
v_c=0; v_r=0; p_r=0;

for k = 1:N
  v_cl(k)  = v_c;
  v_rel(k) = v_r;
  p_cl(k)  = m*v_c;
  p_rel(k) = p_r;

  % Classical: dp/dt = F, v = p/m
  v_c += F/m * dt;

  % Relativistic: dp/dt = F, v = p/sqrt(m²+p²/c²)
  p_r += F * dt;
  v_r = p_r / sqrt(m^2 + (p_r/c)^2);
end

t_arr = (1:N)*dt;

figure(2); clf;
subplot(2,2,1);
plot(t_arr*1e9, v_cl/c, 'r--', t_arr*1e9, v_rel/c, 'b-', 'LineWidth',2);
plot(get(gca,'XLim'),[1 1],'k:');  % c = 1 line
legend({'Classical','Relativistic','Speed of light'});
title('Velocity vs Time'); xlabel('t (ns)'); ylabel('v/c'); grid on;

subplot(2,2,2);
KE_cl  = 0.5*m*(v_cl.^2);
gamma_arr = 1./(1 - (v_rel/c).^2).^0.5;
KE_rel = (gamma_arr - 1)*m*c^2;
plot(t_arr*1e9, KE_cl/1.6e-19, 'r--', t_arr*1e9, KE_rel/1.6e-19, 'b-', 'LineWidth',2);
legend({'Classical KE','Relativistic KE'});
title('Kinetic Energy vs Time'); xlabel('t (ns)'); ylabel('KE (eV)'); grid on;

subplot(2,2,3);
% Verify E² = (pc)² + (mc²)² for relativistic particle
E_total = gamma_arr * m * c^2;
p_check = m * gamma_arr .* v_rel;
E_check = sqrt((p_check*c).^2 + (m*c^2)^2);
semilogy(t_arr*1e9, abs(E_total - E_check)./E_total, 'b-', 'LineWidth',2);
title('E²=(pc)²+(mc²)² residual'); xlabel('t (ns)'); ylabel('|ΔE|/E'); grid on;

subplot(2,2,4);
% Gamma factor evolution
semilogy(t_arr*1e9, gamma_arr, 'm-', 'LineWidth',2); grid on;
title('Lorentz factor γ vs time'); xlabel('t (ns)'); ylabel('γ');

3 — Relativistic Kinematics: Four-Momentum

Relativistic Collisions & Four-MomentumGNU Octave
%% Relativistic kinematics using four-momentum p^μ = (E/c, p)
%% Demonstrates: particle creation, threshold energy, invariant mass

clear; clc;
c = 1;   % natural units c=1
m_p = 0.938;    % proton mass (GeV/c²)
m_pi= 0.135;    % pion mass (GeV/c²)

%% Invariant mass from two four-momenta
function M = inv_mass(p1, p2)
  % p = [E; px; py; pz] (c=1 units)
  ptot = p1 + p2;
  M = sqrt(-ptot(1)^2 + ptot(2)^2 + ptot(3)^2 + ptot(4)^2);
% Note sign: metric (-,+,+,+), p^2 = -m^2 → M=sqrt(-p_tot·p_tot)
end

%% Four-momentum of a particle from mass and 3-momentum
function p4 = four_mom(m, pvec)
  E = sqrt(m^2 + norm(pvec)^2);
  p4 = [E; pvec(:)];
end

%% ─── Compton scattering: γ + e⁻ → γ + e⁻ ──────────────────
m_e = 0.511e-3;  % electron mass in GeV
E_gamma = 0.01;  % 10 MeV photon
theta = 90 * pi/180;  % 90° scattering

% Compton formula: λ' = λ + (h/mₑc)(1-cosθ)
% In energy units: 1/E' = 1/E + (1-cosθ)/mₑ
E_gamma_prime = 1/(1/E_gamma + (1-cos(theta))/m_e);
printf('Compton scattering at 90°:\n');
printf('  Incident photon: %.2f MeV\n', E_gamma*1e3);
printf('  Scattered photon: %.2f MeV\n', E_gamma_prime*1e3);

%% ─── Threshold energy: p + p → p + p + π⁰ ─────────────────
% In COM frame, minimum energy = (2m_p + m_pi)
% In lab frame (target proton at rest):
% Threshold: E_thr = (sum_masses)²/(2m_p) - m_p
m_final = 2*m_p + m_pi;
E_thr = m_final^2/(2*m_p) - m_p/2;  % using s = (2m_p + m_pi)²
E_thr2 = (m_final^2 - 2*m_p^2)/(2*m_p);
printf('\nPion production p+p→p+p+π⁰:\n');
printf('  Threshold beam energy: %.4f GeV = %.1f × m_p c²\n', E_thr2, E_thr2/m_p);

%% ─── Invariant mass scan (e.g. e+e⁻ → hadrons) ────────────
E_beam = linspace(1, 10, 500);   % 1-10 GeV per beam (e+e-)
s = (2*E_beam).^2;                % COM energy squared
W = sqrt(s);                       % invariant mass

% Relativistic Breit-Wigner for J/ψ (M=3.097 GeV, Γ=93 keV)
M_jpsi = 3.097; Gamma = 93e-3;
sigma = Gamma^2 ./ ((W - M_jpsi).^2 + (Gamma/2)^2);

figure(3);
plot(W, sigma/max(sigma), 'c-', 'LineWidth', 2);
title('Breit-Wigner resonance: J/ψ(3097)');
xlabel('√s = invariant mass W (GeV)'); ylabel('σ/σ_max'); grid on;
xlim([3.0 3.2]);  % zoom in
printf('\nJ/ψ mass: %.3f GeV (PDG: 3.097 GeV)\n', M_jpsi);

4 — Schwarzschild Geodesics (GR Orbit)

Schwarzschild Geodesic Integration — Mercury PerihelionGNU Octave
%% Integrate geodesic equations in Schwarzschild spacetime
%% Demonstrates perihelion precession — GR prediction for Mercury

clear; clc;

%% Mercury orbital parameters (SI)
G  = 6.674e-11;
M  = 1.989e30;     % Sun mass
c  = 2.998e8;
rs = 2*G*M/c^2;   % Schwarzschild radius ≈ 2953 m

% Mercury: a=5.79e10 m, e=0.2056, T=87.97 days
a_merc = 5.79e10; e_merc = 0.2056;
rp = a_merc*(1-e_merc);           % perihelion
ra = a_merc*(1+e_merc);           % aphelion
L_sq = G*M*rp*ra*2/(rp+ra);       % specific angular momentum squared approx
h = sqrt(G*M*a_merc*(1-e_merc^2));  % h = L/m

%% Orbit equation in u=1/r form: d²u/dφ² + u = GM/h² + 3GM·u²/c²
%% This is the GR Binet equation (the last term is GR correction)

function du = binet_rhs(phi, u, GM, h, c)
  % u = [1/r; d(1/r)/dphi]
  du = [u(2); GM/h^2 - u(1) + 3*GM/c^2*u(1)^2];
end

%% Integrate for 100 orbits
GM = G*M;
u0 = 1/rp;    % at perihelion: du/dphi = 0
phi_span = linspace(0, 100*2*pi, 5e5);
f = @(phi,u) binet_rhs(phi, u, GM, h, c);
sol = lsode(f, [u0; 0], phi_span);

r_sol = 1./sol(:,1);

%% Find perihelion angles (local minima in r → maxima in u=1/r)
[~, peri_idx] = findpeaks(sol(:,1), 'MinPeakProminence', 1e-14);
if length(peri_idx) >= 2
  dphi = diff(phi_span(peri_idx));         % angular step between perihelions
  precession_per_orbit = mean(dphi) - 2*pi;
  precession_arcsec_per_orbit = precession_per_orbit * 180/pi * 3600;
  T_merc = 87.97 * 86400;                 % Mercury period in seconds
  precession_per_century = precession_arcsec_per_orbit * (100*365.25*86400/T_merc);
  printf('GR prediction: %.2f arcsec/century\n', precession_per_century);
  printf('Observed:      43.0 arcsec/century\n');
end

%% Analytical GR formula: Δφ = 6πGM/(a(1-e²)c²) per orbit
dphi_analytic = 6*pi*GM/(a_merc*(1-e_merc^2)*c^2);
arcsec_anal = dphi_analytic*180/pi*3600*(100*365.25*86400/T_merc);
printf('Analytic formula: %.2f arcsec/century\n', arcsec_anal);

%% Plot orbit (3 revolutions)
n3 = find(phi_span >= 6*pi, 1);
x = r_sol(1:n3) .* cos(phi_span(1:n3)');
y = r_sol(1:n3) .* sin(phi_span(1:n3)');
figure(4); plot(x/1e9, y/1e9, 'c-', 0, 0, 'y*', 'LineWidth', 1.5, 'MarkerSize', 14);
axis equal; grid on;
title('Mercury Orbit with GR Precession (3 revolutions)');
xlabel('x (10⁹ m)'); ylabel('y (10⁹ m)');

5 — Gravitational Wave Signal

Gravitational Wave Inspiral ChirpGNU Octave
%% Gravitational wave strain from a compact binary inspiral
%% Point-particle GR in the quadrupole approximation
%% Models GW150914-like binary black hole merger

clear; clc;
G = 6.674e-11; c = 2.998e8;
M_sun = 1.989e30;

%% Binary parameters (GW150914: ~36+29 Msun)
m1 = 36*M_sun; m2 = 29*M_sun;
M  = m1+m2;                          % total mass
mu = m1*m2/M;                        % reduced mass
Mc = mu^(3/5) * M^(2/5);           % chirp mass
dist = 410e6 * 3.086e22;           % 410 Mpc in metres

printf('Chirp mass: %.2f Msun\n', Mc/M_sun);

%% Time to coalescence τ from orbital separation a
% τ = 5c⁵a⁴/(256G³·M·m1·m2)
% Use f(t) from PN approximation:
% f(τ) = (1/π)(5/256)^(3/8) * (G Mc/c³)^(-5/8) * τ^(-3/8)

tau_arr = linspace(0.5, 0.001, 10000);  % time before merger
GMc_c3 = G*Mc/c^3;
f_gw = (1/pi) * (5/256)^(3/8) * GMc_c3^(-5/8) .* tau_arr.^(-3/8);

%% Phase: Φ(τ) = -2(τ/(5GMc/c³))^(5/8)
Phi_gw = -2 * (tau_arr / (5*GMc_c3)).^(5/8);

%% Strain amplitude h(τ) = (4/dist)*(G Mc/c²)*(π f GMc/c³)^(2/3)
h_amp = (4/dist) * (G*Mc/c^2) .* (pi*f_gw*GMc_c3).^(2/3);

%% h+ polarisation strain
h_plus = h_amp .* cos(2*Phi_gw);

t_phys = -tau_arr;  % time (negative = before merger)

figure(5); clf;
subplot(3,1,1);
plot(t_phys, h_plus/1e-21, 'c-', 'LineWidth', 1.2);
title('Gravitational Wave Chirp (GW150914-like Binary)');
ylabel('h₊ (×10⁻²¹)'); grid on;
xlim([-0.5 0]);

subplot(3,1,2);
plot(t_phys, f_gw, 'm-', 'LineWidth',2); grid on;
ylabel('f_{GW} (Hz)'); xlim([-0.5 0]);
title('Frequency evolution (chirp)');

subplot(3,1,3);
spectrogram(h_plus, 256, 250, 512, 1/(abs(t_phys(2)-t_phys(1))), 'yaxis');
title('Spectrogram (the "chirp")'); ylim([0 300]);
xlabel('Time (s before merger)');

6 — GR Gravitational Redshift & Lensing Deflection

GR Light Bending & Gravitational RedshiftGNU Octave
%% GR light deflection around the Sun and gravitational redshift
%% Compare GR result (1.75") vs Newtonian (0.875") vs observation

clear; clc;
G = 6.674e-11; M_sun = 1.989e30; c = 2.998e8; R_sun = 6.96e8;

%% ─── Light deflection ─────────────────────────────────────
% GR: δ = 4GM/(bc²) — twice Newtonian
% Newtonian: δ = 2GM/(bc²)
b_arr = linspace(1,20,200)*R_sun;  % impact parameter
delta_GR   = 4*G*M_sun./(b_arr*c^2);  % radians
delta_Newt = 2*G*M_sun./(b_arr*c^2);

delta_GR_arcsec   = delta_GR   * 180/pi * 3600;
delta_Newt_arcsec = delta_Newt * 180/pi * 3600;

figure(6); clf;
subplot(1,2,1);
loglog(b_arr/R_sun, delta_GR_arcsec, 'c-', b_arr/R_sun, delta_Newt_arcsec, ...
       'r--', 'LineWidth', 2);
hold on;
plot(1, 1.7505, 'c*', 'MarkerSize', 14);  % GR at solar limb
plot(1, 0.8753, 'r*', 'MarkerSize', 14);  % Newton
legend({'GR (4GM/bc²)','Newton (2GM/bc²)','GR at limb: 1.75"','Newton: 0.88"'});
xlabel('b/R_sun'); ylabel('Deflection (arcseconds)');
title('Light Deflection by the Sun'); grid on;

%% ─── Gravitational redshift ────────────────────────────────
% z = Δλ/λ = GM/(Rc²) (for weak field)
r_arr = linspace(1.01, 10, 500)*R_sun;
rs = 2*G*M_sun/c^2;                 % 2953 m
z_GR   = 1./sqrt(1 - rs./r_arr) - 1;  % exact Schwarzschild
z_weak = G*M_sun./(r_arr*c^2);          % weak-field approx

subplot(1,2,2);
semilogy(r_arr/R_sun, z_GR, 'c-', r_arr/R_sun, z_weak, 'r--', 'LineWidth',2);
legend({'Exact Schwarzschild', 'Weak-field GM/rc²'});
xlabel('r/R_{Sun}'); ylabel('Gravitational redshift z');
title('Solar Gravitational Redshift'); grid on;

printf('Solar surface redshift z = %.2e\n', z_GR(1));
printf('(Solar spectroscopic observed: ~2.1e-6)\n');

XXVII. Interactive Simulations

Simulation I — Interactive Spacetime (Minkowski) Diagram

Adjust the boost velocity \(\beta\) to see how axes, worldlines, and simultaneity planes rotate in Minkowski spacetime. Light always stays at 45°.

Spacetime Diagram — Minkowski Space
0.00
S frame (at rest). Drag β slider to boost.

Simulation II — Lorentz Factor Explorer

Interactively explore time dilation, length contraction, and simultaneity shift as functions of velocity.

Lorentz Factor — γ, Time Dilation, Length Contraction
0.500
γ = 1.155 | Δt′ = 1.155Δt | L′ = 0.866 L₀

Simulation III — Relativistic Energy-Momentum

Plots kinetic energy and momentum vs. velocity, comparing classical and relativistic results. Shows the mass-shell hyperbola.

E² = (pc)² + (mc²)² — Mass Shell

Simulation IV — Schwarzschild Orbit & Precession

Numerically integrates the GR orbit equation. Adjust compactness \(r_s/r_p\) to increase relativistic precession.

GR Orbital Precession — Schwarzschild Geodesic
0.40
10
Precession per orbit: —

Simulation V — Gravitational Wave Polarizations

Animates the effect of \(h_+\) and \(h_\times\) polarizations on a ring of test particles — the "rubber spacetime" deformation.

GW Polarization — Ring of Test Particles
15
3
GW propagating in ẑ direction. Rings deform in x-y plane.