AMATH 475: Introduction to General Relativity

These notes integrate the professor’s typed lecture blocks with spoken explanations from the full lecture series. Video lectures by Eduardo Martín-Martínez are available on YouTube: Online course — Intro to General Relativity.

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Part I — Special Relativity

Chapter 1. Index Notation and the Einstein Summation Convention

Index notation is the algebraic backbone of both special and general relativity. Once learned, it transforms pages of component algebra into a single compact line.

The summation convention. Whenever an index appears exactly twice in a monomial — once as a superscript (contravariant, “upstairs”) and once as a subscript (covariant, “downstairs”) — a sum over all values of that index is implied. One up, one down: sum.

\[ \sum_{\mu=0}^{3} A^\mu B_\mu \;\equiv\; A^\mu B_\mu \]

The repeated index \(\mu\) is a dummy index and may be freely renamed without changing the expression. An index that appears only once is a free index and must match on both sides of an equation.

The Kronecker delta. \(\delta^\mu{}_\nu = 1\) if \(\mu=\nu\), zero otherwise. As a matrix it is the identity. Its key property is index substitution: contracting with \(\delta\) renames an index.

\[ \delta^\mu{}_\nu A^\nu = A^\mu, \qquad \delta^\mu{}_\mu = n \;\text{(dimension of the space)} \]

The Levi-Civita symbol. The fully antisymmetric symbol \(\varepsilon_{\mu_1 \cdots \mu_n}\) takes value \(+1\) if the indices are an even permutation of \(1,2,\ldots,n\), value \(-1\) for an odd permutation, and \(0\) if any two indices repeat. It is not a tensor (it does not transform with \(n\) copies of the Jacobian), but \(\sqrt{|\det g|}\,\varepsilon_{\mu_1\cdots\mu_n}\) is a tensor density — important when we reach integration on manifolds.

\[ \det M = \varepsilon^{i_1 \cdots i_n} M_{1 i_1} M_{2 i_2} \cdots M_{n i_n} \]

The cross product in \(\mathbb{R}^3\): \((v \times w)^i = \varepsilon^{ijk} v_j w_k\). The professor shows how vector identities like \(v\times(w\times u) = w(v\cdot u) - u(v\cdot w)\) become two-line proofs using the \(\varepsilon\)-\(\delta\) contraction identity \(\varepsilon_{ijk}\varepsilon^{ilm} = \delta^l_j \delta^m_k - \delta^m_j \delta^l_k\).

Rules for index notation. The professor drills the following rules repeatedly throughout the course:

1. Every free index must appear exactly once on each side of an equation, in the same position (up or down).
2. A repeated (dummy) index must appear exactly once up and once down in each term.
3. You may never have an index appear three or more times in a single term — this is always an error.
4. Dummy indices may be freely renamed, but only within a single term.
5. Free indices may not be renamed unless you rename them consistently on both sides.

A common source of errors. When working with the Einstein convention, beginners often write expressions like \(A^\mu B^\mu\) (both up) thinking this is a dot product. It is not a valid Einstein-summed expression, and more importantly, it is not invariant under coordinate changes. The invariant contraction is always one up, one down: \(A^\mu B_\mu\).

\[ \epsilon_{\mu_1\cdots\mu_n} = \sqrt{|\det g|}\,\varepsilon_{\mu_1\cdots\mu_n} \]

This transforms as a tensor under diffeomorphisms that preserve orientation (pseudo-tensor under orientation-reversing ones). The professor proves this in Lecture 22 as part of an assignment exercise.

Worked example: Proving the BAC–CAB rule. To prove \(\vec A \times (\vec B \times \vec C) = \vec B(\vec A \cdot \vec C) - \vec C(\vec A \cdot \vec B)\) in index notation:

\[ [\vec A \times (\vec B \times \vec C)]^i = \varepsilon^{ijk} A_j (\vec B \times \vec C)_k = \varepsilon^{ijk} A_j \varepsilon_{klm} B^l C^m \]

Using \(\varepsilon_{ijk}\varepsilon^{ilm} = \delta^l_j\delta^m_k - \delta^m_j\delta^l_k\):

\[ = (\delta^l_j\delta^m_k - \delta^m_j\delta^l_k) A^j B_l C^m = A^j B_j C^i - A^j B^i C_j = (\vec A \cdot \vec B)C^i... \]

Wait, tracking the raising/lowering carefully in flat space where \(\delta_{ij}\) identifies upper and lower indices: \(= B^i(A_j C^j) - C^i(A_j B^j) = \vec B(\vec A \cdot \vec C) - \vec C(\vec A \cdot \vec B)\). QED.

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Chapter 2. Covariance, Contravariance, and the Dual Basis

The professor uses a memorable analogy: “Imagine measuring the size of a room in metres, then changing to centimetres. Your numbers get larger — that is a covariant transformation. The room itself does not change. Contravariant objects go the other way: the components get smaller when the units get larger, because you need fewer big units to cover the same distance.”

If \(\widetilde{\Lambda}^\mu{}_\nu\) is the Jacobian of the coordinate change \(x \to x'\), then:

\[ \text{Contravariant (upper index):}\quad v'^\mu = \widetilde{\Lambda}^\mu{}_\nu\, v^\nu \]\[ \text{Covariant (lower index):}\quad \omega'_\mu = \Lambda^\nu{}_\mu\, \omega_\nu \]

where \(\Lambda = (\widetilde\Lambda^{-1})^\top\). The two matrices are inverses of each other, guaranteeing that the contraction \(\omega_\mu v^\mu\) is coordinate-independent — it is a scalar.

\[ \langle e^a, e_b \rangle = \delta^a{}_b \]

The basis vectors transform with \(\Lambda\); the dual basis transforms with \(\widetilde\Lambda\). Components go the other way: vector components transform with \(\widetilde\Lambda\) (contravariant), one-form components transform with \(\Lambda\) (covariant).

Why the distinction matters physically. In Newtonian mechanics we rarely distinguish contravariant from covariant, because in Euclidean space with Cartesian coordinates the metric is the identity and raising/lowering indices does nothing. The distinction becomes unavoidable in:

1. Curvilinear coordinates (e.g., spherical), where the metric is non-trivial
2. Special relativity, where the metric is indefinite (Minkowski)
3. General relativity, where the metric is position-dependent

Example: Spherical coordinates. In \(\mathbb{R}^3\) with coordinates \((r, \theta, \phi)\), the metric is \(g_{ij} = \operatorname{diag}(1, r^2, r^2\sin^2\theta)\). A vector with contravariant components \(v^r = 1, v^\theta = v^\phi = 0\) has covariant components \(v_r = g_{rr}v^r = 1\), \(v_\theta = g_{\theta\theta}v^\theta = 0\), \(v_\phi = 0\). But a vector with \(v^\theta = 1\) has covariant component \(v_\theta = r^2\). This is why the gradient formula in spherical coordinates looks different from Cartesian — the covariant components of the gradient (which are just partial derivatives, \(\partial_\mu f\)) need to be converted to contravariant components using the inverse metric before they represent the “direction of steepest ascent” in the ordinary sense.

\[ \flat: T_pM \to T^*_pM, \quad v^\mu \mapsto v_\mu = g_{\mu\nu}v^\nu \quad\text{(lowering)} \]\[ \sharp: T^*_pM \to T_pM, \quad \omega_\mu \mapsto \omega^\mu = g^{\mu\nu}\omega_\nu \quad\text{(raising)} \]

These are called the “musical isomorphisms” (\(\flat\) is “flat,” \(\sharp\) is “sharp”). Without a metric, there is no natural identification between vectors and covectors — they genuinely live in different spaces.

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Chapter 3. Special Relativity: Postulates and Spacetime

The two postulates of Special Relativity:

The professor notes that demanding the Lorentz transformation be linear is essentially a hidden third postulate — it encodes the assumption that spacetime is homogeneous and isotropic. The existence of a maximum speed is more fundamental: one can derive that there must be a universal invariant speed (which could be infinite — giving Galilean relativity — or finite, giving special relativity) from the homogeneity and isotropy of spacetime alone. Experiment selects the finite case.

Historical context. The Michelson-Morley experiment (1887) failed to detect the motion of the Earth through the hypothetical luminiferous ether, indicating that the speed of light is the same in all directions regardless of the Earth’s motion. Lorentz and FitzGerald proposed that objects contract along their direction of motion through the ether. Einstein’s insight (1905) was that the ether is superfluous — the constancy of \(c\) is a fundamental property of spacetime itself, not a dynamical effect. The resulting theory abolishes absolute simultaneity while preserving absolute causality.

Deriving the Lorentz transformation. Consider two inertial frames \(S\) and \(S'\), with \(S'\) moving at velocity \(v\) along the \(x\)-axis. The derivation proceeds in five steps:

Step 1 (Linearity): The transformation must be linear — non-linear transformations would map uniform motion in one frame to non-uniform motion in another, violating Postulate 1. So \(t' = \alpha t + \beta x\) and \(x' = \gamma x + \delta t\) for constants \(\alpha, \beta, \gamma, \delta\) to be determined.

Step 2 (Relative velocity): The origin of \(S'\) (\(x' = 0\)) satisfies \(x = vt\) in \(S\). So \(0 = \gamma vt + \delta t\), giving \(\delta = -\gamma v\).

Step 3 (Invariance of the speed of light): A light pulse satisfying \(x = ct\) must also satisfy \(x' = ct'\). Substituting and requiring this for all \(t\) gives a relation between \(\alpha, \beta, \gamma\).

Step 4 (Symmetry): The inverse transformation is obtained by swapping \(v \to -v\). This gives additional constraints.

Step 5 (Normalization): The requirement that the spacetime interval is invariant: \(-c^2 dt'^2 + dx'^2 = -c^2 dt^2 + dx^2\) fixes the remaining freedom uniquely, giving \(\gamma = 1/\sqrt{1-v^2/c^2}\).

The invariance of the spacetime interval \(\Delta s^2 = -c^2 \Delta t^2 + \Delta x^2\) under the transformation uniquely determines the Lorentz boost:

\[ t' = \gamma\!\left(t - \frac{v}{c^2}x\right), \qquad x' = \gamma(x - vt), \qquad \gamma = \frac{1}{\sqrt{1-v^2/c^2}} \]

The spacetime interval. The quantity

\[ \Delta s^2 = -c^2 \Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2 \]

is invariant under all Lorentz transformations. Its sign classifies the separation:

Sign	Name	Physical meaning
\(\Delta s^2 < 0\)	Timelike	Causal connection possible
\(\Delta s^2 = 0\)	Null (lightlike)	Connected by a light ray
\(\Delta s^2 > 0\)	Spacelike	No causal connection

Relativity of simultaneity. Two events that are simultaneous in one inertial frame (\(\Delta t = 0\)) are generally not simultaneous in another frame. This is not an illusion or a measurement artifact — it is a genuine feature of spacetime structure. Simultaneity is relative. Only timelike-separated events have an invariant chronological ordering.

Time dilation. Consider a clock at rest in \(S'\). Two ticks occur at \(x' = 0\), \(t'_1 = 0\) and \(t'_2 = T'\). In \(S\): \(\Delta t = \gamma T'\). Moving clocks run slow.

Length contraction. A rod at rest in \(S'\) with endpoints at \(x'_1 = 0\) and \(x'_2 = L'\). Measured simultaneously in \(S\): \(\Delta x = L'/\gamma\). Moving rods are shorter.

\[ ds^2 = \eta_{\mu\nu}\, dx^\mu\, dx^\nu = -c^2\, dt^2 + dx^2 + dy^2 + dz^2 \]

The sign convention used here is the east coast (mostly plus) convention. Every physical relation in SR has a natural 4-dimensional covariant expression, and the goal of learning index notation is precisely to be able to write those relations.

Worked example: The twin paradox. Alice stays on Earth; Bob travels to a distant star at high speed and returns. When they reunite, Bob is younger. This is not a paradox but a direct consequence of proper time: Alice’s worldline is a straight vertical line in spacetime (she is inertial), and Bob’s is a bent worldline. The invariant proper time is shorter for the bent worldline — the longest proper time between two events is achieved by the inertial (straight) worldline. SR makes a definitive prediction: the non-inertial twin ages less.

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Chapter 4. Proper Time and Four-Velocity

Proper time \(\tau\) is the time measured by a clock carried along a worldline. For a particle moving with velocity \(v(t)\) in an inertial frame:

\[ d\tau = dt\sqrt{1 - v^2/c^2} = \frac{dt}{\gamma}, \qquad \tau = \int_{t_1}^{t_2} \frac{dt}{\gamma(t)} \]

Proper time is invariant — every observer agrees on how much time a given clock has accumulated between two events on its worldline.

Four-velocity. Parametrize a worldline by proper time \(\tau\). The four-velocity is:

\[ u^\mu = \frac{dx^\mu}{d\tau} = \left(\gamma c,\; \gamma \vec{v}\right) \]\[ u^\mu u_\mu = \eta_{\mu\nu} u^\mu u^\nu = -c^2 \]

This is a constraint, not a dynamical equation. It says that all massive particles move through spacetime at the same “speed” — the speed of light — but in different directions through the four-dimensional manifold.

Four-acceleration. \(a^\mu = du^\mu/d\tau\). Differentiating \(u^\mu u_\mu = -c^2\): \(u_\mu a^\mu = 0\) — four-acceleration is always orthogonal to four-velocity. Its magnitude is the proper acceleration: \(a^\mu a_\mu = \alpha^2 \geq 0\). For an inertial observer \(\alpha = 0\); for an accelerating observer \(\alpha \neq 0\).

\[ u^\mu = c(\cosh\xi,\, \sinh\xi,\, 0,\, 0) \]

The utility of rapidity is that rapidities add for collinear boosts: an observer boosted with rapidity \(\xi_1\) relative to the ground, and then boosted with rapidity \(\xi_2\) relative to the first frame, sees the ground moving with rapidity \(\xi_1 + \xi_2\). This is simpler than the relativistic velocity addition formula \(v_{13} = (v_{12}+v_{23})/(1+v_{12}v_{23}/c^2)\).

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Chapter 5. The Minkowski Metric and the Poincaré Group

The Minkowski metric. In Cartesian coordinates \((t, x, y, z)\):

\[ \eta_{\mu\nu} = \operatorname{diag}(-1, +1, +1, +1) \]

This is the east coast convention (also called the “mostly plus” convention). The professor explicitly warns: “Some books use the opposite sign — west coast convention — and you will get all your results with the wrong sign if you mix them. Choose one and be consistent.”

\[ v_\mu = \eta_{\mu\nu} v^\nu, \qquad v^\mu = \eta^{\mu\nu} v_\nu \]\[ \Lambda^\rho{}_\mu\, \eta_{\rho\sigma}\, \Lambda^\sigma{}_\nu = \eta_{\mu\nu} \]

It has four connected components. The proper orthochronous Lorentz group \(SO^+(1,3)\) is the component connected to the identity.

Generators. The six generators split into three rotation generators \(\hat{J}_i\) and three boost generators \(\hat{K}_i\). Boosts are hyperbolic rotations. A boost along \(x\) with rapidity \(\xi = \operatorname{arctanh}(v/c)\):

\[ \Lambda^\mu{}_\nu = \begin{pmatrix} \cosh\xi & -\sinh\xi & 0 & 0 \\ -\sinh\xi & \cosh\xi & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

Rapidities add: \(\xi_{13} = \xi_{12} + \xi_{23}\) for collinear boosts. The professor demonstrates in Mathematica (Lecture 9): computing the product of two non-collinear boosts reveals a rotation in the result — this is Thomas precession, a purely relativistic kinematic effect with real consequences for electron spin in atoms.

\[ \hat{P}_\mu = -i\partial_\mu \quad\text{(four translations)}, \qquad \hat{M}_{\rho\sigma} \quad\text{(six Lorentz generators)} \]\[ [\hat{P}_\mu, \hat{P}_\nu] = 0, \qquad [\hat{M}_{\rho\sigma}, \hat{P}_\mu] = i(\eta_{\sigma\mu}\hat{P}_\rho - \eta_{\rho\mu}\hat{P}_\sigma) \]\[ \Omega_\text{Thomas} = -\gamma^2 \frac{\vec a \times \vec v}{c^2(\gamma+1)} \approx -\frac{\vec a \times \vec v}{2c^2} \quad (\text{for } v \ll c) \]

For an electron orbiting a proton, this gives a precession that halves the naive spin-orbit coupling, explaining the factor of 2 in the hydrogen spectrum fine structure. Thomas precession is a purely kinematic effect of special relativity, requiring no forces beyond the centripetal acceleration.

Discrete symmetries. The full Lorentz group \(O(1,3)\) has four connected components:

• \(L_+^\uparrow\) (proper orthochronous): det\(\Lambda = +1\), \(\Lambda^0{}_0 > 0\) — connected to identity
• \(L_+^\downarrow\) (proper non-orthochronous): time reversal \(T\)
• \(L_-^\uparrow\) (improper orthochronous): parity reversal \(P\)
• \(L_-^\downarrow\) (improper non-orthochronous): combined \(PT\)

The Standard Model of particle physics is invariant under \(CPT\) (charge conjugation times parity times time reversal) but violates \(P\) (weak interactions), \(C\), and \(CP\) separately.

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Chapter 6. Dynamics: Hamilton’s Principle and Noether’s Theorem

Hamilton’s principle (stationary action). The trajectory followed by a physical system is the one that makes the action \(S = \int L\, dt\) stationary under variations with fixed endpoints:

\[ \delta S = 0 \implies \frac{\partial L}{\partial q^i} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}^i} = 0 \quad\text{(Euler-Lagrange equations)} \]

Examples: time-translation invariance → energy; spatial translation invariance → momentum; rotational invariance → angular momentum.

Free particle action. The relativistically invariant action for a free massive particle is proportional to the proper time elapsed:

\[ S = -mc^2 \int d\tau = -mc \int ds = -mc \int \sqrt{-\eta_{\mu\nu}\dot{x}^\mu \dot{x}^\nu}\; d\lambda \]

Varying this action gives the geodesic equation: \(d^2x^\mu/d\tau^2 = 0\) — free particles travel in straight lines in Minkowski spacetime.

Four-momentum and the energy-momentum relation. The four-momentum:

\[ p^\mu = mu^\mu = \left(\frac{E}{c},\; \vec{p}\right), \qquad p^\mu p_\mu = -m^2c^2 \implies E^2 = m^2c^4 + c^2|\vec{p}|^2 \]

For a photon (\(m=0\)): \(E = c|\vec{p}|\) and \(p^\mu p_\mu = 0\) (null four-vector).

Particle collider exercise. Comparing two scenarios for creating a particle of mass \(M\) from two protons of mass \(m_p\):

Scenario A (head-on collision): two beams each with energy \(E\) collide. Total four-momentum squared: \(s = -(p_1 + p_2)^2 = 4E^2/c^2\). Available centre-of-mass energy: \(\sqrt{s}\, c = 2E\).

Scenario B (fixed target): one beam with energy \(E\) hits a proton at rest. Total four-momentum squared: \(s = -(p_1 + p_2)^2 = 2m_p^2c^2 + 2m_p E/c^2\). Centre-of-mass energy: \(\sqrt{s}\, c \approx \sqrt{2m_p c^2 E}\) at high energy.

For Scenario A: \(\sqrt{s} \propto E\) (linear). For Scenario B: \(\sqrt{s} \propto \sqrt{E}\) (square root). At \(E = 10^3\) GeV, Scenario A gives \(\sqrt{s} \approx 2000\) GeV while Scenario B gives \(\sqrt{s} \approx \sqrt{2 \times 0.938 \times 10^3} \approx 43\) GeV. The head-on collider wins by a factor of ~46. This is why the LHC, Tevatron, and LEP all used colliding beams rather than fixed targets for high-energy physics.

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Chapter 7. Four-Acceleration and Rindler Coordinates

Four-acceleration: \(b^\mu = du^\mu/d\tau\). It is always orthogonal to the four-velocity (\(u^\mu b_\mu = 0\)). Its norm gives the proper acceleration: \(b^\mu b_\mu = a^2 \geq 0\).

Rindler coordinates. An observer with constant proper acceleration \(a\) along \(x\) follows the hyperbolic worldline \(-t^2 + x^2 = c^4/a^2\). In Rindler coordinates \((\tau, \xi)\):

\[ ds^2 = -\left(1 + \frac{a\xi}{c^2}\right)^2 c^2\, d\tau^2 + d\xi^2 + dy^2 + dz^2 \]

Acceleration horizon. The surface \(\xi = -c^2/a\) is an acceleration horizon: signals from beyond it can never reach the accelerated observer. This is a purely kinematic horizon — no GR required.

\[ x(\tau) = \frac{c^2}{a}\cosh\!\left(\frac{a\tau}{c}\right), \quad t(\tau) = \frac{c}{a}\sinh\!\left(\frac{a\tau}{c}\right) \]

The worldline satisfies \(x^2 - c^2t^2 = c^4/a^2\) — a hyperbola in spacetime. The proper time and coordinate time are related by \(t = (c/a)\sinh(a\tau/c)\); for small \(v/c\), \(\tau \approx t\) as expected.

\[ \frac{d\tau_2}{d\tau_1} = \frac{1 + a\xi_2/c^2}{1 + a\xi_1/c^2} \]

By the Equivalence Principle, an observer in a gravitational field with local acceleration \(g\) at height \(h\) has the same relation: clocks at greater height run faster. This is gravitational time dilation, derived here from special relativity alone.

Bell’s rocket paradox. Two rockets accelerating identically in the ground frame (same program). A rope connects them. Does it break?

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Part II — Differential Geometry

Chapter 8. Differentiable Manifolds

The charts provide coordinates: if \(p \in U_\alpha\) then \(\phi_\alpha(p) = (x^1(p), \ldots, x^n(p))\). A function \(f: M \to \mathbb{R}\) is smooth if \(\bar{f} := f \circ \phi^{-1}: \mathbb{R}^n \to \mathbb{R}\) is smooth for every chart.

Examples of manifolds.

• \(\mathbb{R}^n\) is trivially a manifold with a single global chart (the identity map). It has a natural flat metric \(\delta_{ij}\) or Minkowski metric \(\eta_{\mu\nu}\).
• The \(n\)-sphere \(S^n\) requires at least two charts (e.g., stereographic projection from north and south poles). It has no global coordinate chart — any single chart must exclude at least one point.
• A cylinder \(\mathbb{R} \times S^1\) can be covered with a single chart with periodicity identification, but it is topologically distinct from \(\mathbb{R}^2\).
• The torus \(T^2 = S^1 \times S^1\) is a 2-dimensional manifold with non-trivial topology.
• Spacetime in General Relativity is a 4-dimensional Lorentzian manifold. Its topological structure (global shape) is a subject of active research in mathematical physics.

The Hausdorff condition. The Hausdorff condition says: for any two distinct points \(p \neq q\) in \(M\), there exist disjoint open sets \(U \ni p\) and \(V \ni q\). This prevents pathologies like two points that cannot be separated by any open sets. All physical spacetimes are Hausdorff.

Transition maps and compatibility. When two charts \((U_\alpha, \phi_\alpha)\) and \((U_\beta, \phi_\beta)\) overlap, the transition map \(\phi_\beta \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap U_\beta) \to \phi_\beta(U_\alpha \cap U_\beta)\) must be smooth. This condition ensures that the notion of “smooth function on \(M\)” does not depend on which chart you use to check it.

Orientability. A manifold is orientable if it has an atlas where all transition maps have positive Jacobian determinant. Physical spacetimes are assumed orientable. The Möbius strip is a classic example of a non-orientable manifold.

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Chapter 9. Tangent Vectors

Curves on a manifold. A smooth curve is a smooth map \(\gamma: \mathbb{R} \to M\).

\[ v: C^\infty(M) \to \mathbb{R}, \qquad v(fg) = f(p)\,v(g) + g(p)\,v(f) \]

Every tangent vector is the tangent to some curve through \(p\): \(v[f] = \frac{d}{dt}(f\circ\gamma)\big|_{t=0}\).

\[ v = v^\mu \partial_\mu, \qquad v^\mu = v[x^\mu] \]

The tangent space \(T_pM\) is an \(n\)-dimensional real vector space.

\[ [v,w][f] = v[w[f]] - w[v[f]], \qquad [v,w]^\mu = v^\nu \partial_\nu w^\mu - w^\nu \partial_\nu v^\mu \]

Coordinate vs. non-coordinate bases. A basis \(\{e_a\}\) is a coordinate basis if and only if \([e_a, e_b] = 0\) for all pairs. The orthonormal polar basis \(\{\hat{e}_r, \hat{e}_\theta\}\) is not a coordinate basis — this is why gradient, divergence, and Laplacian formulas look complicated in polar coordinates.

\[ e_1 = y\partial_x + \partial_y, \qquad e_2 = \partial_y \]\[ [e_1, e_2] = [y\partial_x + \partial_y, \partial_y] = -\partial_x \neq 0 \]

So this is not a coordinate basis, even though it looks like one at first glance. The non-commutativity arises from the \(y\)-dependence of \(e_1\).

Tangent space at a point vs. vector fields. It is crucial to distinguish:

• Tangent space at a point \(p\): \(T_pM\), a fixed \(n\)-dimensional vector space. Its elements are tangent vectors at \(p\) only.
• Tangent bundle: \(TM = \bigsqcup_{p \in M} T_pM\), the collection of all tangent spaces. Its sections are vector fields.
• Vector field: a smooth assignment of a tangent vector to each point.

In GR, equations involving tensors hold as identities of tensor fields — they must hold at every point of the manifold simultaneously.

Integral curves and flows. Given a vector field \(v\) on \(M\), an integral curve is a curve \(\gamma(t)\) such that \(\dot\gamma(t) = v(\gamma(t))\). The existence and uniqueness theorem for ODEs guarantees a unique maximal integral curve through each point. The collection of all integral curves fills the manifold (they cannot cross) and defines the flow \(\phi_t\). In GR: geodesics are integral curves of the geodesic vector field on the tangent bundle \(TM\).

Connection to the geodesic equation. The geodesic equation \(\ddot x^\mu + \Gamma^\mu{}_{\nu\rho}\dot x^\nu\dot x^\rho = 0\) is a second-order ODE on \(M\), but it can be rewritten as a first-order ODE on \(TM\) by introducing velocity as an additional variable: \(\dot x^\mu = v^\mu\), \(\dot v^\mu = -\Gamma^\mu{}_{\nu\rho}v^\nu v^\rho\). The geodesic flow is then the integral curve system on \(TM\). The Hamiltonian formulation uses the co-geodesic flow on \(T^*M\).

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Chapter 10. One-Forms and the Cotangent Space

\[ \langle dx^\mu, \partial_\nu \rangle = \delta^\mu{}_\nu \]\[ \langle df, v \rangle = v[f], \qquad (df)_\mu = \partial_\mu f \]

The gradient is a covector, not a vector. In coordinates: \(df = (\partial_\mu f)\, dx^\mu\). The components of the gradient are the components of the differential of the function.

The gradient is not a vector field. This is a subtle but important point. In elementary calculus, we freely write “the gradient vector \(\nabla f\)” and think of it as a vector. On a manifold without a metric, \(df\) is a one-form — its components \(\partial_\mu f\) transform with \(\Lambda\) (covariant), not \(\widetilde\Lambda\). To get a vector, you must raise the index with the metric: \((\text{grad}\, f)^\mu = g^{\mu\nu}\partial_\nu f\). In flat Cartesian space, \(g^{\mu\nu} = \delta^{\mu\nu}\) and this distinction is invisible — which is why it is never taught in undergraduate calculus.

Non-coordinate (anholonomic) bases. Sometimes it is convenient to use an orthonormal frame \(\{e_a\}\) that is not the coordinate basis. The corresponding dual frame \(\{e^a\}\) satisfies \(\langle e^a, e_b\rangle = \delta^a{}_b\). In an orthonormal frame, the metric components are just the constant matrix \(\eta_{ab} = \operatorname{diag}(-1,+1,+1,+1)\) everywhere — but the connection coefficients (spin connection) are no longer symmetric, because the frame basis vectors do not commute. This is the basis for the tetrad (vierbein) formulation of GR, which is essential for coupling gravity to spinor fields (fermions).

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Chapter 11. Tensors on Manifolds

Components: \(T^{\mu_1\cdots\mu_r}{}_{\nu_1\cdots\nu_s} = T(dx^{\mu_1},\ldots,dx^{\mu_r}, \partial_{\nu_1},\ldots,\partial_{\nu_s})\). Under coordinate change: each upper index transforms with \(\widetilde\Lambda\), each lower index with \(\Lambda\).

Tensor operations:

• Symmetrization: \(T_{(\mu\nu)} = \tfrac{1}{2}(T_{\mu\nu} + T_{\nu\mu})\)
• Antisymmetrization: \(T_{[\mu\nu]} = \tfrac{1}{2}(T_{\mu\nu} - T_{\nu\mu})\)
• Tensor product: \((R \otimes S)\) with components \(R^{\mu_1\cdots}{}_{\nu_1\cdots} S^{\rho_1\cdots}{}_{\sigma_1\cdots}\)
• Contraction: sum over a matched pair of upper and lower indices, reducing type by \((1,1)\)

Scalar fields are type \((0,0)\), vector fields type \((1,0)\), one-form fields type \((0,1)\).

Why the transformation law defines a tensor. A quantity with indices is a tensor if and only if its components transform in the prescribed way under coordinate changes. This is not just a formal definition — it guarantees that tensor equations are coordinate-independent. If \(T^{\mu\nu} = 0\) in one coordinate system and the components transform as a \((2,0)\) tensor, then \(T^{\mu\nu} = 0\) in every coordinate system. This is why the Einstein equations make sense: both \(G_{ab}\) and \(T_{ab}\) are genuine tensors, so their equality is a coordinate-independent statement.

The metric as a tensor. The metric \(g_{\mu\nu}\) is a \((0,2)\) symmetric tensor. Its inverse \(g^{\mu\nu}\) is a \((2,0)\) tensor. They satisfy \(g^{\mu\rho}g_{\rho\nu} = \delta^\mu{}_\nu\). The metric can be used to define an inner product: for two vectors \(u\) and \(v\), \(g(u,v) = g_{\mu\nu}u^\mu v^\nu\). For a Lorentzian metric, this inner product is not positive definite, and vectors can be classified as timelike, null, or spacelike based on the sign of \(g(v,v)\).

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Chapter 12. n-Forms, Exterior Derivative, and Integration

n-forms. An \(n\)-form is a fully antisymmetric covariant tensor of type \((0,n)\).

\[ \alpha \wedge \beta = (-1)^{kp}\,\beta \wedge \alpha \]

for a \(k\)-form \(\alpha\) and \(p\)-form \(\beta\). In particular, \(dx^\mu \wedge dx^\mu = 0\) (no repeated forms). This is why antisymmetry produces the Jacobian determinant in multivariable integration.

\[ (d\sigma)_{\mu_0\cdots\mu_n} = (n+1)\,\partial_{[\mu_0}\sigma_{\mu_1\cdots\mu_n]}, \qquad d^2 = 0 \]

For a function \(f\) (0-form): \(df = \partial_\mu f\, dx^\mu\) (the gradient one-form). For a one-form \(\omega\): \((d\omega)_{\mu\nu} = \partial_\mu \omega_\nu - \partial_\nu \omega_\mu\). For a two-form \(F_{\mu\nu}\): \((dF)_{\mu\nu\rho} = \partial_\mu F_{\nu\rho} + \partial_\nu F_{\rho\mu} + \partial_\rho F_{\mu\nu}\) — this is precisely the Bianchi identity \(\partial_{[\mu}F_{\nu\rho]} = 0\) for the electromagnetic field, equivalent to the homogeneous Maxwell equations.

Exact and closed forms:

• \(\omega\) is exact if \(\omega = d\sigma\); every exact form is closed (\(d\omega = d^2\sigma = 0\))
• \(\omega\) is closed if \(d\omega = 0\); locally every closed form is exact (Poincaré lemma)
• Globally, a closed form need not be exact — the failure measures topology (de Rham cohomology)

de Rham cohomology. The \(k\)-th de Rham cohomology group is \(H^k_{dR}(M) = \ker(d:\ \Omega^k \to \Omega^{k+1}) / \text{im}(d:\ \Omega^{k-1} \to \Omega^k)\), i.e., closed forms modulo exact forms. For \(S^2\): \(H^0 = \mathbb{R}\), \(H^1 = 0\), \(H^2 = \mathbb{R}\). The non-trivial \(H^2\) captures the fact that the area form on \(S^2\) is closed (no 3-forms on a 2-manifold) but not exact (there is no 1-form whose exterior derivative is the area form — which would require \(S^2\) to have a “hole”). The de Rham theorem connects this algebra to the topology of the manifold via integration.

\[ \int_\gamma \omega = \int_{s_1}^{s_2} \omega_\mu \frac{dx^\mu}{ds}\, ds \quad\text{(parameter-independent)} \]

If \(\omega = df\): integral equals \(f(\gamma(s_2)) - f(\gamma(s_1))\) — Fundamental Theorem of Calculus on manifolds.

\[ \operatorname{vol} = \sqrt{|\det g|}\; dx^1 \wedge dx^2 \wedge \cdots \wedge dx^n \]

Integration of an \(n\)-form over an oriented manifold automatically incorporates the correct Jacobian factor from the change of variables.

\[ \int_M d\omega = \int_{\partial M} \omega \]

This unifies the Fundamental Theorem of Calculus, Green’s theorem, the divergence theorem, and the classical Stokes theorem into one statement.

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Chapter 13. Pullbacks, Pushforwards, and Lie Derivatives

Smooth maps. \(\phi: M \to N\) is smooth if the composition with charts is smooth.

Pullback \(\phi^*\). For a function \(f\) on \(N\): \((\phi^* f)(p) = f(\phi(p))\). For a one-form \(\omega\) on \(N\): \((\phi^* \omega)(v) = \omega(\phi_* v)\). Pullback works for all covariant tensors.

Pushforward \(\phi_*\). For a vector \(v \in T_pM\): \((\phi_* v)[f] = v[\phi^* f]\). Pushforward works for contravariant tensors. If \(\phi\) is a diffeomorphism, both operations extend to all tensor types.

\[ \frac{d}{dt}\phi_t(p) = v(\phi_t(p)), \quad \phi_0(p) = p \]

This is just integrating the vector field as a system of differential equations. The flow exists for small \(t\) near every point; if \(v\) has compact support, it is a one-parameter group of diffeomorphisms for all \(t \in \mathbb{R}\).

Lie derivative. The Lie derivative \(\mathcal{L}_v T\) of a tensor \(T\) along a vector field \(v\): drag \(T\) along the flow of \(v\) by an infinitesimal step, compare with the original, divide by the step size. For different tensor types:

\[ \mathcal{L}_v f = v[f], \qquad \mathcal{L}_v w = [v,w], \qquad (\mathcal{L}_v \omega)_\mu = v^\nu \partial_\nu \omega_\mu + \omega_\nu \partial_\mu v^\nu \]\[ (\mathcal{L}_v T)^\mu{}_\nu = v^\rho \partial_\rho T^\mu{}_\nu - T^\rho{}_\nu \partial_\rho v^\mu + T^\mu{}_\rho \partial_\nu v^\rho \]

The general rule: one \(-\partial v\) term (with the same structure as the tensor’s lower index) for each lower index, and one \(+\partial v\) term (with the appropriate structure) for each upper index.

Cartan’s magic formula (for forms): \(\mathcal{L}_v \omega = d(i_v \omega) + i_v(d\omega)\), where \(i_v\) is contraction with \(v\) (interior product, inserting \(v\) into the first slot of \(\omega\)).

Diffeomorphism invariance. A crucial property of GR is diffeomorphism invariance: if \((M, g)\) is a solution of the Einstein equations and \(\phi: M \to M\) is any diffeomorphism, then \((M, \phi^* g)\) is also a solution (with the stress-energy tensor pulled back accordingly). This means that solutions are only defined up to diffeomorphism — a profound statement about the physical content of GR. The Lie derivative \(\mathcal{L}_\xi g = 0\) is the condition for \(\xi\) to generate a diffeomorphism symmetry (isometry), which by Noether’s theorem corresponds to a conserved charge along geodesics — exactly the Killing equation.

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Chapter 14. Affine Connection and Covariant Derivative

\[ (\nabla_\mu V)^\nu = \partial_\mu V^\nu + \Gamma^\nu{}_{\mu\rho}\, V^\rho \]

For a one-form: \((\nabla_\mu \omega)_\nu = \partial_\mu \omega_\nu - \Gamma^\rho{}_{\mu\nu}\, \omega_\rho\).

The pattern: \(+\Gamma\) for each upper index, \(-\Gamma\) for each lower index.

Torsion-free connections: \(\Gamma^\rho{}_{\mu\nu} = \Gamma^\rho{}_{\nu\mu}\) (symmetric in lower indices). In a coordinate basis, this means \([\nabla_\mu, \nabla_\nu]f = 0\) for scalars.

Motivation for the covariant derivative. The professor motivates the need for a new derivative operator through polar coordinates. Consider a constant vector field \(V = \partial_x\) in Cartesian coordinates. In polar coordinates, \(V = \cos\theta\,\partial_r - (\sin\theta/r)\,\partial_\theta\). The components \(V^r = \cos\theta\) and \(V^\theta = -\sin\theta/r\) are not constant, yet \(V\) itself has not changed. The ordinary partial derivative \(\partial_r V^\theta = \sin\theta/r^2 \neq 0\) is telling us about the change in basis vectors, not any intrinsic change in \(V\). The covariant derivative corrects for this: \(\nabla_r V^\theta = \partial_r V^\theta + \Gamma^\theta{}_{r\theta} V^\theta + \Gamma^\theta{}_{rr} V^r\). For flat space in polar coordinates, the Christoffel symbols are \(\Gamma^r{}_{\theta\theta} = -r\), \(\Gamma^\theta{}_{r\theta} = \Gamma^\theta{}_{\theta r} = 1/r\), all others zero. Substituting: \(\nabla_\mu V^\nu = 0\) — as expected for a constant vector field.

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Chapter 15. Parallel Transport and Geodesics

\[ \nabla_{\dot\gamma} V = 0 \implies \frac{dV^\mu}{d\lambda} + \Gamma^\mu{}_{\nu\rho}\,\dot\gamma^\nu\, V^\rho = 0 \]

This is a system of ODEs: given \(V\) at one point, it propagates uniquely along the curve.

Geodesics are the “straightest possible curves.” In Riemannian geometry they also extremize arc length. In GR, free particles follow timelike geodesics.

The professor illustrates with walking on a sphere keeping the nose forward without rotating the head: return to the starting point and the nose points in a different direction. This holonomy is governed by the Riemann tensor.

\[ \delta V^\rho = R^\rho{}_{\sigma\mu\nu}\, V^\sigma\, \delta\Sigma^{\mu\nu} \]

This is the definition of curvature in terms of holonomy — entirely equivalent to the commutator definition. For a sphere of radius \(R\), a vector transported around a latitude circle at colatitude \(\theta\) rotates by \(2\pi\cos\theta\). At the equator (\(\theta = \pi/2\)): no rotation. At the pole (\(\theta = 0\)): rotation by \(2\pi\) (vector makes one full turn). The holonomy angle is \(2\pi(1-\cos\theta)\), equal to the solid angle subtended divided by \(R^2\) — directly related to the Gaussian curvature \(K = 1/R^2\).

Geodesic completeness. A spacetime is geodesically complete if every geodesic can be extended to arbitrary values of its affine parameter. Physical spacetimes are in general incomplete — geodesics that reach a singularity terminate. The Penrose-Hawking singularity theorems (1960s) prove that under reasonable energy conditions, gravitational collapse inevitably produces geodesic incompleteness — singularities are generic in GR, not just special cases like the Schwarzschild solution.

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Chapter 16. Curvature: The Riemann Tensor

\[ [\nabla_a, \nabla_b]\,\omega_c = R^d{}_{abc}\,\omega_d \]\[ R^\rho{}_{\sigma\mu\nu} = \partial_\mu \Gamma^\rho{}_{\nu\sigma} - \partial_\nu \Gamma^\rho{}_{\mu\sigma} + \Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} - \Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma} \]

Symmetries (for Levi-Civita connection):

\[ R_{abcd} = -R_{bacd} = -R_{abdc}, \qquad R_{[abc]d} = 0, \qquad \nabla_{[e}R_{ab]cd} = 0 \]

For \(n=4\): 20 algebraically independent components. The last identity is the differential Bianchi identity; contracted, it gives \(\nabla^a G_{ab} = 0\).

Ricci tensor: \(R_{ac} = R^b{}_{abc}\) (symmetric). Ricci scalar: \(R = g^{ac}R_{ac}\).

\[ R_{abcd} = C_{abcd} + \frac{1}{n-2}\left(g_{ac}R_{bd} - g_{bc}R_{ad} - g_{ad}R_{bc} + g_{bd}R_{ac}\right) - \frac{R}{(n-1)(n-2)}\left(g_{ac}g_{bd} - g_{ad}g_{bc}\right) \]

The Weyl tensor encodes tidal deformation of shapes without volume change. In vacuum (where \(R_{ab} = 0\)), the Weyl tensor is the only nonzero part of the curvature — it carries gravitational wave content and tidal forces. The Schwarzschild metric has zero Ricci tensor but nonzero Weyl tensor; the gravitational field of a star in vacuum is pure Weyl curvature.

\[ G_{ab} = R_{ab} - \frac{1}{2}Rg_{ab}, \qquad \nabla^a G_{ab} = 0 \;\text{(contracted Bianchi identity)} \]

The contracted Bianchi identity \(\nabla^a G_{ab} = 0\) is an identity — it holds for any metric, without needing the Einstein equations. When the Einstein equations are imposed (\(G_{ab} = 8\pi G_N T_{ab}\)), it immediately implies \(\nabla^a T_{ab} = 0\): the stress-energy tensor is automatically conserved. This is not a separate assumption (Postulate 2) but rather a consequence of the geometric identity plus the Einstein equations.

\[ \frac{D^2\xi^\mu}{d\tau^2} = R^\mu{}_{\nu\rho\sigma}\,u^\nu\, u^\rho\, \xi^\sigma \]

The Riemann tensor is responsible for tidal forces — the geometric description of gravity. In Newtonian gravity, tidal acceleration between two nearby particles separated by \(\xi^i\) is \(\ddot\xi^i = -\partial^i\partial_j\Phi\,\xi^j\) (the tidal tensor is the second derivative of the potential). In GR, the Riemann tensor generalises this: the four-dimensional curvature encodes both the Newtonian tidal tensor and the magnetic part of the gravitational field (frame-dragging effects, gravitational waves).

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Chapter 17. Metric Manifolds and the Levi-Civita Connection

Pseudo-Riemannian metric. A symmetric non-degenerate \((0,2)\) tensor field \(g_{ab}\). For GR: Lorentzian signature \((-,+,+,+)\). Provides inner products (not positive definite), index raising/lowering, and lengths.

Metric-compatibility means parallel transport preserves inner products. Geodesics remain timelike, spacelike, or null throughout.

Proof of uniqueness. Suppose \(\nabla\) is torsion-free and metric-compatible. Write three copies of \(\nabla_\rho g_{\mu\nu} = 0\) with cyclically permuted indices: \begin{align} \partial_\rho g_{\mu\nu} &= \Gamma^\sigma{}{\rho\mu}g{\sigma\nu} + \Gamma^\sigma{}{\rho\nu}g{\mu\sigma}\ \partial_\mu g_{\nu\rho} &= \Gamma^\sigma{}{\mu\nu}g{\sigma\rho} + \Gamma^\sigma{}{\mu\rho}g{\nu\sigma}\ \partial_\nu g_{\rho\mu} &= \Gamma^\sigma{}{\nu\rho}g{\sigma\mu} + \Gamma^\sigma{}{\nu\mu}g{\rho\sigma} \end{align}

Take the combination (second + third − first), use torsion-freeness \(\Gamma^\sigma{}_{\mu\nu} = \Gamma^\sigma{}_{\nu\mu}\), and solve for \(\Gamma^\rho{}_{\mu\nu}\) to obtain the formula above. The derivation shows that the metric completely determines the connection — there are no free parameters.

Geodesics as proper-length extrema. For a Riemannian metric (positive definite), geodesics minimise arc length between nearby points. For a Lorentzian metric, timelike geodesics maximise proper time — the straightest worldline has the most elapsed proper time. This is why the twin who stays home (straight worldline) ages more than the twin who travels (bent worldline). For spacelike geodesics, the arc length is a saddle point in general.

\[ L[\gamma] = \int \sqrt{|g_{\mu\nu}\dot x^\mu \dot x^\nu|}\, d\lambda \]

This is reparametrisation-invariant. The convention is to use the arc length parameter (proper time \(\tau\) for timelike curves) so that \(g_{\mu\nu}u^\mu u^\nu = -c^2\) (our normalisation). The geodesic equation \(\nabla_{\dot\gamma}\dot\gamma = 0\) is equivalent to the Euler-Lagrange equations for this functional, with the affine parameter as a special choice that simplifies the resulting equation.

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Chapter 18. Isometries and Killing Vectors

\[ \nabla_a k_b + \nabla_b k_a = 0 \]

For torsion-free connections, this is equivalent to \(\mathcal{L}_k g = 0\).

\[ Q = k_a v^a = \text{constant along the geodesic} \]\[ \frac{d}{d\tau}(k_\mu u^\mu) = u^\nu \nabla_\nu(k_\mu u^\mu) = u^\nu(\nabla_\nu k_\mu)u^\mu + k_\mu u^\nu \nabla_\nu u^\mu = u^\nu u^\mu \nabla_{(\nu}k_{\mu)} + 0 \]

The second term vanishes because \(u^\mu\) is geodesic (\(\nabla_u u = 0\)), and the first vanishes by the Killing equation (\(\nabla_{(\nu}k_{\mu)} = 0\)). QED.

This theorem is the backbone of practical calculations in GR. For the Schwarzschild metric, there are two Killing vectors (time translation \(\partial_t\) and azimuthal rotation \(\partial_\phi\)) giving two conserved quantities (energy \(E\) and angular momentum \(L\)) that reduce the geodesic equation to a one-dimensional problem.

Killing vectors of Minkowski spacetime (all 10):

Generator	Symmetry	Conserved quantity
\(\partial_t\)	Time translation	Energy
\(\partial_x, \partial_y, \partial_z\)	Spatial translations	Momentum
\(y\partial_z - z\partial_y\), etc.	Rotations	Angular momentum
\(x\partial_t + t\partial_x\), etc.	Boosts	Centre-of-mass velocity

The professor (Lecture 22) finds all 10 Killing vectors systematically by solving the Killing equation for the Minkowski metric \(\eta_{\mu\nu}\). The general solution is \(k_\mu = a_\mu + b_{\mu\nu}x^\nu\) where \(a_\mu\) is an arbitrary constant (4 parameters — translations) and \(b_{\mu\nu} = -b_{\nu\mu}\) is antisymmetric (6 parameters — Lorentz generators). This gives exactly 10 independent Killing vectors.

An \(n\)-dimensional manifold has at most \(n(n+1)/2\) independent Killing vectors. For \(n=4\): maximum 10, achieved by Minkowski spacetime and de Sitter/anti-de Sitter spacetimes (maximally symmetric).

Stationary spacetimes: a timelike Killing vector exists, so the metric does not depend on \(t\). Static spacetimes: additionally \(g_{0i} = 0\) — no rotational dragging. The Schwarzschild metric is static (outside the horizon). The Kerr metric (rotating black hole) is stationary but not static: the frame dragging means there is no timelike Killing vector that is also hypersurface-orthogonal.

Conformal Killing vectors. A vector field \(\xi\) is a conformal Killing vector if \(\mathcal{L}_\xi g_{\mu\nu} = \lambda\, g_{\mu\nu}\) for some function \(\lambda\). It generates a conformal isometry — a diffeomorphism that scales the metric by a position-dependent factor. Minkowski spacetime has 15 conformal Killing vectors (10 Poincaré + 4 special conformal transformations + 1 dilation), forming the conformal group \(SO(2,4)\).

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Part III — General Relativity

Chapter 19. The Postulates of General Relativity

The Weak Equivalence Principle (WEP): all objects fall identically in a gravitational field, regardless of their composition. Einstein elevated this to a structural principle: locally, in a freely falling frame, all laws of physics look like special relativity. There is no local experiment that distinguishes free fall in a gravitational field from inertial motion in empty space.

The WEP has been tested to one part in \(10^{13}\) by Eötvös-type experiments. The MICROSCOPE satellite experiment (2022) confirmed it to \(10^{-15}\). General Relativity assumes it is exact. Extensions of GR (scalar-tensor theories like Brans-Dicke) can violate the Strong Equivalence Principle (which adds that gravitational self-energy also falls the same way) while preserving WEP.

Tidal forces and the need for local frames. There is a subtle but essential qualification to the equivalence principle: it can only hold locally. Consider a large enough falling elevator — particles near the floor and near the ceiling experience slightly different gravitational accelerations, since gravity weakens with distance. These tidal forces, caused by the non-uniformity of the field, are not removable by any choice of freely-falling frame. Two nearby freely-falling particles that begin at rest relative to each other will slowly drift apart (or together) as they fall, revealing the curvature lurking beneath the locally flat appearance of spacetime. Tidal forces measure the second derivative of the gravitational potential, which is precisely the Riemann curvature tensor. This is why the equivalence principle must be stated carefully: in a sufficiently small, local freely-falling frame, spacetime looks flat to first order. The word “local” carries the entire weight.

\[ \lambda_B = \lambda_A\!\left(1 - \frac{gh}{c^2}\right) \]

The signal is blue-shifted: the lower observer receives more cycles per second than the upper one sends. By Einstein’s Strong Equivalence Principle, a rocket stationary in a uniform gravitational field \(g\) must behave identically. A photon climbing out of a gravitational well is therefore red-shifted by the factor \(\Delta\lambda/\lambda \approx gh/c^2\). On the Earth’s surface this is of order \(10^{-15}\). Pound and Rebka measured exactly this shift in 1959 using the Mössbauer effect, confirming the prediction to 10% precision. The result is profound: time itself runs slower deeper in a gravitational field, and this follows from nothing more than the equivalence principle.

Early attempts at a relativistic theory of gravity. Before arriving at general relativity in November 1915, Einstein had to navigate a thicket of plausible-but-wrong theories. The history is instructive because it shows that the tensor field equations were not logically inevitable — and that the key discriminating experiment was the bending of light.

\[ \Box\,\phi = 4\pi G\, T^\alpha{}_\alpha \]

where \(\Box = \eta^{\mu\nu}\partial_\mu\partial_\nu\) is the Minkowski d’Alembertian. The theory is linear and, problematically, gravity does not self-gravitate. Nordstrom revised it in 1913 to the nonlinear equation \(\Box\phi/\phi = 4\pi G\,T^\alpha{}_\alpha\), which at least admits self-interaction. In 1914, Einstein and Fokker showed this is equivalent to the geometric scalar theory \(R = \tilde T^\alpha{}_\alpha\) — a genuinely covariant and geometric gravity theory. Yet all of these scalar theories share a fatal flaw. Light has a trace-free stress-energy tensor, \(T^\alpha{}_\alpha = 0\) for electromagnetic radiation, so scalar-gravity predicts no light bending at all. Since the equivalence principle itself requires light to bend — the Pound-Rebka argument shows photons must gain and lose energy in a gravitational field — any theory with zero light bending is ruled out. Einstein’s full tensor theory arrived in 1915, and Eddington’s 1919 measurement of light deflection during a solar eclipse confirmed it decisively.

The lesson is clean: the source of gravity cannot be merely the trace of the stress-energy tensor. The full symmetric tensor \(T_{\mu\nu}\) must source the geometry, which is why the Einstein equations couple the Ricci tensor — not a scalar — to the complete stress-energy content of matter.

\[ d\tau^2 = \left(1 + \frac{2\phi}{c^2}\right)c^2\,dt^2 - \left(1 - \frac{2\phi}{c^2}\right)\!\left(dx^2 + dy^2 + dz^2\right) \]

The \(g_{00}\) component encodes gravitational time dilation; the spatial components encode a position-dependent length scale. For \(\phi = -G_N M/r\), this is the \(R_s/r \ll 1\) limit of the Schwarzschild metric — a consistency check, and a hint that the exact solution must “exponentiate” the weak-field perturbation.

From WEP to curved spacetime. If freely-falling observers experience flat spacetime locally, then the geodesics of freely-falling test particles determine the geometry. Gravity is not a force but a manifestation of spacetime curvature: a ball thrown upward follows a geodesic in curved spacetime, and what we perceive as the “force of gravity” is really the curvature of that spacetime causing nearby geodesics to converge. The geodesic deviation equation captures this precisely: \(D^2\xi^\mu/d\tau^2 = R^\mu{}_{\nu\rho\sigma}u^\nu u^\rho \xi^\sigma\) is the GR version of Newton’s tidal force equation \(\ddot\xi^i = (\partial_i\partial_j\Phi)\xi^j\).

The left side is purely geometric; the right side encodes matter and energy. Matter tells spacetime how to curve; curved spacetime tells matter how to move.

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Chapter 20. The Stress-Energy Tensor and Energy Conditions

\[ T_{ab} = (\rho + p)\,u_a u_b + p\,g_{ab} \]

Conservation \(\nabla_b T^{ab} = 0\) gives the relativistic Euler equations. For dust (\(p=0\)): particles follow geodesics, confirming Postulate 0.

Physical interpretation. The stress-energy tensor \(T^{ab}\) has a direct physical meaning in any orthonormal frame:

• \(T^{00}\) = energy density measured by an observer with four-velocity \(u^\mu\)
• \(T^{0i}\) = momentum flux in the \(i\)-direction (also energy flux per unit area per unit time)
• \(T^{ij}\) = stress tensor (momentum flux in \(j\)-direction through a surface with normal in \(i\)-direction)

For a perfect fluid at rest, \(T^{ij} = p\delta^{ij}\) (isotropic pressure). For radiation (photon gas), the equation of state is \(p = \rho/3\), giving \(T = T^\mu{}_\mu = -\rho + 3p = 0\) — the trace vanishes. This is a consequence of conformal invariance for massless particles.

Why \(8\pi G_N\) in the Einstein equations? The coefficient is fixed by requiring the Newtonian limit. For a non-relativistic source with density \(\rho\) and small metric perturbation \(g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}\) (with \(|h_{\mu\nu}| \ll 1\)), the \(00\)-component of the Einstein equations reduces to \(\nabla^2 h_{00} = -8\pi G_N \rho\). Comparing with the Poisson equation \(\nabla^2\Phi = 4\pi G_N \rho\) and the identification \(h_{00} = -2\Phi/c^2\) fixes the coefficient to be \(8\pi G_N/c^4\) (restoring \(c\)). The factor of 2 mismatch is precisely what gives the factor of 8 rather than 4\(\pi G_N\).

Stress-energy tensor examples:

• Scalar field of mass \(m\): \(T_{ab} = \nabla_a\phi\nabla_b\phi - \tfrac{1}{2}g_{ab}(\nabla^c\phi\nabla_c\phi + m^2\phi^2)\)
• Electromagnetic field: \(T_{ab} = \tfrac{1}{4\pi}(F_{ac}F_b{}^c - \tfrac{1}{4}g_{ab}F_{cd}F^{cd})\)

Energy conditions (not part of GR itself, but imposed to exclude pathological solutions):

Implication chain: \(\text{DEC} \Rightarrow \text{WEC} \Rightarrow \text{NEC}\), and \(\text{SEC} \Rightarrow \text{NEC}\).

Quantum fields generically violate the WEC — the Casimir effect provides a laboratory example of negative energy density. This is why quantum gravity is intimately connected to exotic solutions like wormholes.

Energy conditions in cosmology. The accelerated expansion of the universe (discovered 1998) requires the dark energy to violate the strong energy condition: \(\rho + 3p < 0\), i.e., \(w < -1/3\). A cosmological constant has \(w = -1\), violating SEC while preserving WEC. Phantom fields with \(w < -1\) violate even WEC, and could lead to a “Big Rip” where the scale factor diverges in finite proper time, tearing apart all bound structures.

Hawking radiation preview. The four laws of black hole thermodynamics are:

Law	Thermodynamic	Black hole
0th	Temperature constant in equilibrium	Surface gravity \(\kappa\) constant on horizon
1st	\(dE = T\,dS + \text{work}\)	\(dM = \frac{\kappa}{8\pi G_N}dA + \Omega\,dJ + \Phi\,dQ\)
2nd	Entropy never decreases	Horizon area never decreases (classical)
3rd	Cannot reach \(T = 0\)	Cannot reach \(\kappa = 0\) (extremal BH)

Hawking radiation promotes the 0th and 2nd laws to actual equalities: black hole temperature \(T_H = \hbar\kappa/(2\pi k_B c)\) and Bekenstein-Hawking entropy \(S_{BH} = k_B A/(4\ell_P^2)\).

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Chapter 21. The Schwarzschild Metric and Weak-Field Gravity

Birkhoff’s theorem. The unique vacuum, spherically symmetric solution of the Einstein equations (up to diffeomorphism) is the Schwarzschild metric. Birkhoff’s theorem also implies that the exterior field of any spherically symmetric body is Schwarzschild — even if the star is pulsating. There is no spherically symmetric gravitational radiation. This is the GR analogue of Newton’s shell theorem.

\[ ds^2 = -\left(1 - \frac{R_s}{r}\right)c^2\,dt^2 + \left(1 - \frac{R_s}{r}\right)^{-1}dr^2 + r^2 d\Omega^2 \]

where \(d\Omega^2 = d\theta^2 + \sin^2\theta\,d\phi^2\). The metric is asymptotically flat: as \(r\to\infty\) it becomes Minkowski.

Derivation. It is worth understanding where this metric comes from — Schwarzschild obtained it within weeks of Einstein’s November 1915 paper, while serving in the German army on the Eastern Front, before dying of illness the following year. The derivation is a beautiful illustration of how symmetry alone constrains a solution.

\[ ds^2 = -e^{2\alpha(r)}\,dt^2 + e^{2\beta(r)}\,dr^2 + r^2\,d\Omega^2 \]\[ R_{tt} = 0 \implies e^{2(\alpha-\beta)}\!\left(\alpha'' + (\alpha')^2 - \alpha'\beta' + \frac{2\alpha'}{r}\right) = 0 \]\[ R_{rr} = 0 \implies -\alpha'' - (\alpha')^2 + \alpha'\beta' + \frac{2\beta'}{r} = 0 \]\[ \alpha(r) = -\beta(r) \]\[ \partial_r\!\left(r\, e^{2\alpha}\right) = 1 \]

Integrating: \(r\,e^{2\alpha} = r - R_s\) for an integration constant \(R_s\), so \(e^{2\alpha} = 1 - R_s/r\), and the metric follows. The constant is fixed by the weak-field limit: \(g_{tt} \approx -(1 + 2\phi/c^2)\) with \(\phi = -G_N M/r\) gives \(R_s = 2G_N M/c^2\). Birkhoff’s theorem guarantees this is the unique vacuum spherically symmetric solution — even for a pulsating source.

\[ R_s = \frac{2G_N M}{c^2} \]

Relativistic correction strength \(2|\phi|/c^2\): \(\sim 10^{-9}\) on Earth’s surface, \(\sim 10^{-6}\) on the Sun, \(\sim 0.1\) on a neutron star.

Geodesic equation from variational principle. For a particle following a timelike geodesic in Schwarzschild geometry, vary \(S = -mc\int ds\). The Killing vectors give conserved quantities:

\[ E = -\left(1 - \frac{R_s}{r}\right)\dot t \quad\text{(energy)}, \qquad L = r^2\dot\phi \quad\text{(angular momentum)} \]\[ \frac{1}{2}\dot r^2 + V(r) = \frac{1}{2}E^2, \qquad V(r) = \frac{1}{2} - \frac{G_N M}{r} + \frac{L^2}{2r^2} - \frac{G_N M L^2}{r^3} \]

The last term is the GR correction; the rest matches Newtonian mechanics.

\[ \nabla_V(\xi^\mu p_\mu) = (\nabla_V \xi^\mu)p_\mu + \xi^\mu(\nabla_V p_\mu) = V^\alpha V^\beta \nabla_\alpha \xi_\beta + 0 \]

where the second term vanishes by the geodesic equation. The first term vanishes because \(\nabla_\alpha \xi_\beta\) is antisymmetric (Killing equation) while \(V^\alpha V^\beta\) is symmetric — their contraction is zero. Thus \(\xi^\mu p_\mu = \text{const}\) along any photon worldline.

\[ u^\mu = \left(1 - \frac{R_s}{r}\right)^{-1/2}\!\xi^\mu \]\[ E(r) = \left(1 - \frac{R_s}{r}\right)^{-1/2}\!(-\xi^\mu p_\mu) \]\[ \frac{E_1}{E_2} = \sqrt{\frac{1 - R_s/r_2}{1 - R_s/r_1}} \]\[ \frac{E_2}{E_1} = \sqrt{\frac{1 - R_s/r_1}{1 - R_s/r_2}} \]

Since \(r_1 < r_2\), we have \(E_2 < E_1\): the photon loses energy climbing out of the gravitational well. Equivalently, \(\lambda_2 > \lambda_1\) — the wavelength is stretched, the light is red-shifted. In the weak-field limit \(R_s \ll r_{1,2}\), this reduces to \(E_2/E_1 \approx 1 - R_s(1/r_1 - 1/r_2)/2 \approx 1 + \Delta\phi/c^2\), recovering the Pound-Rebka result.

GPS correction: satellite clocks at \(r_2 \approx 4.2\times 10^7\) m run \(\approx 0.53\) ns/s faster than Earth clocks due to gravitational time dilation. There is also a competing special-relativistic effect: the satellite’s orbital speed (\(v \approx 3.9\) km/s) causes time dilation that slows the clock by \(\approx -0.22\) ns/s. The net correction is \(+0.31\) ns/s. Accumulated over 24 hours: \(\approx 27 \mu\)s. At 30 cm per 1 ns timing error, ignoring GR would cause GPS to accumulate a position error of \(\sim 8\) km per day. GPS would be useless within hours without relativistic corrections.

\[ \Delta\phi = \frac{6\pi G_N M}{(1-e^2)a} \]

For Mercury: \(\Delta\phi \approx 43''\)/century. This matched the long-standing anomaly (the total observed precession is \(574''\)/century, of which \(531''\) is due to perturbations from other planets; the remaining \(43''\) was unexplained for decades) and became the first quantitative confirmation of GR.

Derivation of perihelion precession. The result \(\Delta\phi = 6\pi G_N M/[(1-e^2)a]\) is not obvious — it emerges from a perturbative analysis of the radial geodesic equation. Let us trace the argument.

\[ E = \left(1 - \frac{R_s}{r}\right)\dot t, \qquad L = r^2\dot\phi \]\[ -E^2 + \dot r^2 + \frac{L^2}{r^2}\!\left(1 - \frac{R_s}{r}\right) = -\!\left(1 - \frac{R_s}{r}\right) \]\[ \frac{1}{2}\dot r^2 + V(r) = \mathcal{E}, \qquad V(r) = \frac{1}{2} - \frac{R_s}{2r} + \frac{L^2}{2r^2} - \frac{R_s L^2}{2r^3}, \qquad \mathcal{E} = \frac{1}{2}E^2 \]\[ \frac{d^2 x}{d\phi^2} - 1 + x = \frac{3}{4}\frac{R_s^2}{L^2}\,x^2 \]\[ x \approx 1 + e\cos\!\left[(1 - \alpha)\phi\right], \qquad \alpha = \frac{3}{4}\frac{R_s^2}{L^2} \approx \frac{3R_s}{(1-e^2)a} \]

The perihelion advances by \(\Delta\phi = 2\pi\alpha\) per orbit, giving the formula above. The calculation is structurally identical to the secular perturbation theory used in celestial mechanics, but the perturbation now comes from the geometry of spacetime rather than from other planets.

Stable and unstable circular orbits. Setting \(dV/dr = 0\) and \(d^2V/dr^2 > 0\) for the effective potential, stable circular orbits in Schwarzschild exist for \(r > 3R_s\). The innermost stable circular orbit (ISCO) is at \(r_\text{ISCO} = 3R_s = 6G_N M/c^2\). Below this radius, only unstable circular orbits (saddle points) or plunging trajectories exist. The ISCO is crucial for accretion disk physics — matter spirals in and falls off the ISCO into the black hole, radiating away energy. The efficiency of accretion onto a Schwarzschild black hole is \(\eta = 1 - \sqrt{8/9} \approx 5.7\%\) of rest mass energy, far greater than nuclear fusion (\(\sim 0.7\%\)).

For null geodesics, the analogous unstable circular orbit (photon sphere) is at \(r = 3R_s/2\).

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Chapter 22. Orbital Mechanics and Deflection of Light

Light deflection. Light follows null geodesics (\(g_{\mu\nu}\dot x^\mu\dot x^\nu = 0\)). The same effective potential analysis applies with \(m=0\). The deflection angle for a ray grazing the Sun:

\[ \Delta\phi = \pi + \frac{4G_N M}{bc^2} + \mathcal{O}(G_N^2) \]

where \(b = L/E\) is the impact parameter. The Newtonian prediction gives only half this angle. Eddington’s 1919 solar eclipse measurement confirmed the GR value, making Einstein world-famous. The deflection for rays grazing the Sun is \(\approx 1.75''\).

\[ \frac{1}{b^2} = \frac{\dot r^2}{L^2} + W_\text{eff}(r), \qquad W_\text{eff}(r) = \frac{1}{r^2}\!\left(1 - \frac{R_s}{r}\right) \]\[ \Delta\phi = 2\int_{r_1}^{\infty} \frac{dr}{r^2}\!\left(\frac{1}{b^2} - W_\text{eff}(r)\right)^{-1/2} \]\[ \delta\phi = \frac{2R_s}{b} = \frac{4G_N M}{bc^2} \]

The factor of 2 relative to the Newtonian result arises because both the \(g_{tt}\) and \(g_{rr}\) components of the metric contribute equally to the deflection — the spatial curvature doubles the answer. Newtonian gravity only modifies \(g_{tt}\) (the gravitational potential) but leaves spatial geometry flat; GR curves both time and space, giving the full deflection. Eddington’s eclipse measurement was the first direct test of this spatial curvature contribution, and it gave a result consistent with GR and inconsistent with the Newtonian (half) value.

\[ \theta_E = \sqrt{\frac{4G_N M}{c^2}\frac{D_{LS}}{D_L D_S}} \]

Gravitational lensing is used to map dark matter distributions in galaxy clusters, detect exoplanets via microlensing, and measure the Hubble constant through time delays.

\[ V_\text{null}(r) = \frac{L^2}{2r^2}\left(1 - \frac{R_s}{r}\right) \]

The maximum of \(V_\text{null}\) occurs at \(r = 3R_s/2\) — the photon sphere, where circular null geodesics exist. A photon with \(b < 3\sqrt{3}\,R_s/2\) is captured; with \(b > 3\sqrt{3}\,R_s/2\) it escapes. The photon sphere is the edge of the black hole shadow visible to distant observers.

Einstein–Newton race. Two observers — one jumping through a tunnel through the Earth, one flying in a sub-orbital arc — who arrives first? Newtonian gravity: both take approximately 42 minutes (a remarkable coincidence). GR predicts the tunnel traveller arrives slightly later by a few microseconds — a non-intuitive result driven by the greater proper time accumulated in stronger curvature.

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Chapter 23. Schwarzschild Black Holes

\[ K = R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma} = \frac{12 R_s^2}{r^6} \]

At \(r = 0\): \(K \to \infty\) — genuine curvature singularity. At \(r = R_s\): \(K\) is finite — the divergence in Schwarzschild coordinates is a coordinate singularity (removable by changing coordinates, like the North Pole in longitude-latitude).

Eddington-Finkelstein coordinates. The coordinate singularity at \(r = R_s\) can be removed by a clever change of variables. The idea, due to Eddington (1924) and rediscovered by Finkelstein (1958), is to trade the Schwarzschild time \(t\) for a coordinate adapted to ingoing null geodesics.

\[ v = t + r + R_s\ln\!\left|\frac{r}{R_s} - 1\right| \]\[ ds^2 = -\!\left(1 - \frac{R_s}{r}\right)dv^2 + 2\,dv\,dr + r^2\,d\Omega^2 \]\[ g_{\alpha\beta} = \begin{pmatrix}-(1-R_s/r) & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & r^2 \end{pmatrix} \]\[ \left(1 - \frac{R_s}{r}\right)dv^2 - 2\,dv\,dr = 0 \implies dv\!\left[\left(1 - \frac{R_s}{r}\right)dv - 2\,dr\right] = 0 \]

The two families of null rays are: (i) \(dv = 0\), i.e., \(v = \text{const}\) — these are the ingoing rays; (ii) \((1-R_s/r)\,dv = 2\,dr\), which integrates to \(v = 2r + 2R_s\ln|r - R_s| + \text{const}\). Outside the horizon (\(r > R_s\)) the second family moves outward. At the horizon (\(r = R_s\)) the equation becomes \(0 = 2\,dr\), so \(r = R_s\) is itself a null surface — the trapped null ray that hovers at the horizon forever. Inside (\(r < R_s\)), both families move toward decreasing \(r\). There is no way for a light signal emitted at \(r < R_s\) to escape to \(r > R_s\). The surface \(r = R_s\) is a one-way membrane — the event horizon — and EF coordinates make this causally transparent without any coordinate pathology.

The EF chart covers only the exterior region and the black hole interior (\(r > 0\)). It does not naturally describe the white hole region. To expose the full maximal extension — all four quadrants of the Kruskal diagram — one needs the Kruskal-Szekeres coordinates.

\[ ds^2 = \frac{4R_s^3}{r}e^{-r/R_s}\left(-dT^2 + dX^2\right) + r^2 d\Omega^2 \]

The horizon \(r = R_s\) corresponds to \(T = \pm X\) — perfectly regular.

Three-observer analysis:

• Alice (infalling): reaches \(r = R_s\) in finite proper time \(\Delta\tau = \frac{2}{3}\frac{R_s}{c}\left(\frac{r_0}{R_s}\right)^{3/2}\) from initial radius \(r_0\), continues inside. At the horizon, nothing special happens locally — tidal forces for a stellar black hole (\(M \sim 10 M_\odot\)) are already enormous at the horizon; for a supermassive black hole (\(M \sim 10^9 M_\odot\)), they are negligible at the horizon.
• Bob (at infinity): sees Alice’s light increasingly redshifted, her image frozen at the horizon, exponentially dimming. The time constant of dimming is \(\tau \sim R_s/c\). From Bob’s perspective, Alice never crosses — but the signal becomes exponentially faint.
• Carol (stationary at finite \(r\)): also sees Alice’s light redshift, but can actually see her pass through the horizon in Carol’s proper time if \(r\) is large enough and Carol watches long enough. The coordinate description hides this — Kruskal coordinates reveal it clearly.

Inside the horizon. For \(r < R_s\), \(\partial_r\) is timelike and \(\partial_t\) is spacelike. The singularity at \(r=0\) is not in “another place” but in Alice’s future — every timelike geodesic inside reaches \(r = 0\) in finite proper time. The singularity is inevitable.

\[ \Delta\tau \approx \frac{\pi M}{c}\left(\frac{M}{M_\odot}\right) \times 10^{-5}\text{ s} \]

For the most massive known black holes (\(\sim 10^{10} M_\odot\)): approximately 2.5 days of interior experience.

Penrose diagrams. A conformal compactification that brings infinity to finite distance, preserving causal structure. Light rays always travel at 45°.

Definition of a black hole. In an asymptotically flat spacetime, a black hole is a region from which null geodesics cannot reach future null infinity \(\mathcal{I}^+\). The event horizon is the boundary of this region.

\[ T_H = \frac{\hbar c^3}{8\pi G_N M k_B} \approx \frac{6 \times 10^{-8}\,\text{K}}{M/M_\odot} \]

For stellar black holes (\(M \sim 10 M_\odot\)), \(T_H \sim 10^{-9}\,\text{K}\) — far below the cosmic microwave background temperature. Hawking radiation is currently undetectable, but it has profound implications: black holes slowly evaporate, and the information paradox (does information fall in or come back out?) remains one of the deepest open problems in physics.

Ergosphere (Kerr black holes). Rotating black holes (Kerr metric, 1963) have an ergosphere — a region outside the horizon where no stationary observer can exist (the frame-dragging is so strong that you cannot remain at rest). The Penrose process extracts energy from the black hole’s rotation by splitting a particle in the ergosphere: one fragment falls in carrying negative energy (as measured at infinity), the other escapes with more energy than the original particle had. The maximum extractable fraction of a Kerr black hole’s mass is \(1 - 1/\sqrt{2} \approx 29\%\).

\[ ds^2 = -\left(1 - \frac{R_s r}{\Sigma}\right)c^2 dt^2 - \frac{2aR_s r \sin^2\theta}{\Sigma}c\, dt\, d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\, d\theta^2 + \left(r^2 + a^2 + \frac{aR_s r\sin^2\theta}{\Sigma}\right)\sin^2\theta\, d\phi^2 \]

where \(\Sigma = r^2 + a^2\cos^2\theta\), \(\Delta = r^2 - R_s r + a^2\), and \(a = J/(Mc)\) is the specific angular momentum. The First Image of a Black Hole (Event Horizon Telescope, M87*, 2019) is consistent with the Kerr metric.

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Part IV — Cosmology

Chapter 24. The FLRW Metric and Friedmann Equations

The dark night sky and the end of the static universe. Up until the early twentieth century, scientists assumed the universe to be infinite, static, and eternal — a vast unchanging stage for local events. The first serious challenge to this picture came not from measurement but from a simple observation: the night sky is dark.

If the universe were infinite, uniform, and eternal, then every line of sight from Earth would eventually intersect the surface of a star. The night sky would be as bright as the surface of the Sun in every direction. This is Olbers’ paradox, first formulated clearly in 1826, and it rules out the classical infinite static universe. One resolution is that the universe had a beginning — stars have finite ages and their light has had finite time to reach us — but in 1826 this seemed philosophically unacceptable.

The observational breakthrough came in stages. In 1912, Slipher measured Doppler shifts of nearby galaxies and found most were receding from us. In 1929, Hubble extended Slipher’s work to greater distances, using Cepheid variable stars as standard candles, and found a striking linear relationship: the recession velocity of a galaxy is proportional to its distance. Written as \(v = H_0 d\), this is Hubble’s law, with Hubble’s original constant \(H_0 \approx 500\) km/s/Mpc (now revised to \(67\)–\(73\) km/s/Mpc). The implication was unavoidable: the space between galaxies is growing, all galaxies are moving away from each other, and the universe is not static. Running the film backward, everything must have been much closer together in the past — the universe has a beginning.

At this point, general relativity became the natural tool for studying the evolution of the universe as a whole. To apply it, one needs a metric that describes a homogeneous, isotropic, and expanding spacetime. The symmetry analysis that produces this metric also teaches us what “homogeneous and isotropic” means in the language of Killing vectors.

The cosmological principle. The universe is:

1. Homogeneous: no preferred locations
2. Isotropic: no preferred directions (for comoving observers)

These two demands, together with the requirement that the spacetime be time-orientable, strongly constrain the metric. Observational evidence comes from: (1) the nearly perfect isotropy of the cosmic microwave background (CMB) to one part in \(10^5\), (2) the large-scale galaxy surveys showing homogeneity on scales above \(\sim 300\) Mpc, and (3) the Copernican principle (we are not in a special location). The cosmological principle is an approximation — on small scales the universe is highly inhomogeneous (stars, galaxies, voids).

Comoving coordinates and proper distance. In FLRW, comoving coordinates \((r, \theta, \phi)\) are fixed to the matter distribution — they label galaxies by their positions at some reference time. The proper distance between two comoving points at time \(t\) is \(d(t) = a(t)r\). The comoving distance \(r\) is constant in time; the proper distance grows with the expansion. The Hubble law \(v = Hd\) arises from differentiating: \(\dot d = \dot a r = (a/a) ar = Hd\). The recession velocity can exceed \(c\) for large distances — this is not a violation of relativity because it is not a motion through space but an expansion of space itself. No information travels faster than light.

\[ 1 + z = \frac{\lambda_r}{\lambda_e} = \frac{a_r}{a_e} \]

The redshift \(z\) directly measures the ratio of scale factors, not a Doppler shift (though both descriptions are valid in the limit of small distances). The CMB was emitted at \(z \approx 1100\), when the universe was \(\approx 380\,000\) years old and the scale factor was \(1/1101\) of its present value.

\[ R_{\alpha\beta\gamma\delta} = \frac{k}{n(n-1)}\left(g_{\alpha\gamma}g_{\beta\delta} - g_{\alpha\delta}g_{\beta\gamma}\right) \]

For 3-dimensional space, the maximum is \(3 \cdot 4/2 = 6\) Killing vectors (3 rotations + 3 translations). The three maximally symmetric spaces are the flat Euclidean space \(\mathbb{R}^3\) (with \(k=0\)), the round 3-sphere \(S^3\) (with \(k > 0\)), and the hyperbolic space \(\mathcal{H}^3\) (with \(k < 0\)).

Homogeneity (6 translational Killing vectors from isotropy at every point, and 3 rotational ones) combined with isotropy forces the spatial sections to be maximally symmetric. This is not an assumption about the universe’s large-scale geometry — it is a consequence of requiring that space looks the same from every location in every direction. The three possibilities are exactly \(k \in \{-1, 0, +1\}\), up to rescaling.

Derivation of the FLRW metric. Starting from the most general metric and imposing homogeneity and isotropy systematically:

1. Isotropy forces \(g_{0i} = 0\) (no preferred direction in space)
2. Homogeneity forces the spatial sections to have constant curvature \(k \in \{-1, 0, +1\}\)
3. The time dependence is encoded in a single function \(a(t)\) — the scale factor

The Friedmann–Lemaître–Robertson–Walker (FLRW) metric:

\[ ds^2 = -c^2\,dt^2 + a(t)^2\left[\frac{dr^2}{1-kr^2} + r^2 d\Omega^2\right] \]

Three cases:

\(k\)	Spatial geometry	Topology
\(+1\)	Closed (3-sphere)	Finite volume
\(0\)	Flat (Euclidean)	Infinite (likely)
\(-1\)	Open (hyperbolic)	Infinite

Current observations favour \(k \approx 0\).

The scale factor \(a(t)\). Physical distances scale as \(a(t)\). The Hubble parameter \(H(t) = \dot a/a\) measures the rate of expansion. Today: \(H_0 \approx 67\)–\(73\) km/s/Mpc.

Friedmann equations. Apply Einstein equations to FLRW with a perfect fluid:

\[ \left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G_N}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} \]\[ \frac{\ddot a}{a} = -\frac{4\pi G_N}{3}\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3} \]

The first is the Friedmann equation; the second is the Raychaudhuri acceleration equation.

Equations of state and evolution. The matter content of the universe is modelled as a perfect fluid with equation of state \(P = w\rho\). Each component has a characteristic \(w\):

Component	\(w\)	\(\rho \propto\)	\(a(t) \propto\)
Dust (cold dark matter)	\(0\)	\(a^{-3}\)	\(t^{2/3}\)
Radiation	\(1/3\)	\(a^{-4}\)	\(t^{1/2}\)
Cosmological constant	\(-1\)	\(\text{const}\)	\(e^{Ht}\) (de Sitter)
“Curvature”	\(-1/3\)	\(a^{-2}\)	\(t\)

The \(\rho \propto a^{-3(1+w)}\) scaling follows from stress-energy conservation \(\nabla_\mu T^{\mu\nu} = 0\) in the FLRW background, which reduces to \(\dot\rho + 3(\dot a/a)(\rho + P) = 0\). The physical interpretations are transparent: for dust, the energy density dilutes as the volume \(a^3\); for radiation, there is an extra factor of \(a^{-1}\) from the cosmological redshift of each photon’s energy; for \(\Lambda\), the vacuum energy density is a constant property of spacetime itself.

Since \(\rho_R \propto a^{-4}\) and \(\rho_M \propto a^{-3}\), radiation dominates at early times (small \(a\)) and matter dominates at later times. Eventually, as \(a\) grows, the cosmological constant wins. The universe passes through three eras: radiation-dominated \(\to\) matter-dominated \(\to\) \(\Lambda\)-dominated — and we live near the transition between the latter two.

\[ q(t) = -\frac{a\ddot a}{\dot a^2} = -\frac{\ddot a/a}{H^2} \]

A universe decelerating under gravity has \(q > 0\); one accelerating under dark energy has \(q < 0\). The Raychaudhuri equation shows \(q = \frac{1}{2}(1 + 3w)\Omega\) in a flat single-component universe — so matter (\(w=0\)) gives \(q = 1/2\) (decelerating), radiation (\(w=1/3\)) gives \(q = 1\) (more decelerating), and a cosmological constant (\(w=-1\)) gives \(q = -1\) (exponential acceleration). Current measurements give \(q_0 \approx -0.55\), confirming the accelerating expansion discovered by Riess, Perlmutter, and Schmidt in 1998.

\[ \frac{k}{a_0^2} = \frac{8\pi G}{3}\rho_0 + \frac{\Lambda}{3}, \qquad 0 = -4\pi G(\rho_0 + 3P_0) + \Lambda \]

For dust (\(P_0 = 0\)) the second equation gives \(\Lambda = 4\pi G\rho_0 > 0\), and the first then gives \(k = 1\) (a closed 3-sphere). The Einstein universe is a compact space with topology \(\mathbb{R} \times S^3\), held in equilibrium by the cosmological constant exactly balancing the gravitational attraction. Einstein introduced \(\Lambda\) for precisely this purpose — to permit a static solution. The solution is unstable: a small perturbation causes either unbounded expansion or collapse. When Hubble showed \(\dot a \neq 0\), Einstein called the cosmological constant “the greatest blunder of my life.” It returned in 1998 — with the opposite implication.

Fate of the universe. The Raychaudhuri equation shows: The Raychaudhuri equation shows:

• Without \(\Lambda\): matter and radiation cause \(\ddot a < 0\) — decelerating expansion
• With positive \(\Lambda\): dark energy makes \(\ddot a > 0\) for sufficiently large \(a\) — accelerating expansion (observed since 1998)

Dark energy and energy condition violation. The cosmological constant with \(p_\Lambda = -\rho_\Lambda c^2\) violates the Strong Energy Condition (\(\rho + 3p < 0\)). This is what allows accelerated expansion. More exotic dark energy models (quintessence, phantom fields) can violate even the WEC, connecting cosmology to the quantum field theory violation of energy conditions.

Solving the Friedmann equations. For a flat universe (\(k=0\)) with a single component of equation of state \(p = w\rho c^2\):

• Energy conservation \(\nabla_\mu T^{\mu 0} = 0\) gives \(\dot\rho + 3H(\rho + p/c^2) = 0\), so \(\rho \propto a^{-3(1+w)}\)
• Substituting into the Friedmann equation: \(H^2 = (8\pi G_N/3)\rho \propto a^{-3(1+w)}\)
• This integrates to \(a(t) \propto t^{2/3(1+w)}\) for \(w \neq -1\)

Component	\(w\)	\(a(t)\)
Matter (dust)	\(0\)	\(t^{2/3}\)
Radiation	\(1/3\)	\(t^{1/2}\)
Cosmological constant	\(-1\)	\(e^{Ht}\) (de Sitter)

The cosmological constant problem. The observed value of \(\Lambda\) corresponds to an energy density \(\rho_\Lambda \approx 7 \times 10^{-27}\) kg/m³. Quantum field theory predicts zero-point energy contributions of order \(\rho_\text{vac} \sim E_\text{Planck}^4/(\hbar c)^3 \approx 10^{97}\) kg/m³. The ratio is \(10^{124}\) — the worst fine-tuning problem in all of physics. No satisfactory resolution is known.

\[ \Omega_\text{total} - 1 = \frac{kc^2}{a^2 H^2} \]

For a matter-dominated universe, the right side grows as \(\propto t^{2/3}\), meaning any initial deviation from flatness is amplified. The observed flatness (\(|\Omega - 1| \lesssim 0.01\)) requires extraordinary fine-tuning at early times unless there is a period of inflation (\(w < -1/3\), so \(\ddot a > 0\)) that drives \(\Omega \to 1\) exponentially fast.

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Part V — Advanced Topics

Chapter 25. Gravitational Waves and Linearised Gravity

\[ h_{\mu\nu} \to h_{\mu\nu} - \partial_\mu \xi_\nu - \partial_\nu \xi_\mu \]\[ \Box \bar h_{\mu\nu} = -\frac{16\pi G_N}{c^4} T_{\mu\nu} \]

where \(\bar h_{\mu\nu} = h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}h\) is the trace-reversed perturbation and \(\Box = \eta^{\mu\nu}\partial_\mu\partial_\nu\) is the flat-space d’Alembertian. This is a wave equation!

\[ h_{\mu\nu} = \text{Re}\left[A_{\mu\nu} e^{ik_\rho x^\rho}\right], \qquad k^\mu k_\mu = 0 \]\[ h_+\begin{pmatrix}0&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&0\end{pmatrix}e^{i\omega(t-z/c)} + h_\times\begin{pmatrix}0&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&0\end{pmatrix}e^{i\omega(t-z/c)} \]

The two polarisations \(h_+\) and \(h_\times\) squeeze and stretch space perpendicular to the propagation direction.

\[ \ddot\xi^x = \frac{1}{2}\ddot h_{xx}\xi^x, \qquad \ddot\xi^y = \frac{1}{2}\ddot h_{yy}\xi^y = -\frac{1}{2}\ddot h_{xx}\xi^y \]

A ring of test particles in the \(x\)-\(y\) plane oscillates in the characteristic \(+\) pattern for the \(h_+\) polarisation. LIGO/Virgo detectors measure this strain: \(h = \Delta L/L \sim 10^{-21}\) for binary black hole mergers at cosmological distances.

\[ P = \frac{G_N}{5c^5}\left\langle \dddot Q_{ij}\dddot Q^{ij}\right\rangle \]

where \(Q_{ij} = \int \rho(x_i x_j - \frac{1}{3}\delta_{ij}r^2)\,d^3x\) is the mass quadrupole moment. This is analogous to the electric quadrupole radiation in electromagnetism (there is no gravitational dipole radiation because the dipole moment is the centre of mass, which is conserved by momentum conservation).

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Appendix: Key Reference Formulas

\[ \Gamma^\rho{}_{\mu\nu} = \frac{1}{2}g^{\rho\sigma}\bigl(\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu}\bigr) \]\[ R^\rho{}_{\sigma\mu\nu} = \partial_\mu \Gamma^\rho{}_{\nu\sigma} - \partial_\nu \Gamma^\rho{}_{\mu\sigma} + \Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} - \Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma} \]

Ricci tensor: \(R_{\mu\nu} = R^\rho{}_{\mu\rho\nu}\) (symmetric)

Einstein tensor: \(G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}\), satisfies \(\nabla^\mu G_{\mu\nu} = 0\)

\[ G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G_N\, T_{\mu\nu} \]

Schwarzschild radius: \(R_s = 2G_N M/c^2\)

Geodesic equation: \(\ddot x^\mu + \Gamma^\mu{}_{\nu\rho}\dot x^\nu \dot x^\rho = 0\)

Killing equation: \(\nabla_a k_b + \nabla_b k_a = 0\)

Covariant derivative (vector): \(\nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu{}_{\mu\rho}V^\rho\)

Covariant derivative (one-form): \(\nabla_\mu \omega_\nu = \partial_\mu \omega_\nu - \Gamma^\rho{}_{\mu\nu}\omega_\rho\)

Four-velocity norm: \(g_{\mu\nu}u^\mu u^\nu = -c^2\) (massive particle)

FLRW metric: \(ds^2 = -c^2 dt^2 + a(t)^2[dr^2/(1-kr^2) + r^2 d\Omega^2]\)

Geodesic deviation: \(D^2\xi^\mu/d\tau^2 = R^\mu{}_{\nu\rho\sigma}u^\nu u^\rho \xi^\sigma\)

Bianchi identity: \(\nabla_{[\mu}R_{\nu\rho]\sigma\lambda} = 0\) (differential); contracted: \(\nabla^\mu G_{\mu\nu} = 0\)

Palatini identity: \(\delta R^\mu{}_{\nu\rho\sigma} = \nabla_\rho(\delta\Gamma^\mu{}_{\sigma\nu}) - \nabla_\sigma(\delta\Gamma^\mu{}_{\rho\nu})\)

Linearised gravity (harmonic gauge): \(\Box\bar h_{\mu\nu} = -16\pi G_N T_{\mu\nu}/c^4\) where \(\bar h_{\mu\nu} = h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}h\)

Quadrupole formula: \(P = \frac{G_N}{5c^5}\langle\dddot Q_{ij}\dddot Q^{ij}\rangle\)

Bekenstein-Hawking entropy: \(S_{BH} = k_B A/(4\ell_P^2)\), \(\ell_P = \sqrt{\hbar G_N/c^3} \approx 1.6 \times 10^{-35}\) m

Hawking temperature: \(T_H = \hbar c^3/(8\pi G_N M k_B)\)

Hubble parameter: \(H(t) = \dot a/a\); today \(H_0 \approx 67\)–\(73\) km/s/Mpc

Friedmann equation: \(H^2 = \frac{8\pi G_N}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}\)

Scale factor evolution: \(a(t) \propto t^{2/3}\) (matter), \(a(t) \propto t^{1/2}\) (radiation), \(a(t) \propto e^{Ht}\) (de Sitter / \(\Lambda\)-dominated)

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Summary: The Logical Architecture of GR

General Relativity is built in layers. Understanding the logical dependencies helps navigate the subject:

Layer 0: Index notation. All tensor equations are index equations. Mastery of Einstein summation, Kronecker delta, and Levi-Civita symbol is prerequisite for everything else.

Layer 1: Special Relativity. The Minkowski metric \(\eta_{\mu\nu} = \operatorname{diag}(-1,+1,+1,+1)\) encodes the causal structure of flat spacetime. Four-vectors, four-velocity, four-momentum, and the energy-momentum relation arise from Lorentz symmetry. The Euler-Lagrange formalism applied to the free-particle action produces the flat-space geodesic equation as a special case.

Layer 2: Differential Geometry. Manifolds provide the arena; tangent and cotangent spaces provide the mathematical home for vectors and one-forms; tensors of type \((r,s)\) unify these. The exterior derivative \(d\) and pullback \(\phi^*\) allow calculus without coordinates. The affine connection \(\nabla\) extends differentiation to curved spaces; the Riemann tensor measures curvature; the Levi-Civita connection is the unique metric-compatible torsion-free connection.

Layer 3: General Relativity.

• Postulate 0 (geodesic hypothesis): gravity is spacetime curvature, free particles follow geodesics
• Postulate 1 (causality): causal structure via the metric
• Postulate 2 (conservation): \(\nabla_b T^{ab} = 0\) follows from the Bianchi identity once the Einstein equations hold
• Postulate 3 (Einstein equations): \(G_{ab} + \Lambda g_{ab} = 8\pi G_N T_{ab}\) — ten second-order PDEs for the metric

Layer 4: Solutions. The Schwarzschild metric is the unique vacuum spherically symmetric solution. Its geodesics yield all the classical tests (time dilation, gravitational redshift, Mercury precession, light deflection). Black hole structure (horizon, singularity, Kruskal extension, Penrose diagrams) reveals the global causal structure. The FLRW metric describes homogeneous isotropic cosmology; the Friedmann equations govern the scale factor.

Layer 5: Gravitational waves. Linearised GR predicts wave solutions propagating at \(c\) with two polarisations. The quadrupole formula gives the radiated power for a slowly-moving source. LIGO/Virgo confirmed GR to high precision with binary black hole mergers (GW150914, 2015) and neutron star mergers (GW170817, 2017) — the latter also detected in electromagnetic waves, inaugurating multi-messenger astronomy.

The road not taken in this course. Kaluza-Klein theory (extra dimensions), string theory, loop quantum gravity, the full Penrose-Hawking singularity theorems, exact gravitational wave solutions (Bondi-Sachs), and numerical relativity required for LIGO waveform templates. These form graduate GR courses.

Why GR matters. Beyond its beauty, GR has technological consequences: GPS (requires relativistic corrections, \(\sim 27\,\mu\)s/day), gravitational lensing (used to map dark matter and detect exoplanets), gravitational wave astronomy (LIGO/Virgo detected binary mergers and neutron star collisions, enabling multi-messenger astronomy), cosmological observations (the CMB and large-scale structure are explained by FLRW + inflation + ΛCDM), and black hole imaging (Event Horizon Telescope, 2019). The subject of GR is not a historical curiosity — it is a working precision tool of 21st-century science.

Open problems at the frontier:

1. Quantisation of gravity: No complete, consistent quantum theory of gravity exists. String theory and loop quantum gravity are candidates but neither is experimentally confirmed.
2. Information paradox: If black holes evaporate via Hawking radiation, does information about infalling matter come back out? Unitary quantum mechanics says yes; naive semiclassical GR says no. Recent progress (island formula, Page curve) suggests unitarity is preserved, but the mechanism is not understood.
3. Dark matter and dark energy: \(\sim 27\%\) of the universe’s energy budget is unknown dark matter (detected only gravitationally), and \(\sim 68\%\) is the cosmological constant or some dynamical dark energy field. Neither has been directly detected by non-gravitational means.
4. Initial singularity: Classical GR predicts the Big Bang singularity (\(a \to 0\)), where the theory breaks down. A quantum theory of gravity is needed to describe the Planck epoch (\(t \lesssim 10^{-43}\) s).

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Lecture Video Index

Lecture	Topic
1	Index notation and Einstein summation
2	Covariance and contravariance
4	Special relativity postulates
5	Proper time
6	Minkowski metric
7	Poincaré group: translations
8	Lorentz group generators
9	Lorentz group in Mathematica (Thomas precession)
10	Hamilton’s principle and Noether’s theorem
11	Free particle dynamics, four-momentum
12	Collision examples (particle colliders)
13	Rindler coordinates and Bell’s paradox
14	Introduction to manifolds
15	Curves and tangent vectors
16	One-forms and cotangent space
17	n-forms, exterior derivative, and integration
18	Pullbacks, pushforwards, and Lie derivatives
19	Affine connection and covariant derivative
20	Parallel transport, geodesics, and curvature
21	Riemannian manifolds and Levi-Civita connection
22	Isometries, Killing vectors, and Levi-Civita symbol
23	GR postulates
24	Stress-energy tensor and energy conditions
25	Schwarzschild metric
26	Orbital mechanics in GR
27	Einstein–Newton race (exam problem)
28	Black holes
29	Cosmology I: FLRW metric
30	Cosmology II: Friedmann equations