Classical Physics Foundations: Geometry, Tensors and Dynamics

Geometric Viewpoint of Classical Physics

the supplied physics reference, Applications of Classical Physics — Chapters 1–2 Companion to Modern Classical Physics (Princeton, 2017)

/20 Summary

• Core thesis: physical laws are geometric relations between tensors that exist independently of any coordinate system or reference frame. Components are merely projections; the tensor itself is real.
• Newtonian (Ch. 1): absolute time + 3-D Euclidean space → 3-tensors, orthonormal bases, dot product with $\delta_{ij}$.
• Special Relativity (Ch. 2): 4-D spacetime with Minkowski metric $\eta_{\alpha\beta} = \text{diag}(-1,+1,+1,+1)$ → 4-tensors, Lorentz transformations, no absolute simultaneity.
• Three operational principles for the entire book:
  1. Write equations as tensor equalities (frame-independent).
  2. Choose the frame that simplifies (rest frame, lab frame, comoving).
  3. Project to components only at the end.
• Key payoff: every later branch (fluids, plasma, GR, optics) reuses the same tensorial machinery — stress, stress-energy, Maxwell tensor, fluid 4-velocity.

Master Map

mindmap
  root((Foundations))
    Geometric objects
      Scalars
      Vectors
      Tensors of rank n
      Frame independence
    Newtonian (Ch.1)
      Absolute time
      Euclidean 3-space
      Orthonormal basis
      delta_ij metric
      Stress tensor T_ij
      Newton's 2nd law
      EM in 3-vector form
    Special Relativity (Ch.2)
      4-D spacetime
      Minkowski metric eta
      Light cones
      Proper time tau
      4-velocity u^alpha
      4-momentum p^alpha
      Stress-energy T^alphabeta
      Lorentz transforms
      EM tensor F^alphabeta
    Conservation laws
      Mass continuity
      Momentum (div T)
      Energy-momentum (div T^alphabeta)
      Charge (div J)
    Differential operators
      Gradient
      Divergence
      Curl (3D)
      d'Alembertian (4D)

Newtonian Physics: Geometric Viewpoint

Foundational Concepts

Geometric Objects — Tensors Without Coordinates

Definition (rank-$n$ tensor): a multilinear map $$T: \underbrace{V^* \times \cdots \times V^*}_{p \text{ times}} \times \underbrace{V \times \cdots \times V}_{q \text{ times}} \to \mathbb{R}, \quad n = p+q$$ where $V$ is the vector space (here $\mathbb{E}^3$) and $V^*$ its dual.

• Scalar (rank-0): number; invariant. Examples: mass $m$, temperature $T$, charge $q$.
• Vector (rank-1): arrow; transforms as $V^i = R^i{}_j V^j$ under rotations.
• Rank-2 tensor: machine eating two vectors, outputting a scalar. Examples: stress $T_{ij}$, moment of inertia $I_{ij}$, strain $e_{ij}$.

Key operations (all coordinate-free):

Component Representation

Pick an orthonormal basis $\{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3\}$ with $\mathbf{e}_i \cdot \mathbf{e}_j = \delta_{ij}$:

$$\mathbf{V} = V^i \mathbf{e}_i, \qquad T = T^{ij} \mathbf{e}_i \otimes \mathbf{e}_j$$

Einstein summation convention: repeated index ⇒ sum.

Index gymnastics in Euclidean space: because $g_{ij} = \delta_{ij}$, up and down indices are interchangeable; we write everything with subscripts.

• Dot product: $\mathbf{A}\cdot\mathbf{B} = A_i B_i$
• Magnitude: $|\mathbf{V}|^2 = V_i V_i$
• Cross product: $(\mathbf{A}\times\mathbf{B})_i = \epsilon_{ijk} A_j B_k$
• Determinant: $\det M = \epsilon_{ijk} M_{1i} M_{2j} M_{3k}$

Useful identity (epsilon-delta): $$\epsilon_{ijk}\epsilon_{ilm} = \delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl}$$

Orthogonal Transformations of Bases

Change basis: $\mathbf{e}_{i'} = R_{i'j} \mathbf{e}_j$ with $R^T R = I$, i.e. $R_{i'j} R_{i'k} = \delta_{jk}$.

Components transform contragredient/cogredient:

Two parities:

• True (polar) tensors: position, velocity, force.
• Pseudo (axial) tensors: angular velocity, magnetic field, torque — pick up extra $\det R$ under improper rotations (reflections).

Differentiation of Tensor Fields

Gradient operator $\boldsymbol\nabla = \mathbf{e}_i \partial_i$ (with $\partial_i = \partial/\partial x_i$).

Key identities (memorize):

∇·(∇×A) = 0           (div curl = 0)
∇×(∇φ) = 0            (curl grad = 0)
∇×(∇×A) = ∇(∇·A) − ∇²A
∇·(φA) = φ∇·A + A·∇φ
∇×(φA) = φ∇×A + ∇φ × A
∇·(A×B) = B·(∇×A) − A·(∇×B)
(A·∇)B = A_j ∂_j B_i   (note: not a scalar; vector)
∇(A·B) = (A·∇)B + (B·∇)A + A×(∇×B) + B×(∇×A)

Volumes, Integration, Integral Conservation Laws

Three integral theorems (Stokes' theorem in disguise):

flowchart LR
    A["Gradient Thm:\n∫∇φ · dl = φ_b − φ_a"]
    B["Divergence Thm:\n∫_V ∇·A dV = ∮_∂V A · dS"]
    C["Curl / Stokes Thm:\n∫_S (∇×A) · dS = ∮_∂S A · dl"]

Generic conservation-law template:

For any density $\rho$ with flux $\mathbf{F}$:

$$\boxed{\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{F} = \sigma}$$

where $\sigma$ is source/sink (zero ⇒ strict conservation). Integrate over fixed volume $V$:

$$\frac{d}{dt}\int_V \rho\, dV = -\oint_{\partial V} \mathbf{F}\cdot d\mathbf{S} + \int_V \sigma\, dV$$

Applications (all from this template):

The Stress Tensor — Linchpin of All Continuum Physics

Definition: $T_{ij}$ is the $i$-th component of the force per unit area transmitted across a surface whose outward normal points in the $+j$ direction.

$$\mathrm{d}F_i = T_{ij}\, \mathrm{d}S_j$$

Properties:

• Symmetric, $T_{ij} = T_{ji}$ (proof: angular-momentum conservation; vanishing torque on infinitesimal cube as $L\to 0$).
• 3 eigenvalues = principal stresses; 3 mutually perpendicular principal axes.
• Pressure is the isotropic part: $p = -\tfrac{1}{3} T_{ii}$ (trace/3, negative because compression).
• Deviatoric stress $\tau_{ij} \equiv T_{ij} + p\,\delta_{ij}$ is the traceless shear part.

Momentum conservation (Newton 2nd law for continua):

$$\boxed{\frac{\partial (\rho v_i)}{\partial t} + \partial_j\!\left(\rho v_i v_j + T_{ij}\right) = f_i^{\text{ext}}}$$

(Here I include the convective momentum flux $\rho v_i v_j$ inside the divergence; some books separate it.)

Special cases of $T_{ij}$:

Electromagnetism in Geometric (3-D) Form

Maxwell's equations (Gaussian units, vacuum):

$$\nabla\cdot\mathbf{E} = 4\pi\rho_e$$ $$\nabla\cdot\mathbf{B} = 0$$ $$\nabla\times\mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{B}}{\partial t}$$ $$\nabla\times\mathbf{B} = \frac{4\pi}{c}\mathbf{J} + \frac{1}{c}\frac{\partial \mathbf{E}}{\partial t}$$

Lorentz force: $\mathbf{F} = q\!\left(\mathbf{E} + \tfrac{\mathbf{v}}{c}\times\mathbf{B}\right)$

Charge conservation (follows automatically from $\nabla\cdot$ of Ampère + $\partial_t$ of Gauss): $$\partial_t \rho_e + \nabla\cdot\mathbf{J} = 0$$

EM stress-energy (a teaser for Ch. 2):

• Energy density: $u_{EM} = (E^2 + B^2)/(8\pi)$
• Poynting flux: $\mathbf{S} = (c/4\pi)\,\mathbf{E}\times\mathbf{B}$
• Momentum density: $\mathbf{g} = \mathbf{S}/c^2$

Geometric viewpoint summary (Ch. 1 wrap-up): $\mathbf{E}$ is a true vector, $\mathbf{B}$ is a pseudovector — meaning Maxwell theory is only formally tidy when we promote them to a rank-2 antisymmetric tensor $F_{\mu\nu}$ — which is exactly what Ch. 2 does.

Special Relativity: Geometric Viewpoint

Foundational Postulates

1. Principle of Relativity: all inertial frames are physically equivalent.
2. Constancy of $c$: speed of light is the same finite invariant in every inertial frame.

Consequences (instantly):

• Simultaneity is frame-dependent.
• Time dilation: $\Delta t = \gamma\, \Delta\tau$.
• Length contraction: $L = L_0/\gamma$.
• Velocity addition is nonlinear.

Define $\beta = v/c$, $\gamma = 1/\sqrt{1-\beta^2}$.

Spacetime and Four-Vectors

Spacetime $\mathbb{M}^4$: 4-D manifold; an event is a point with coordinates $x^\alpha = (ct, x, y, z) = (x^0, x^1, x^2, x^3)$.

Conventions used throughout B&T:

• Greek indices $\alpha,\beta,\ldots \in \{0,1,2,3\}$ (spacetime).
• Latin $i,j,\ldots \in \{1,2,3\}$ (space).
• Signature $(-,+,+,+)$ — the "mostly plus" convention (matches the MTW / Thorne school).
• Often set $c=1$ in formal manipulations.

Four-vectors: geometric arrows in $\mathbb{M}^4$. Examples:

• Displacement $\Delta x^\alpha$
• 4-velocity $u^\alpha = dx^\alpha/d\tau$
• 4-momentum $p^\alpha = m u^\alpha$
• 4-acceleration $a^\alpha = du^\alpha/d\tau$
• 4-current $J^\alpha = (c\rho_e, \mathbf{J})$
• 4-wavevector $k^\alpha = (\omega/c, \mathbf{k})$

The Metric — The Heart of SR

$$\eta_{\alpha\beta} = \mathrm{diag}(-1,+1,+1,+1)$$

Spacetime interval (Lorentz-invariant):

$$ds^2 = \eta_{\alpha\beta}\, dx^\alpha dx^\beta = -c^2 dt^2 + dx^2 + dy^2 + dz^2$$

Classification of intervals:

Proper time: for a timelike worldline, $d\tau^2 = -ds^2/c^2$. Proper time is the clock reading on the particle itself.

Light cone structure (essential picture):

              future
        \   timelike  /
         \    ^     /
          \   |    /
   null --→ \ |  / ← null
              \|/
   ----elsewhere-----elsewhere----
              /|\
   null --→ / | \ ← null
          /   |   \
         /    v    \
        /  timelike \
              past

Index Raising/Lowering & Dual Vectors

In SR (with $\eta$), indices do not simply commute as in Euclidean space — the time component flips sign:

• Up-vector (contravariant) $V^\alpha$
• Down-vector (covariant) $V_\alpha = \eta_{\alpha\beta} V^\beta$

So if $V^\alpha = (V^0, V^1, V^2, V^3)$ then $V_\alpha = (-V^0, V^1, V^2, V^3)$.

Inner product: $$A\cdot B = \eta_{\alpha\beta} A^\alpha B^\beta = A_\alpha B^\alpha = -A^0 B^0 + \mathbf{A}\cdot\mathbf{B}$$

Inverse metric: $\eta^{\alpha\beta} = \mathrm{diag}(-1,+1,+1,+1)$ (same matrix in this convention).

Lorentz Transformations

A Lorentz transformation $\Lambda$ preserves the metric: $\eta_{\alpha\beta} = \Lambda^\mu{}_\alpha \Lambda^\nu{}_\beta \eta_{\mu\nu}$.

Standard boost along $x$ with velocity $v=\beta c$:

$$\Lambda^\alpha{}_\beta = \begin{pmatrix} \gamma & -\gamma\beta & 0 & 0 \\ -\gamma\beta & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$

Explicit form: $$ct' = \gamma(ct - \beta x), \quad x' = \gamma(x - \beta ct), \quad y' = y, \quad z' = z$$

Rapidity parameterization: $\beta = \tanh\eta$, $\gamma = \cosh\eta$, $\gamma\beta = \sinh\eta$. Boosts then add linearly in rapidity — composition of two collinear boosts = boost with $\eta_1+\eta_2$.

Full Lorentz group $O(1,3)$:

• 6 parameters: 3 boost rapidities + 3 spatial rotation angles.
• Subgroups: proper orthochronous $SO^+(1,3)$ (excludes parity P and time-reversal T).

Velocity addition (collinear): $$u' = \frac{u-v}{1 - uv/c^2}$$

Velocity addition (general 3-vector): see B&T §2.2. The component parallel to the boost transforms like collinear; perpendicular divides by $\gamma(1-uv_\|/c^2)$.

Particle Kinematics

4-velocity: $u^\alpha = dx^\alpha/d\tau = \gamma(c, \mathbf{v})$, with $u^\alpha u_\alpha = -c^2$ (always; geometric identity).

4-momentum: $p^\alpha = m u^\alpha = (E/c, \mathbf{p})$ where

• $E = \gamma m c^2$ (relativistic energy)
• $\mathbf{p} = \gamma m \mathbf{v}$ (relativistic 3-momentum)

On-shell condition (mass shell): $$\boxed{p^\alpha p_\alpha = -m^2 c^2 \quad\Leftrightarrow\quad E^2 = (pc)^2 + (mc^2)^2}$$

Limits:

• Massive at rest: $E = mc^2$
• Massless (photon): $E = pc$, $m=0$, moves on null geodesic
• Non-relativistic: $E \approx mc^2 + \tfrac{1}{2}mv^2 + \tfrac{3}{8}mv^4/c^2 + \cdots$

4-acceleration: $a^\alpha = du^\alpha/d\tau$. Always orthogonal to 4-velocity: $a^\alpha u_\alpha = 0$ (differentiate $u\cdot u = -c^2$).

Relativistic Collisions — Cheatsheet Logic

Workflow for any scattering problem:

flowchart TD
    A[Identify particles, assign 4-momenta] --> B[Write conservation: Σp_in = Σp_out]
    B --> C{Best frame?}
    C -->|Threshold/decay| D[CM frame: total p = 0]
    C -->|Lab measurement| E[Lab frame: one particle at rest]
    D --> F[Use invariants: s, t, u]
    E --> F
    F --> G[Solve in chosen frame]
    G --> H[Boost back if needed]

Mandelstam invariants (2→2 process, $1+2\to 3+4$):

• $s = -(p_1+p_2)^2 = $ (CM energy)$^2/c^2$
• $t = -(p_1-p_3)^2 = $ momentum transfer
• $u = -(p_1-p_4)^2$
• Identity: $s+t+u = \sum_i m_i^2 c^2$

Threshold for endothermic reaction in lab frame (target at rest): $$E_1^{\text{th}} = \frac{(\sum m_f c^2)^2 - (m_1 c^2)^2 - (m_2 c^2)^2}{2 m_2 c^2}$$

Compton scattering (foundational example): $$\lambda' - \lambda = \frac{h}{m_e c}(1-\cos\theta)$$

The Stress-Energy Tensor $T^{\alpha\beta}$

The single most important object in continuum + relativistic physics.

$T^{\alpha\beta}$ is symmetric, rank-2; its components are interpreted as the flux of $\alpha$-momentum across a surface of constant $x^\beta$:

Master conservation law: $$\boxed{\partial_\beta T^{\alpha\beta} = f^\alpha_{\text{ext}}}$$

— a single 4-equation that contains both energy conservation ($\alpha=0$) and momentum conservation ($\alpha=i$). Zero on the right ⇒ closed system.

Stress-energy of canonical systems:

Properties forced by physics:

• Symmetric ($T^{\alpha\beta} = T^{\beta\alpha}$): angular-momentum conservation.
• Real eigenvalue $-\rho c^2$ with timelike eigenvector $u^\alpha$ defines rest-frame energy density.
• Energy conditions (used heavily in GR Ch. 24–28):
  • Weak: $T^{\alpha\beta} u_\alpha u_\beta \ge 0$ (energy density $\ge 0$ for all observers).
  • Strong: ditto with $+$ trace constraint.
  • Dominant: energy flux is timelike or null.

Relativistic Fluid Dynamics (preview of Part V)

Perfect fluid:

$$T^{\alpha\beta} = (\rho + p/c^2) u^\alpha u^\beta + p\,\eta^{\alpha\beta}$$

Apply $\partial_\beta T^{\alpha\beta} = 0$:

• Project parallel to $u^\alpha$ ⇒ relativistic continuity / energy equation: $$u^\alpha \partial_\alpha \rho + (\rho + p/c^2)\,\partial_\alpha u^\alpha = 0$$

• Project orthogonal to $u^\alpha$ (using projector $P^{\alpha\beta} = \eta^{\alpha\beta} + u^\alpha u^\beta /c^2$) ⇒ relativistic Euler equation: $$(\rho + p/c^2)\,u^\beta\partial_\beta u^\alpha = -P^{\alpha\beta}\partial_\beta p$$

In the non-relativistic limit ($v\ll c$, $p\ll\rho c^2$) these collapse to standard continuity + Euler.

Electromagnetism Reformulated — The Field Tensor $F^{\alpha\beta}$

4-potential: $A^\alpha = (\phi/c, \mathbf{A})$, with gauge $A^\alpha \to A^\alpha + \partial^\alpha\Lambda$.

Field tensor: $F_{\alpha\beta} = \partial_\alpha A_\beta - \partial_\beta A_\alpha$, antisymmetric (6 independent components).

Components (Gaussian; with metric signature $(-,+,+,+)$, careful with signs):

$$F^{\alpha\beta} = \begin{pmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & B_z & -B_y \\ E_y/c & -B_z & 0 & B_x \\ E_z/c & B_y & -B_x & 0 \end{pmatrix}$$

(Exact signs differ across textbooks — B&T uses Gaussian with $(-,+,+,+)$. Always check conventions before plugging into formulas.)

Maxwell's equations — 2 lines:

$$\partial_\beta F^{\alpha\beta} = \frac{4\pi}{c} J^\alpha \qquad \text{(sources: Gauss + Ampère)}$$

$$\partial_{[\alpha} F_{\beta\gamma]} = 0 \qquad \text{(no monopoles + Faraday)}$$

The second is equivalent to $\partial_\alpha \tilde F^{\alpha\beta} = 0$ where $\tilde F^{\alpha\beta} = \tfrac{1}{2}\epsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}$ is the dual.

Lorentz force law (geometric): $$\frac{dp^\alpha}{d\tau} = \frac{q}{c} F^{\alpha\beta} u_\beta$$

Invariants of the EM field (frame-independent):

1. $\tfrac{1}{2}F_{\alpha\beta}F^{\alpha\beta} = B^2 - E^2/c^2$
2. $\tfrac{1}{4}F_{\alpha\beta}\tilde F^{\alpha\beta} = -\mathbf{E}\cdot\mathbf{B}/c$

So a pure-E field in one frame stays pure-E in all frames only if $\mathbf{E}\cdot\mathbf{B}=0$ AND $B^2 < E^2/c^2$; similarly for pure-B.

Field transformation (boost along $x$):

$$E'_\| = E_\|, \quad E'_\perp = \gamma(\mathbf{E} + \mathbf{v}\times\mathbf{B})_\perp$$ $$B'_\| = B_\|, \quad B'_\perp = \gamma(\mathbf{B} - \tfrac{\mathbf{v}}{c^2}\times\mathbf{E})_\perp$$

Memorize: "$\mathbf{E}$ and $\mathbf{B}$ mix under boosts." A pure electric field in one frame has magnetic components in another. Magnetism is a relativistic effect of moving charges.

Workflow / Process — Solving any "Foundations" problem

flowchart TD
    A[Identify physical quantity] --> B{Scalar / Vector / Tensor?}
    B --> C[Frame-independent statement first]
    C --> D{Newtonian or Relativistic?}
    D -->|Newtonian| E[Use 3-tensors, ortho basis, δ_ij]
    D -->|Relativistic| F[Use 4-tensors, η_αβ, Lorentz]
    E --> G[Identify conservation: ∂_tρ + ∇·F = 0]
    F --> H[Identify conservation: ∂_β T^αβ = 0]
    G --> I[Pick best frame/basis to compute]
    H --> I
    I --> J[Project to components, compute]
    J --> K[Verify invariants/scalars match]

Comparison Tables

Newtonian vs Special Relativistic

Coordinate vs Geometric formulation

Tensor types by transformation behavior

Common Mistakes

• ❌ Confusing up/down indices in SR. In Euclidean space $V^i = V_i$; in SR $V^0 = -V_0$. Losing a sign here changes physics.
• ❌ Treating $\mathbf{B}$ as a true vector. It's a pseudovector — flips sign under parity differently from $\mathbf{E}$. This is the motivation for $F_{\alpha\beta}$.
• ❌ Forgetting that velocity addition is non-Galilean. Two boosts of 0.6c and 0.6c do not give 1.2c; they give 0.88c.
• ❌ Using $E = mc^2$ as the total energy for a moving particle. That's $\gamma mc^2$. The rest-energy interpretation requires $\mathbf{v}=0$.
• ❌ Setting up problems in the wrong frame. Threshold problems → CM frame. Single-target experiments → lab frame. Pick before computing.
• ❌ Confusing time dilation and length contraction directions. Moving clocks tick slow; moving rods contract along motion.
• ❌ Misusing $T^{\alpha\beta}$ symmetry. $T^{0i} = T^{i0}$ is a physical statement (energy flux ↔︎ momentum density × $c^2$), not just notation.
• ❌ Forgetting that $u^\alpha u_\alpha = -c^2$ is an identity. It constrains 4-velocity to a hyperboloid — only 3 independent components, not 4.
• ❌ Writing relativistic Euler with $\rho$ instead of $\rho + p/c^2$. Pressure contributes to inertia in SR.
• ❌ Plugging into a formula without checking sign convention (mostly-plus vs mostly-minus, Gaussian vs SI). B&T uses mostly-plus + Gaussian.

Expert Insights

Tensors are bookkeeping for invariance. Once you understand a law as a tensor equation, you've understood it in all frames simultaneously — there's no separate "derivation" for each observer.

The stress tensor is the same idea in Newtonian fluids, elastic solids, and the EM field — just different ingredients. Pattern-match across chapters: if you've seen $T_{ij}$ once, every later occurrence is a variation on the theme.

$T^{\alpha\beta}$ is what gravitates in GR — Einstein's equations are $G_{\alpha\beta} = (8\pi G/c^4) T_{\alpha\beta}$. Mastering it in Ch. 2 pays directly in Ch. 24+.

Conservation laws come from symmetries (Noether). Time translation → energy, space translation → momentum, rotation → angular momentum, Lorentz boost → moment-of-energy. In tensor form: $\partial_\beta T^{\alpha\beta}=0$ encodes the first ten.

Magnetism = relativistic correction to electrostatics. A current is a moving line of charge; the second-order Lorentz correction in another frame is the magnetic field. The $F_{\alpha\beta}$ formalism makes this manifest.

The "$(\rho+p/c^2)$" inertia is non-negotiable. In any relativistic continuum (fluid, plasma, radiation), pressure contributes to the effective inertia. Forgetting it gives qualitatively wrong sound speeds, instability growth rates, accretion solutions.

Choose frames mercilessly. The CM frame turns 4-body kinematics into 1-D algebra. The fluid rest frame turns $T^{\alpha\beta}$ into a diagonal matrix. The "boost away" tactic saves more pages than any algebraic identity.

The dual $\tilde F^{\alpha\beta}$ is not optional decoration — it's how you write the "homogeneous" Maxwell pair in tensor form. Half the field-theory texts hide this; B&T puts it front and center.

Frame-independent statements first; components last. If your derivation depends on coordinates from line 1, you're doing it wrong. State the physics covariantly, then pick coordinates to evaluate.

Watch out for unit systems. Gaussian (B&T) puts factors of $4\pi$ and $c$ in Maxwell's equations; SI hides them in $\mu_0$, $\epsilon_0$. Translate carefully — many "errors" in derivations are unit-system collisions.

Troubleshooting

Cheatsheet

=== GEOMETRIC BASICS ===
Tensor      = multilinear map; coord-free
Indices     = projections onto a chosen basis
Einstein    = repeated index ⇒ summation
Up/down     = contravariant/covariant; metric raises/lowers

=== NEWTONIAN (Ch.1) ===
Metric:     δ_ij           (up/down equivalent)
Cross:      (A×B)_i = ε_ijk A_j B_k
ε-δ:        ε_ijk ε_ilm = δ_jl δ_km − δ_jm δ_kl

Stress:     dF_i = T_ij dS_j ;  T_ij = T_ji
Pressure:   p = −T_ii/3
Mom. cons:  ∂_t(ρv_i) + ∂_j(ρv_i v_j + T_ij) = f_i

Maxwell (Gaussian):
  ∇·E = 4πρ_e
  ∇·B = 0
  ∇×E = −(1/c)∂_t B
  ∇×B = (4π/c)J + (1/c)∂_t E
Lorentz:    F = q(E + v/c × B)

=== SPECIAL RELATIVITY (Ch.2) ===
Metric:     η = diag(−1,+1,+1,+1)
Interval:   ds² = −c²dt² + dx²
Proper τ:   dτ² = −ds²/c²
γ:          1/√(1−β²)
Rapidity:   β = tanh η, γ = cosh η

4-velocity: u^α = γ(c, v),  u·u = −c²
4-momentum: p^α = (E/c, p),  p·p = −m²c²
Mass shell: E² = (pc)² + (mc²)²

Lorentz boost (x-dir):
  t' = γ(t − vx/c²)
  x' = γ(x − vt)
  y' = y,  z' = z

Velocity addn (collin): u' = (u−v)/(1−uv/c²)

Stress-energy:  ∂_β T^αβ = f^α
Perfect fluid:  T^αβ = (ρ+p/c²)u^α u^β + p η^αβ
Dust:           T^αβ = ρ u^α u^β
EM:             T^αβ = (1/4π)(F^αμ F^β_μ − ¼ η^αβ F²)

EM tensor:      F_αβ = ∂_α A_β − ∂_β A_α
Maxwell:        ∂_β F^αβ = (4π/c) J^α
                ∂_[α F_βγ] = 0
Lorentz force:  dp^α/dτ = (q/c) F^αβ u_β
EM invariants:  ½F·F = B² − E²/c²
                ¼F·F̃ = −E·B/c

Field transforms (boost along x):
  E_∥' = E_∥,  E_⊥' = γ(E + v×B)_⊥
  B_∥' = B_∥,  B_⊥' = γ(B − v×E/c²)_⊥

Glossary

• 4-velocity ($u^\alpha$) — Tangent vector to a worldline, parameterized by proper time. Magnitude $-c^2$.
• Affine parameter — Parameter along a null geodesic; substitute for proper time when $d\tau = 0$.
• Contraction — Summation over one up and one down index, reducing tensor rank by 2.
• Contravariant — Tensor with upper indices; transforms with $\Lambda$.
• Covariant — Tensor with lower indices; transforms with $\Lambda^{-1}$.
• Dust — Pressureless fluid; $T^{\alpha\beta} = \rho u^\alpha u^\beta$.
• Einstein summation — Repeated index implies sum over its range.
• Eikonal / null — Lightlike, $ds^2 = 0$.
• Energy-momentum (4-momentum) — $p^\alpha = (E/c, \mathbf{p})$; mass shell $p\cdot p = -m^2c^2$.
• Eulerian description — Field values at fixed spatial points (vs Lagrangian = follow the particle).
• Field tensor ($F_{\alpha\beta}$) — Antisymmetric rank-2 tensor encoding $\mathbf{E}$ and $\mathbf{B}$.
• Frame — Choice of observer + clock + ruler; equivalently, choice of basis $\{e_\alpha\}$.
• Galilean transformation — Newtonian frame change; $\mathbf{x}' = \mathbf{x} - \mathbf{v}t$, $t'=t$.
• Inertial frame — Free particles move in straight lines at constant speed (Newton's 1st).
• Levi-Civita — Totally antisymmetric tensor $\epsilon_{ijk}$ (3-D) or $\epsilon_{\alpha\beta\gamma\delta}$ (4-D).
• Lorentz group ($O(1,3)$) — 6-parameter group preserving $\eta$.
• Lorentz transformation — Linear isometry of Minkowski space.
• Mass shell — $p\cdot p = -m^2c^2$; 3-D hyperboloid in 4-momentum space.
• Metric — Inner-product structure; $\delta_{ij}$ (Newtonian) or $\eta_{\alpha\beta}$ (SR).
• Minkowski space $\mathbb{M}^4$ — Flat 4-D pseudo-Riemannian manifold.
• Mostly-plus signature — $(-,+,+,+)$, the B&T/MTW convention.
• Null — $ds^2 = 0$, on the light cone.
• Orthochronous — Lorentz transformation preserving direction of time.
• Pseudotensor / pseudovector — Tensor that picks up extra $\det\Lambda$ under improper transformations.
• Poincaré group — Lorentz group + translations; 10-parameter symmetry of SR.
• Polar vector — True vector (flips under parity).
• Proper time ($\tau$) — Time measured by a clock comoving with the worldline.
• Rapidity ($\eta$) — Additive parameter for collinear boosts; $\beta=\tanh\eta$.
• Rest frame — Frame in which a body's 3-momentum is zero.
• Spacelike — $ds^2 > 0$; causally disconnected events.
• Stress tensor ($T_{ij}$) — Symmetric rank-2 tensor; force per area transmitted across surfaces.
• Stress-energy tensor ($T^{\alpha\beta}$) — Relativistic generalization; flux of $\alpha$-momentum across $x^\beta$ surface.
• Tetrad / vierbein — Orthonormal basis $\{e_\alpha\}$ at a point.
• Timelike — $ds^2 < 0$; causally connectable.
• Worldline — Path of particle through spacetime, parameterized by $\tau$.

Final Takeaways

1. The whole book is a single recursive idea: write physics as tensor equations, then specialize. Every Part below (statistical, optics, elasticity, fluids, plasma, GR) re-applies Ch. 1–2 tooling.
2. Geometric viewpoint = invariance + structure. Components are scaffolding; the tensor is the physics.
3. Conservation laws are divergence equations. Newtonian: $\partial_t\rho + \nabla\cdot\mathbf{F}=0$. Relativistic: $\partial_\beta T^{\alpha\beta}=0$. Same idea, different number of dimensions.
4. Master the stress / stress-energy tensor early. It is the universal currency of continuum + relativistic physics.
5. Choose frames aggressively. The right frame collapses a problem; the wrong one buries it in algebra.
6. Maxwell's equations are simpler in $F^{\alpha\beta}$ form. $\mathbf{E}$ and $\mathbf{B}$ are projections of a single object.
7. Don't confuse Newtonian intuition (absolute simultaneity, Galilean velocity addition) with SR. Boost-mixing of $\mathbf{E}$/$\mathbf{B}$, of energy/momentum, of space/time is the correct picture.
8. Pressure contributes to inertia in SR ($\rho + p/c^2$). Forget this only when $p\ll \rho c^2$ — never in radiation-dominated, neutron-star, or accretion-disk regimes.
9. Set sign and unit conventions before computing. B&T = mostly-plus + Gaussian; check against any external formula source.
10. The geometric viewpoint is the bridge to GR. Ch. 1–2 prepare you so that when curvature enters in Part VII, only the metric changes — the tensorial machinery is already yours.

Next: Part II — Statistical Physics (Random Processes, Kinetic Theory, Transport, Fluctuation-Dissipation). Built directly on the geometric language above.