Geometry Review for Electrodynamics
Conventions and Notation
- Signature: \((-,+,+,+)\) (convention)
- Greek indices: \(\mu,\nu = 0,1,2,3\)
- Einstein summation implied
- Time coordinate: \(x^0 = ct\)
Metric: \[ \eta_{\mu\nu} = \mathrm{diag}(-1,+1,+1,+1) \]
The choice of signature is a convention: one may equivalently use \((+,-,-,-)\) provided it is applied consistently. Physical predictions are independent of this choice.
Instead of setting \(c=1\), we define the time coordinate as \(x^0 = ct\), so that all components of \(x^\mu\) have dimensions of length. In this formulation, \[ \mathrm{d}s^2 = -c^2 \mathrm{d}t^2 + \mathrm{d}x^2 + \mathrm{d}y^2 + \mathrm{d}z^2 \] which is algebraically equivalent to working in units with \(c=1\). The two approaches differ only by a rescaling of the time coordinate.
Relativistic Framework
Motivation and Space-Time Structure
- Postulates of special relativity
- Lorentz invariance
- Space-time as a unified geometric object
- Worldlines
The postulates (constancy of \(c\) and equivalence of inertial frames) require a unified treatment of space and time. Particle histories are described by worldlines, making causal relations explicit.
Lorentz transformations: \[ x'^{\mu} = \Lambda^{\mu}_{\ \nu} x^{\nu} \]
Minkowski Space
- Vector space \(\mathbb{R}^4\) with indefinite metric
Interval: \[ \mathrm{d}s^2 = \eta_{\mu\nu} \mathrm{d}x^{\mu} \mathrm{d}x^{\nu} \]
Invariance: \[ \mathrm{d}s'^2 = \mathrm{d}s^2 \]
The invariant interval replaces separate notions of spatial distance and time separation.
Euclidean Comparison
- Positive-definite inner product
\[ \|\mathbf{a}\|^2 = \mathbf{a}\cdot \mathbf{a} \]
Inequalities: \[ |\mathbf{a}\cdot \mathbf{b}| \le \|\mathbf{a}\|\|\mathbf{b}\|, \quad \|\mathbf{a}+\mathbf{b}\| \le \|\mathbf{a}\| + \|\mathbf{b}\| \]
These properties rely on positive definiteness and do not directly extend to Minkowski space.
Minkowski Geometry and Causality
Norm: \[ A^{\mu} A_{\mu} = \eta_{\mu\nu} A^{\mu} A^{\nu} \]
Classification:
- \(A^{\mu} A_{\mu} < 0\) — timelike
- \(A^{\mu} A_{\mu} > 0\) — spacelike
- \(A^{\mu} A_{\mu} = 0\) — null
Light cone: \[ ds^2 = 0 \]
Regions:
- future: \(x^0 > 0\), \(ds^2 < 0\)
- past: \(x^0 < 0\), \(ds^2 < 0\)
- spacelike: \(ds^2 > 0\)
Timelike separation allows causal influence; spacelike does not.
Proper Time
Definition: \[ \mathrm{d}\tau^2 = -\mathrm{d}s^2 \]
Worldline: \[ au = \int \sqrt{-\eta_{\mu\nu} \mathrm{d}x^{\mu} \mathrm{d}x^{\nu}} \]
Proper time is invariant and depends on the trajectory.
Lorentz Transformations
Definition: \[ x'^{\mu} = \Lambda^{\mu}_{\ \nu} x^{\nu} \]
Constraint: \[ \eta_{\alpha\beta}\Lambda^{\alpha}_{\ \mu}\Lambda^{\beta}_{\ \nu} = \eta_{\mu\nu} \]
Boost (along \(x\)): \[ \begin{aligned} t' &= \gamma (t - vx) \\ x' &= \gamma (x - vt) \end{aligned} \]
Four-Vectors
Position: \[ x^{\mu} = (t,x,y,z) \]
Four-velocity: \[ u^{\mu} = \frac{dx^{\mu}}{d\tau}, \quad u^{\mu}u_{\mu} = -1 \]
Momentum: \[ p^{\mu} = m u^{\mu} \]
Minkowski Inner Product and Inequalities
Definition: \[ A^{\mu} B_{\mu} = \eta_{\mu\nu} A^{\mu} B^{\nu} \]
Lorentz invariant.
For timelike, same time orientation: \[ (A^{\mu} B_{\mu})^2 \ge (A^{\mu} A_{\mu})(B^{\nu} B_{\nu}) \]
Inequalities require causal restrictions.
Tensor and Differential Structure
Index Notation
- Contravariant: \(A^{\mu}\)
- Covariant: \(A_{\mu}\)
Lowering: \[ A_{\mu} = \eta_{\mu\nu} A^{\nu} \]
Raising: \[ A^{\mu} = \eta^{\mu\nu} A_{\nu} \]
Vectors and Covectors
- Tangent space: \(V\)
- Dual space: \(V^*\)
Pairing: \[ \omega(v) \]
Gradient: \[ \partial_{\mu} \]
Covectors act linearly on vectors; \(\partial_\mu\) is naturally covariant.
Metric as Isomorphism
Mapping: \[ v^{\mu} \leftrightarrow v_{\mu} \]
The metric identifies \(V\) and \(V^*\) and defines contractions.
Tensors
Transformation: \[ T'^{\mu}_{\ \nu} = \Lambda^{\mu}_{\ \alpha} \Lambda^{\beta}_{\ \nu} T^{\alpha}_{\ \beta} \]
Tensor equations with fully contracted indices are invariant.
Differential Operators
Four-gradient: \[ \partial_{\mu} \]
d’Alembertian: \[ \Box = \partial_{\mu}\partial^{\mu} \]
Lorentz-invariant wave operator.
Differential Forms and Electrodynamics
Differential Forms
Basis: \[ \mathrm{d}x^{\mu} \]
1-form: \[ \omega = \omega_{\mu} \mathrm{d}x^{\mu} \]
Exterior product: \[ \mathrm{d}x^{\mu} \wedge \mathrm{d}x^{\nu} = -\mathrm{d}x^{\nu} \wedge \mathrm{d}x^{\mu} \]
Electromagnetic 2-Form
\[ F = \frac{1}{2} F_{\mu\nu} \mathrm{d}x^{\mu} \wedge \mathrm{d}x^{\nu} \]
Encodes electric and magnetic fields covariantly.
Hodge Dual
Mapping: \[ \star: \Omega^p \to \Omega^{4-p} \]
Property: \[ \star\star \omega = (-1)^{p(4-p)+1}\,\omega \]
Components: \[ (\star F)_{\mu\nu} = \frac{1}{2} \epsilon_{\mu\nu\alpha\beta} F^{\alpha\beta} \]
Integration and Orientation
- Integration of \(p\)-forms
- Orientation
Volume element: \[ \mathrm{d}^4x \]
Defines invariant integration measures.
Stokes’ Theorem
\[ \int_{\partial M} \omega = \int_M \mathrm{d}\omega \]
Unifies classical integral theorems.
Covariant Electrodynamics
Potential: \[ A = A_{\mu} \mathrm{d}x^{\mu} \]
Field: \[ F = \mathrm{d}A \]
Components: \[ F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu} \]
Gauge symmetry: \[ A \mapsto A + \mathrm{d}\lambda \]
Maxwell Equations
Homogeneous: \[ \mathrm{d}F = 0 \]
With sources: \[ \mathrm{d}\star F = J \]
Integral Laws
Gauss: \[ \int_{\partial V} \star F = \int_V J \]
Faraday: \[ \int_{\partial S} F = 0 \]
Ampère–Maxwell: \[ \int_{\partial S} \star F = \int_S J \]
Charge Conservation
\[ \partial_{\mu} J^{\mu} = 0 \]
Follows from: \[ \mathrm{d}(\mathrm{d}\star F) = 0 \Rightarrow \mathrm{d}J = 0 \]
Structural Summary
Geometric Chain
- Metric \(\rightarrow\) inner product
- Inner product \(\rightarrow\) duality
- Duality \(\rightarrow\) Hodge star
- \(d\) and \(\star\) \(\rightarrow\) Maxwell equations
- Stokes \(\rightarrow\) integral laws
Exercises
Conceptual and Computational
- Classify intervals (timelike, spacelike, null) with explicit examples.
- Verify invariance of \(ds^2\) under a boost.
- Compute \(F_{\mu\nu}\) from a given \(A_{\mu}\) and identify \(\mathbf{E}\) and \(\mathbf{B}\).
- Show \(dF=0\) in components and relate to Faraday’s law.
- Derive Gauss’ law from \(d\star F = J\).
- Compute \(\star F\) for a plane wave and interpret components.