Unit 8: Physical Applications: Continuum Mechanics, Electromagnetism & Gravitation

1. Cauchy Stress Tensor in Continuum Mechanics

In classical 3D continuum mechanics, internal contact forces distributed across an infinitesimal area element $d\mathbf{S} = \mathbf{n} \, dS$ are characterized by the Cauchy stress tensor $\sigma^{ij}$:

• Conservation of Linear Momentum: In the absence of body forces:

where $\nabla_j$ is the covariant derivative associated with the spatial Riemannian metric $g_{ij}$.

• Conservation of Angular Momentum: By requiring the net torque on an infinitesimal volume element to vanish, the stress tensor must be symmetric:

2. The Relativistic Stress-Energy-Momentum Tensor $T^{\mu\nu}$

In 4-dimensional pseudo-Riemannian spacetime $(M, g_{\mu\nu})$, energy, momentum density, shear stress, and isotropic pressure are unified into a symmetric rank-2 contravariant tensor field $T^{\mu\nu}$:

Perfect Fluid Stress-Energy Tensor:

For an ideal fluid characterized by proper energy density $\rho(x)$, isotropic pressure $p(x)$, and 4-velocity field $u^\mu = \frac{dx^\mu}{d\tau}$ (normalized such that $g_{\mu\nu} u^\mu u^\nu = -c^2$):

3. Local Conservation Laws

In special relativity (flat Minkowski spacetime $\eta_{\mu\nu}$), conservation of energy and momentum is expressed as $\partial_\mu T^{\mu\nu} = 0$. According to the principle of minimal gravitational coupling (equivalence principle), the physical conservation law in curved spacetime is obtained by replacing ordinary derivatives with covariant derivatives:

Fundamental Conservation Law:

Projecting along and orthogonal to the 4-velocity $u^\nu$ yields the relativistic continuity equation and the relativistic Euler equation of fluid dynamics!

1. The 4-Potential and Field Strength Tensor

Classical electrodynamics is governed by electric and magnetic vector fields $\mathbf{E}$ and $\mathbf{B}$, which mix under Lorentz coordinate transformations. Tensor analysis unifies them into a single antisymmetric rank-2 covariant tensor field $F_{\mu\nu}$, called the Faraday (electromagnetic field) tensor.

Let $A_\mu = (-\phi/c, \mathbf{A})$ be the covariant electromagnetic 4-potential.

Definition 8.1 (Electromagnetic Field Tensor):

Because the connection coefficients $\Gamma^\lambda_{\mu\nu} = \Gamma^\lambda_{\nu\mu}$ are symmetric, the Christoffel symbols cancel identically!

In matrix form with metric signature $(-, +, +, +)$:

By raising indices with $g^{\mu\alpha} g^{\nu\beta}$, we obtain the contravariant tensor $F^{\mu\nu}$.

2. Covariant Maxwell Equations

Theorem 8.1 (Maxwell's Equations in Curved Spacetime): The four classical Maxwell equations reduce to two elegant, manifestly covariant tensor equations:

1. Inhomogeneous Maxwell Equations (Sources):

where $J^\nu = (\rho c, \mathbf{J})$ is the 4-current density.

1. Homogeneous Maxwell Equations (Bianchi Identity):

Automatic Charge Conservation:

Taking the covariant divergence of the inhomogeneous equation:

because $F^{\mu\nu}$ is antisymmetric while the commutator contraction is symmetric. Thus, electric charge conservation $\nabla_\nu J^\nu = 0$ is an exact geometric identity!

3. The Lorentz Force Density

The electromagnetic 4-force density acting on charged matter is:

which unifies Coulomb electric force and magnetic Lorentz force $\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})$.

1. The Geodesic Hypothesis

In Einstein's theory of General Relativity, gravity is not a physical Newtonian force, but the curvature of spacetime. Free particles experiencing no non-gravitational forces follow geodesics of the spacetime metric:

where $\tau$ is the proper time experienced by a clock moving along the worldline:

2. Gravitational Time Dilation

Consider a static gravitational field with metric:

For a stationary observer situated at fixed spatial coordinates ($dx^i = 0$):

Let two clocks be stationed at positions $A$ and $B$ where the metric components are $g_{00}(A)$ and $g_{00}(B)$:

In a weak Newtonian gravitational potential $\Phi(x)$, $g_{00} \approx -\left( 1 + \frac{2\Phi}{c^2} \right)$:

Clocks situated deeper in a gravitational well ($\Phi < 0$) run measurably slower than clocks higher up!

3. Gravitational Redshift of Light

A light wave emitted at position $A$ with frequency $\nu_A$ and received at position $B$ undergoes a gravitational frequency shift:

For light escaping from the surface of a star of mass $M$ and radius $R$ ($g_{00} = -\left(1 - \frac{2GM}{c^2 R}\right)$) to a distant observer ($g_{00}(\infty) = -1$):

The light is redshifted ($\Delta \lambda > 0$), directly confirming the curvature of the metric tensor!

1. The Search for the Gravitational Field Equations

In Newtonian gravity, the gravitational potential $\Phi$ satisfies Poisson's equation:

Einstein sought a tensorial equation of the form:

where $\mathcal{G}_{\mu\nu}$ is a symmetric rank-2 geometric tensor constructed from the metric $g_{\mu\nu}$ and its first and second derivatives.

Essential Constraints:

1. Since energy-momentum is conserved ($\nabla^\mu T_{\mu\nu} = 0$), the geometric tensor must satisfy the divergence-free identity:

1. In 1915, David Hilbert and Albert Einstein proved that the unique symmetric tensor containing at most second derivatives of the metric that satisfies this condition (Lovelock's theorem in 4D) is:

2. The Einstein Field Equations

Definition 8.2 (The Einstein Field Equations):

where:

• $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu}$ is the Einstein tensor.
• $\Lambda$ is the cosmological constant.
• $G = 6.674 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}$ is Newton's gravitational constant.
• $c$ is the speed of light.
• $\kappa = \frac{8\pi G}{c^4} \approx 2.076 \times 10^{-43} \, \text{N}^{-1}$ is Einstein's gravitational coupling constant.

3. Trace-Reversed Form and Vacuum Equations

Taking the trace with $g^{\mu\nu}$:

Substituting back into $G_{\mu\nu}$:

Vacuum Field Equations ($\Lambda = 0, T_{\mu\nu} = 0$):

In empty spacetime outside mass distributions:

Spacetimes satisfying $R_{\mu\nu} = 0$ are called Ricci-flat. Crucially, $R_{\mu\nu} = 0$ does NOT imply flat spacetime! The full Riemann tensor $R^\rho_{\mu\sigma\nu}$ can still be non-zero through its Weyl tensor component, propagating gravitational waves and mediating gravitational attraction!

1. The Schwarzschild Solution (1916)

Karl Schwarzschild derived the exact vacuum solution ($R_{\mu\nu} = 0$) for a static, spherically symmetric mass $M$:

Definition 8.3 (Schwarzschild Metric):

where $r_s = \frac{2GM}{c^2}$ is the Schwarzschild radius (event horizon). For the Sun: $r_s \approx 2.95 \text{ km}$; for the Earth: $r_s \approx 8.87 \text{ mm}$.

2. Relativistic Planetary Orbits and Perihelion Precession

Consider motion confined to the equatorial plane $\theta = \frac{\pi}{2}$. From the geodesic equations, energy per unit mass $E$ and angular momentum per unit mass $L$ are conserved:

Substituting into the metric $g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu = -c^2$:

where the relativistic effective potential is:

The third term $-\frac{GML^2}{c^2 r^3}$ is the general relativistic correction to Newtonian gravity.

Using $u = 1/r$, the orbital Binet equation becomes:

By perturbation theory, an elliptical orbit precesses by an angle per revolution:

For Mercury: $a = 5.79 \times 10^{10} \text{ m}, e = 0.2056$. This yields $\Delta\phi = 42.98'' \text{ per century}$, matching the observed anomalous 43 arcseconds per century unexplained by Newtonian planetary perturbations!

3. Gravitational Deflection of Starlight

For null geodesics (photons, $ds^2 = 0$), the orbital equation with impact parameter $b$ gives:

Integrating the perturbation from $\phi = -\pi/2$ to $\pi/2$ for light grazing the Sun at radius $R$:

Deflection Angle Formula:

For the Sun ($M = M_\odot, R = R_\odot$):

Exactly twice the naive Newtonian corpuscular value ($0.875''$)! This famous prediction was confirmed by Arthur Eddington's 1919 solar eclipse expedition, providing historic observational confirmation of general relativity.