Differential Equations for Relativity

Ordinary differential equations enter relativity as geodesic equations, background evolution, radial structure, separated wave equations, and perturbative transport. The essential skill is not memorizing solution forms: it is identifying the problem type, stating the data that make it well posed, choosing an appropriate analytic or numerical method, and checking the result. This module develops that skill through focused examples.

Required background. Calculus and Linear Algebra for Relativity supplies differentiation, eigenstructure, linear systems, and basis changes.

You should leave able to solve representative initial-value, boundary-value, and asymptotic ODEs and to justify a method in a gravitational application.

Differential-equations diagnostic

1. Solve $y'+2y=e^{-t}$ with $y(0)=0$, and verify the differential equation directly.
2. For $y''+\omega^2y=0$ on $0\le x\le L$, explain the difference between prescribing $y(0),y'(0)$ and prescribing $y(0),y(L)$. Under the boundary conditions $y(0)=y(L)=0$, for which $\omega$ are nonzero solutions possible?
3. Use a Frobenius ansatz $y=x^s\sum_{n=0}^\infty a_nx^n$ to find the indicial exponents of $$x^2y''+xy'-\nu^2y=0.$$
4. Linearize $\dot z=\lambda z-z^3$ about every equilibrium and classify its stability for $\lambda\gt0$.

Hints and solutions

Hints for 1

Use the integrating factor $e^{2t}$. Keep the homogeneous constant until imposing the initial condition.

Solution to 1

Multiplication by $e^{2t}$ gives

$$\frac{d}{dt}(e^{2t}y)=e^t.$$

Hence $e^{2t}y=e^t+C$, so $y=e^{-t}+Ce^{-2t}$. The initial value gives $C=-1$:

$$y(t)=e^{-t}-e^{-2t}.$$

Then $y'=-e^{-t}+2e^{-2t}$ and $y'+2y=e^{-t}$, which checks both the forcing and the homogeneous decay.

Hints for 2

Initial data determine the two integration constants locally. Boundary data at separated points instead impose a compatibility condition on those constants and, sometimes, on the parameter $\omega$.

Solution to 2

For fixed $\omega\gt0$,

$$y=A\cos(\omega x)+B\sin(\omega x).$$

$y(0)$ and $y'(0)$ determine $A$ and $B$ directly, so this is an initial-value problem. The boundary condition $y(0)=0$ gives $A=0$; then $y(L)=B\sin(\omega L)=0$. A nonzero $B$ exists only if

$$\omega L=n\pi, \qquad n=1,2,\ldots.$$

The boundary problem has selected a discrete spectrum. Treating two-point boundary data as if they were ordinary initial data misses this compatibility condition.

Hints for 3

The smallest power of $x$ comes from the $n=0$ terms. Set its coefficient to zero before deriving any recurrence.

Solution to 3

For the leading term $a_0x^s$,

$$x^2y''+xy'-\nu^2y =a_0\bigl[s(s-1)+s-\nu^2\bigr]x^s+\cdots.$$

Therefore the indicial equation is $s^2-\nu^2=0$, with exponents

$$s=\pm\nu.$$

The result identifies the two local scalings. When their difference is an integer, the second independent solution may contain a logarithm; the indicial roots alone do not settle that question.

Hints for 4

Find the zeros of $f(z)=\lambda z-z^3$. The linearized perturbation obeys $\dot\eta=f'(z_*)\eta$.

Solution to 4

For $\lambda\gt0$, the equilibria are $z_*=0$ and $z_*=\pm\sqrt\lambda$. Since $f'(z)=\lambda-3z^2$,

$$f'(0)=\lambda\gt0, \qquad f'(\pm\sqrt\lambda)=-2\lambda\lt0.$$

Thus $z=0$ is linearly unstable and the two nonzero equilibria are linearly stable. The sign follows from $\eta(t)\sim e^{f'(z_*)t}$; it is not determined by the sign of $z_*$.

Interpret the result

• Secure: you distinguish initial, boundary, spectral, singular-point, and stability questions; all four solutions include a direct check.
• Uncertain: the equation class is identified but one method needs prompting. Review that method, change a coefficient or datum, and solve again.
• Not yet: data are imposed without counting integration constants, a boundary spectrum is treated as an arbitrary parameter, or linear stability is inferred without linearization. Complete the focused review before returning to a route that uses geodesics or separated perturbations.

Focused review

First-order and linear systems

For $y'+p(t)y=q(t)$, an integrating factor converts the left side into one derivative. For a system $\dot X=A(t)X+b(t)$, existence and uniqueness are local statements tied to regularity of the coefficients. When $A$ is constant, eigenvalues organize modes, but defective matrices require generalized eigenvectors.

Initial data, boundary data, and spectra

An $n$th-order equation generally needs $n$ independent conditions, but their location matters. Initial data at one regular point define evolution. Conditions at separate endpoints define a boundary-value problem and can fail to have a solution, have several solutions, or select eigenvalues. State the interval and endpoint behavior before choosing a solver.

Singular points and asymptotics

Near a regular singular point, a Frobenius expansion exposes allowed local powers. At an irregular singular point, exponential or asymptotic scales may replace a convergent power series. An asymptotic series must state its limit and error ordering; it is not automatically convergent.

Stability and controlled approximation

Linearization classifies perturbations only when the linear term is decisive. Zero eigenvalues require higher-order or center-manifold analysis. In perturbation theory, substitute the approximation back into the original equation and state the residual order. These distinctions recur in Affine and Variational Geodesics and in separated black-hole wave equations.

Transfer task: tidal separation

A component of the separation between nearby freely falling trajectories obeys the constant-curvature model

$$\frac{d^2X}{d\tau^2}+\kappa X=0, \qquad X(0)=X_0, \qquad X'(0)=0.$$

Solve for $\kappa\gt0$, $\kappa=0$, and $\kappa\lt0$. State what the first zero or the late-time behavior says, and identify one limitation of this model.

Hint 1

Treat the three signs separately. For $\kappa\lt0$, write $\kappa=-\alpha^2$ with $\alpha\gt0$.

Hint 2

Use both initial conditions. Check dimensions: $\sqrt{|\kappa|}\,\tau$ must be dimensionless.

Solution

For $\kappa\gt0$,

$$X(\tau)=X_0\cos(\sqrt\kappa\,\tau),$$

with first zero at $\tau=\pi/(2\sqrt\kappa)$. In the one-component model this represents focusing of that separation direction. For $\kappa=0$, $X=X_0$ because the initial relative velocity vanishes. For $\kappa=-\alpha^2\lt0$,

$$X(\tau)=X_0\cosh(\alpha\tau),$$

so the separation grows exponentially at late time. Direct differentiation verifies every case. In an actual spacetime, the tidal matrix can vary with $\tau$, couple components, and depend on the chosen geodesic; a single constant $\kappa$ does not establish global focusing or instability.

Common errors

• Solving the homogeneous equation and forgetting the forcing term.
• Counting boundary conditions without checking their independence or compatibility.
• Calling every singular coefficient a physical singularity rather than examining the equation and chosen variable.
• Reporting a numerical curve without a residual, convergence check, or preserved invariant.

Return to your route

After the transfer task, return to whichever route sent you here: Graduate General Relativity Self-Study, Compact Binaries and Gravitational Waves, Computational Relativity Self-Study, the First Graduate General Relativity course, or Gravitational Radiation and Compact Binaries. If your difficulty is specifically with wave propagation or causal Green functions, continue first to PDEs and Green Functions for Gravitation.