advanced28 min studyLesson 53 in Physics

Metrics and geodesic motion

Read proper time from a metric, derive the geodesic equation, and separate coordinate effects from physical acceleration.

Before you begin

On this page

The question for today

What you’ll understand

Use a Lorentzian metric to classify intervals and compute proper time.
Construct connection coefficients from metric derivatives.
Recognize coordinate curvature in a physically straight trajectory.

A metric turns coordinate changes into measurable intervals. Once a spacetime metric is supplied, freely falling test bodies follow geodesics, while light follows null geodesics. The equation can look like a force law, but its connection coefficients also arise in curvilinear coordinates on completely flat space. Learning to tell these situations apart prevents a common mistake: interpreting every bending line on a coordinate plot as gravitational curvature.

Coordinates label, the metric measures

Write the interval as $ds^2=g_{\mu\nu}dx^\mu dx^\nu$ with signature $(+,-,-,-)$ . A timelike trajectory has $ds^2>0$ and proper time $d\tau=ds/c$ . A null interval has zero norm and is tangent to the local light cone. A spacelike separation has negative norm and cannot be the infinitesimal displacement of a massive observer. These classifications survive smooth changes of coordinates, although the component values do not.

The metric is not determined by the coordinate names alone. For example, spatial polar coordinates on a Euclidean plane give $d\ell^2=dr^2+r^2d\varphi^2$ . Equal changes in angle correspond to larger lengths farther from the origin. The factor of r records this geometry of the coordinate grid; it does not by itself imply intrinsic curvature. Checking a curvature tensor, rather than inspecting a metric coefficient’s position dependence, resolves the issue.

c^2d\tau^2=g_{\mu\nu}dx^\mu dx^\nu,\qquad \Gamma^\rho_{\mu\nu}=\frac12g^{\rho\sigma}(\partial_\mu g_{\sigma\nu}+\partial_\nu g_{\sigma\mu}-\partial_\sigma g_{\mu\nu})

The proper-time formula applies on timelike curves. The connection shown is the unique torsion-free connection compatible with the metric, called the Levi-Civita connection.↔ Scroll wide equations horizontally.

Stationary proper time gives free fall

Vary a timelike worldline while keeping its endpoints fixed. Stationarity of its proper time produces the geodesic equation. Proper time is an affine parameter for a massive freely falling particle; a null curve instead needs another affine parameter because its proper time vanishes. Using an arbitrary coordinate time as the parameter generally adds a term proportional to the tangent vector, so an equation copied without specifying its parameter can be wrong.

A timelike geodesic locally maximizes proper time under suitable nearby endpoint comparisons, but this does not imply a unique global maximum for arbitrary distant events. Conjugate points and multiple paths complicate global statements. For practical integration, specify an initial event and a normalized initial tangent, then integrate the second-order equations. The metric normalization should remain constant and provides a valuable numerical check on the computed trajectory.

\frac{d^2x^\rho}{d\lambda^2}+\Gamma^\rho_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}=0,\qquad \frac{d}{d\lambda}\left(g_{\mu\nu}\dot x^\mu\dot x^\nu\right)=0

Dots denote differentiation with respect to an affine parameter lambda. Choose lambda equal to proper time for a massive particle; null geodesics retain zero tangent norm.↔ Scroll wide equations horizontally.

Use symmetry before solving every component

If metric coefficients do not depend on a coordinate, the corresponding canonical momentum is conserved along an affine geodesic. This follows from the geodesic Lagrangian, not from treating every coordinate velocity as constant. A cyclic angular coordinate in an axisymmetric geometry supplies angular momentum. A stationary time coordinate supplies an energy-like constant. Whether that constant is an energy measured by a particular local observer requires an additional conversion using the observer’s four-velocity.

In flat polar coordinates the radial equation contains a term $-r\dot\varphi^2$ , and the angular equation contains $2\dot r\dot\varphi/r$ . These are the same effects familiar from polar-coordinate kinematics. A Cartesian straight line can therefore have nonzero radial coordinate acceleration while its proper acceleration vanishes. At a coordinate origin the polar chart fails, but a regular Cartesian chart still describes the particle smoothly; a chart failure is not a physical singularity.

Work it through

A straight line seen through polar coordinates

On a flat plane use $r(\lambda)=\sqrt{\lambda^2+1}$ and $\varphi(\lambda)=\operatorname{atan2}(1,\lambda)$ . These coordinates describe $x=\lambda$ , $y=1$ . Check the polar geodesic equations at $\lambda=0$ .

At the evaluation point, $r=1$ , $\dot r=0$ , and $\ddot r=1$ . Differentiating the angle gives $\dot\varphi=-1/(1+\lambda^2)$ , so $\dot\varphi=-1$ there.
The radial equation is $\ddot r-r\dot\varphi^2=1-1=0$ . A nonzero $\ddot r$ is therefore compatible with zero covariant acceleration.
The second angular derivative vanishes at zero, so $\ddot\varphi+2\dot r\dot\varphi/r=0+0=0$ . Both coordinate equations describe the same Cartesian straight line.

The result

The connection terms cancel the apparent coordinate acceleration. This example has no intrinsic curvature and no gravitational force despite its nonconstant metric components.

The living notebook

Explore the model

Interactive lab

Interactive lab · spacetime

Check causal classification first

Vary β and ct in the flat Lorentz diagram. The spatial coordinate of the original event stays x = 0.6, and c = 1. Identify timelike, null, and spacelike intervals from the origin before changing observer.

Original eventTransformed eventLight cone

Lorentz factor γ: 1.12
Event x′: 0.168
Event ct′: 0.817

Relative velocity β0.45Event time ct1

t = 0 s

A fixed x = 0.6 event is Lorentz transformed with c = 1. Playback highlights the relation between coordinate descriptions; it does not move the event. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThe lab checks interval invariance between flat inertial frames. It does not integrate the geodesic equation for a user-supplied metric or plot curved-spacetime trajectories.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Connection coefficients are not tensor components and can be nonzero solely because the coordinate basis changes with position. Curvature requires the specific derivative and quadratic combination entering the Riemann tensor.

Your turn

Try it yourself

For flat polar motion derive the conserved angular quantity from $\ddot\varphi+2\dot r\dot\varphi/r=0$ . Evaluate it at $r=2$ with $\dot\varphi=3$ in consistent length and affine-parameter units.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Multiply the angular equation by the appropriate power of r and recognize a product derivative.

Compare with the solution

Multiplying by $r^2$ gives $r^2\ddot\varphi+2r\dot r\dot\varphi=d(r^2\dot\varphi)/d\lambda=0$ . Thus $\ell=r^2\dot\varphi$ is constant, and its specified value is $2^2(3)=12$ length-squared per affine-parameter unit. The angular velocity itself need not stay constant because r can change. For a massive particle parametrized by proper time, this is angular momentum per unit mass in the corresponding flat-space problem.

Keep the idea

Take it with you

A metric supplies invariant interval measurements while coordinates only label events.
Free-fall equations require an affine parameter and an initial tangent normalization.
Position-dependent metric coefficients and nonzero connections do not alone establish curvature.

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