Free fall and the straightest path

5.1 Coordinate acceleration and accelerometer readings#

Standing on the ground, you assign yourself constant spatial coordinates. A dropped ball’s coordinates accelerate downward. Everyday language calls you unaccelerated and the ball accelerated.

Now ask an accelerometer. Yours reads approximately the local gravitational acceleration because the floor pushes upward on you. An ideal accelerometer falling with the ball reads zero, ignoring air resistance and finite-size effects. Its case and internal proof mass follow free fall together, so the proof mass does not deflect relative to the case.

GR organizes its local inertial physics around that second distinction. Free fall is zero proper acceleration. Remaining at a fixed altitude near Earth requires a nongravitational force.

This does not make Newton’s description useless or make gravity imaginary. It distinguishes a coordinate acceleration from a physical acceleration measured along a worldline. Later, tidal effects will reveal gravitational structure even when every individual accelerometer reads zero.

5.2 The equivalence principle#

Newton’s equation can distinguish two roles for mass. Inertial mass $m_{\rm I}$ determines how much acceleration a force produces. Passive gravitational mass $m_{\rm G}$ determines the force exerted on the body by a given gravitational potential. Write

Universality of free fall says the ratio $m_{\rm G}/m_{\rm I}$ is independent of a sufficiently small test body’s composition and internal constitution, under the appropriate idealizations. A universal proportionality can be absorbed into the definition of the gravitational coupling, leaving $m_{\rm G}=m_{\rm I}$. Bodies with the same initial position and velocity then follow the same trajectory.

The Einstein equivalence principle extends the idea to local nongravitational physics: freely falling laboratories obey special relativity locally, and local nongravitational experiments do not acquire different laws merely from the laboratory’s velocity or location. The strong equivalence principle extends the scope to gravitational experiments and self-gravitating bodies. The distinctions and their experimental roles are carefully organized in Clifford Will’s review of tests of gravitation.

To apply these principles, keep track of the size of the laboratory and the kind of body being modeled.

First, local matters. In freely falling coordinates the metric can be Minkowskian and its first derivatives zero at an event. Curvature can still produce measurable effects across a finite laboratory or after a finite time. Making the laboratory smaller suppresses such effects; it does not declare curvature nonexistent.

Second, the test-particle model neglects the body’s effect on the surrounding gravity and treats its size as negligible. Rotation, an uneven mass distribution, or an appreciable gravitational influence of the body itself can require additional terms in its motion law. Here we are deriving the simpler limit in which those effects can be neglected.

Third, the equivalence principle does not uniquely imply Einstein’s field equation. Multiple theories can use a metric, respect the local free-fall picture, and supply different dynamics for that metric. We have learned how the local measuring system behaves; we have not yet derived what creates the gravitational field.

5.3 An accelerating laboratory in flat spacetime#

Can clocks held at different positions run at different rates even in flat spacetime? An accelerating array of clocks gives a concrete example. We will construct its coordinates from the inertial coordinates we already know.

We will use $\sinh\chi=(e^\chi-e^{-\chi})/2$ and $\cosh\chi=(e^\chi+e^{-\chi})/2$, introduced in §3.3. Differentiating these definitions gives $d(\sinh\chi)/d\chi=\cosh\chi$ and $d(\cosh\chi)/d\chi=\sinh\chi$. Their identity $\cosh^2\chi-\sinh^2\chi=1$ will simplify the interval.

Start in flat spacetime with inertial coordinates $(T,X,Y,Z)$. Introduce an accelerating chart $(t,X,Y,z)$ through

with $a>0$ and $z>-c^2/a$. These are Rindler coordinates on a region of Minkowski spacetime. We are explicitly using $t$, rather than $ct$, as the time coordinate in this chart.

To differentiate these expressions, abbreviate $L=c^2/a+z$ and $\chi=at/c$. Then $dL=dz$ and $d\chi=(a/c)dt$, giving

In $-[d(cT)]^2+dZ^2$, the two mixed terms cancel. The $dz^2$ coefficient becomes $\cosh^2\chi-\sinh^2\chi=1$ and the $dt^2$ coefficient becomes $-a^2L^2/c^2$. Restoring $L$ and adding the unchanged sideways terms gives

A clock at fixed $z,X,Y$ measures

Stationary clocks at different heights in this accelerating chart accumulate different proper times per coordinate time. Yet spacetime is exactly flat: we constructed the metric by changing coordinates in Minkowski space.

We can also calculate each clock’s accelerometer reading. At fixed $z$, keep using $L=c^2/a+z$ and $\chi=at/c$. The clock rule above gives $d\chi/d\tau=c/L$. Differentiating the inertial coordinates with respect to proper time gives the four-velocity components $u^0=d(cT)/d\tau$ and $u^Z=dZ/d\tau$,

Differentiate once more. The two four-acceleration components are $(c^2/L)\sinh\chi$ and $(c^2/L)\cosh\chi$. Their squared spacetime norm is $(c^2/L)^2$, because $\cosh^2\chi-\sinh^2\chi=1$. Taking its positive square root gives the proper acceleration

At $z=0$ the result is $a$, explaining the constant used in the coordinate transformation. Higher clocks have smaller proper acceleration. An array that maintains fixed separations therefore needs different accelerometer readings at different heights. Relativity puts restrictions on the notion of an accelerating rigid elevator.

The metric varies with $z$ and the clock rates differ, but the spacetime is flat: we constructed the whole example by relabeling Minkowski coordinates. A clock-rate difference alone therefore does not establish curvature. For a complementary derivation of these accelerating coordinates, see Tong’s treatment of the equivalence principle and Rindler motion.

5.4 An action for free fall#

In flat spacetime, Chapter 3 showed that an inertial clock records the greatest proper time between fixed departure and reunion events. We now seek a motion law that uses the same local clock measurements when the metric varies from place to place.

An action assigns a number to an entire candidate history by integrating a chosen expression along it. We postulate that the actual free-particle history makes its first-order change vanish when the endpoints are fixed. For a massive test particle described only by its position, the standard model uses

The action has units of energy times time. When we write $S=\int L\,dt$, the integrand $L$ is called the Lagrangian. In flat spacetime, this particle model gives $L=-mc^2\sqrt{1-v^2/c^2}$. Chapter 0’s square-root expansion gives $-mc^2+mv^2/2$ at low speed. The constant term has the same integral for every path between fixed endpoint times, leaving the familiar positive kinetic-energy term. The fixed overall factor does not affect the free trajectory for $m\ne0$.

Why this form? Proper time is a scalar quantity attached to the path, and the integral is unchanged if we relabel points along that path. With no additional internal structure or higher-derivative couplings, it is the simplest local relativistic free-particle action. We are adopting this model for structureless test particles. Bodies whose internal motion or gravitational influence matters can need additional terms.

“Stationary” means that the first-order action change vanishes for every sufficiently small change of the path that leaves its endpoints fixed. The comparison is a calculation we perform; the particle does not need to explore alternative paths. The next section translates this condition into a differential equation at each point of its motion.

A timelike geodesic is the free-fall path described by this model; the next section derives its equation. Sufficiently short segments locally maximize proper time. A long geodesic need not give the greatest elapsed time among every possible connecting path. Chapter 22 studies how families of such paths focus and why global maximizing claims need extra conditions.

5.5 Deriving the geodesic equation, one operation at a time#

Choose an arbitrary increasing parameter $\lambda$ along a timelike path. A dot in this subsection means $d/d\lambda$. Define

Then $S_{\rm particle}=-mc\int\ell\,d\lambda$. Because the nonzero constant $-mc$ does not change the stationary paths, vary $I=\int\ell\,d\lambda$ instead.

Perturb the curve by

where $s$ is a small number and $\eta^\mu$ vanishes at both endpoints. We take the derivative with respect to $s$ at $s=0$ and denote it by $\delta$.

Two things change: the metric is sampled at a slightly different point, and the tangent to the path changes. Specifically,

Differentiate the square root using $\delta\sqrt{Q}=\delta Q/(2\sqrt Q)$. Metric symmetry combines the two tangent variations into one term:

The variation contains a derivative of the arbitrary deformation $\eta$. Recall integration by parts: integrate the product rule $(F\eta)'=F'\eta+F\eta'$ and rearrange to obtain $\int F\eta'=[F\eta]-\int F'\eta$. This transfers a derivative from the deformation to its coefficient:

The boundary term vanishes because the endpoint events are fixed. Inside the interval, each $\eta^\rho$ can be chosen freely. The only way the integral can vanish for every such deformation is for its coefficient to vanish:

This is an example of the Euler–Lagrange equation. For a general integrand $L(x,\dot x,\lambda)$, the same integration by parts gives

The partial derivatives treat position and tangent as separate inputs of $L$. Our calculation used $L=\ell$. To put that result into a motion equation, multiply by $\ell$ and expand the ordinary derivative:

The product $\dot x^\mu\dot x^\nu$ is symmetric in $\mu,\nu$. Therefore the second term is unchanged if we replace its coefficient by its symmetrized version:

Multiply by $g^{\alpha\rho}$ to remove the metric in front of $\ddot x^\nu$. The combination of metric derivatives that emerges is denoted

These are the Christoffel symbols of the Levi-Civita connection. For now they are the coefficients our variation has produced. Chapter 7 will explain their geometric origin and why they transform differently from tensor components.

Our result is

If we choose $\lambda=\tau$, then $\ell=c$, so its derivative vanishes. We obtain

This is the timelike geodesic equation. It contains an ordinary coordinate acceleration plus a correction describing how the local coordinate basis and geometry change along the path. The whole combination is the covariant acceleration, which is zero in free fall.

Notice what disappeared: the test particle’s mass. That is the universality of free fall appearing in the variational description.

5.6 Affine parameters#

The same curve can be labeled in different ways. Numbering its points by a parameter $\lambda$ determines how quickly the coordinate functions change with $\lambda$, even when the geometric route stays fixed.

The geodesic equation with zero right-hand side chooses an affine parameter. For a timelike geodesic, proper time is affine, and so is

for constants $A\ne0$ and $B$. An increasing parameter uses $A>0$. A general nonlinear relabeling introduces a term parallel to the tangent, as the preceding derivation showed. Such a term changes the parameter’s rate along the curve while leaving the geodesic route unchanged.

One-dimensional flat-space example: the straight line $x(s)=s$ has $d^2x/ds^2=0$. Relabel it using $s=e^\lambda$. Now $x(\lambda)=e^\lambda$ and $d^2x/d\lambda^2=dx/d\lambda\ne0$. The line did not become geometrically curved. Its parameter became nonaffine.

The same issue occurs if coordinate time $t$ is used to parameterize a relativistic geodesic. In a general spacetime it need not be affine, so one must transform the equation correctly including the extra term when $t$ is nonaffine. These parameter subtleties and the null case are discussed in Sean Carroll’s notes, in the geodesics section.

WORKED EXAMPLE

A straight path can have an accelerating label

Can relabeling points on a straight worldline create a nonzero second derivative?

Builds onChain rule Interval

See the idea

Label a straight inertial worldline first with its clock time, then with a nonlinear function of that time. The same events get new numerical labels. Derivatives with respect to those labels change, although the clock and its motion have not. We can calculate the extra term using the chain rule.

Work it out

1. 01
   Begin with the simplest affine path

   In Cartesian coordinates on flat spacetime, let $x^a(\lambda)=v^a\lambda+b^a$. Its second derivative vanishes. Reparameterize using a smooth monotonic function $\lambda(\sigma)$ with nonzero derivative.

   $$\frac{dx^a}{d\sigma}=v^a\frac{d\lambda}{d\sigma}.$$

   Why this step works Changing the rate of traversal changes the tangent normalization.

2. 02
   Differentiate once more

   The chain rule gives $d^2x^a/d\sigma^2=v^a d^2\lambda/d\sigma^2$. Express $v^a$ using the first derivative. In a general chart the same calculation includes the connection term.

   $$\frac{d^2x^a}{d\sigma^2}+\Gamma^a{}_{bc}\frac{dx^b}{d\sigma}\frac{dx^c}{d\sigma}=\frac{d^2\lambda/d\sigma^2}{d\lambda/d\sigma}\frac{dx^a}{d\sigma}.$$

   Why this step works The extra acceleration is parallel to the tangent and changes the parameter speed.

3. 03
   Identify which changes remain affine

   The right-hand side vanishes for $\lambda=A\sigma+B$, where A is a nonzero constant. For $\lambda=\sqrt\sigma$ on $\sigma>0$, it instead equals $-1/(2\sigma)$ times the tangent.

   $$\frac{\lambda''}{\lambda'}=-\frac1{2\sigma}.$$

   Why this step works A nonlinear relabeling generally does not preserve affine normalization.

Go deeper

For a timelike geodesic, proper time is affine and the physical four-acceleration $du^a/d\tau+\Gamma^a{}_{bc}u^bu^c$ vanishes when $u^a=dx^a/d\tau$. The tangent with respect to an arbitrary parameter is not automatically this normalized four-velocity. A null curve has no advancing proper time, so use an affine parameter fixed up to scale and origin by suitable physical normalization, such as an emission energy. The vanishing proper-time length of a null curve does not remove its geometry.

Further readingDavid Tong · General relativity

Test the idea

FIRST, PREDICT

A straight inertial worldline has nonzero second coordinate derivative after a nonlinear reparameterization. Has a force appeared?

Yes. Every nonzero second derivative is proper acceleration.The path is no longer the same curve.No. Check the parameter and normalize the four-velocity.

Compare the reasoning

Yes. Every nonzero second derivative is proper acceleration.

Proper acceleration uses the proper-time-normalized tangent and the connection.

The path is no longer the same curve.

A monotonic relabeling changes traversal labels, not the events on the path.

No. Check the parameter and normalize the four-velocity.

The new term can be entirely parallel to the tangent and caused by the parameter choice.

A hint

Separate the set of events from the rate at which the parameter visits them.

NOW CHANGE THE EXAMPLE

For $\lambda=\sqrt\sigma$ with dimensionless $\sigma>0$, calculate $\lambda''/\lambda'$ at $\sigma=4$.

A hint

Use $-1/(2\sigma)$.

Work through the solution

$-1/(2\times4)=-1/8$.

A curve, an affine parameter, and an observer’s normalized four-velocity are different pieces of information.

5.7 Following light with a null geodesic#

For a light ray, $ds^2=0$ and $d\tau=0$. Proper time cannot parameterize the path. We therefore use an affine parameter $\lambda$ and write

with

We use this law for vacuum light when its wavelength is much smaller than the distances over which the geometry changes appreciably. In this geometric-optics approximation, a narrow wave packet follows a ray. Wave effects at longer wavelengths and propagation through matter require further analysis.

For a null geodesic, changing the affine scale changes $k^\mu$ without changing the route. The ray’s geometry alone therefore does not fix a photon’s energy. To identify the tangent with a physically normalized wave-vector or momentum, supply frequency or energy data from an observer.

Simply putting $m=0$ into $-mc^2\int d\tau$ makes the action vanish and yields no equation. A correct variational shortcut is the quadratic functional

Here the partial derivatives of the integrand are $\partial L/\partial\dot x^\rho=g_{\rho\nu}\dot x^\nu$ and $\partial L/\partial x^\rho=\tfrac12\partial_\rho g_{\mu\nu}\dot x^\mu\dot x^\nu$. Substituting them into the Euler–Lagrange equation gives the affine geodesic equation from §5.5. Choose an initial tangent with $g_{\mu\nu}k^\mu k^\nu=0$; the motion equation preserves this value. Indeed, differentiating $g_{\mu\nu}k^\mu k^\nu$ and substituting the displayed Christoffel formula makes its derivative zero. Do not restrict the entire family of varied curves to have identically zero integrand first and then expect varying zero to provide dynamics.

Further calculation: imposing the null condition with a multiplier

We can impose the null condition within the variation itself. Introduce a positive function $e(\lambda)$ and vary it as well as the path. It is an auxiliary field: an extra quantity used to impose a condition, rather than a new moving particle. Consider

$$I_0=\frac12\int e^{-1}g_{\mu\nu}\dot x^\mu\dot x^\nu\,d\lambda.$$

The integrand contains $e$ but no derivative of $e$, so its equation is simply

$$\frac{\partial L}{\partial e} =-\frac{1}{2e^2}g_{\mu\nu}\dot x^\mu\dot x^\nu=0.$$

This imposes nullness. Varying the path with the same Euler–Lagrange procedure gives

$$\ddot x^\mu+\Gamma^\mu{}_{\alpha\beta}\dot x^\alpha\dot x^\beta =\frac{\dot e}{e}\dot x^\mu.$$

To see how the parameter can absorb $e$, set $d\sigma/d\lambda=e(\lambda)/e_0$ for a positive constant $e_0$. The action in the new parameter has the same form with $e$ replaced by $e_0$. Its derivative term vanishes, leaving an affine geodesic equation. The auxiliary function has enforced nullness while allowing us to choose a convenient parameter.

The parameter still is not time on a clock traveling with light. As in Chapter 3, a photon has no inertial rest frame; its energy is specified by a physical observer’s measurement.

5.8 Recovering Newton’s law of motion#

Chapter 12 will relate the weak gravitational field to the Newtonian potential. Preview its slowly varying, weak-field result in a suitable nearly Cartesian chart:

where $|\Phi|/c^2\ll1$, particle speeds obey $v^2/c^2\ll1$, and spatial-metric corrections affect the calculation below only at higher combined order. The potential $\Phi$ has units of velocity squared; for a localized spherical mass, $\Phi=-G_NM/r$ when its zero is chosen at infinity.

The clock rate along a slow timelike path is

Insert this into the particle action:

For fixed endpoint times, the constant rest-energy term does not affect the path variation. The remaining Lagrangian is $L=mv^2/2-m\Phi=K-U$. For each Cartesian component, $\partial L/\partial v^i=mv^i$ and $\partial L/\partial x^i=-m\partial_i\Phi$. Substitution into the Euler–Lagrange equation gives

The relativistic action has combined what Newton separated into kinetic and potential terms. This is not a derivation of the gravitational field equation: we supplied the appropriate weak-field metric as a preview. It is a consistency check that, given that metric, the relativistic motion law has the expected limit.

One should also resist the slogan “objects fall toward slower time” as a universal replacement for GR. It can convey part of slow motion in a static weak field. It does not contain spatial curvature, frame dragging, null propagation in full generality, or the behavior of time-dependent geometries.

5.9 Relative acceleration and tides#

Take two nearby freely falling particles in Newtonian gravity, separated by $\xi^i$. Their individual accelerations are approximately $a^i=-\delta^{ij}\partial_j\Phi$. Subtract their equations and Taylor-expand the acceleration of the second particle about the first:

The gradient of $\Phi$ controls the common acceleration. Its Hessian, the matrix of second derivatives, controls the relative acceleration. A falling frame can cancel the former at its origin; it cannot generally cancel the latter throughout its neighborhood.

For $\Phi=-G_NM/r$,

Two radially separated falling particles therefore have relative radial acceleration

They stretch apart radially because the lower one falls more strongly. Neighboring side-by-side particles instead converge toward the central mass. A cloud of freely falling beads changes shape even though each bead’s ideal accelerometer reads zero.

This supplies an operational distinction:

The next chapters build the language needed to express the last row without borrowing a preferred Newtonian frame. The connection will tell us how to compare directions. Curvature will tell us why those comparisons can fail to fit together around a loop. Einstein’s equation will then say how that geometry is related to matter, energy, momentum, and stress.