From geometry to measurements

24.1 From a spacetime model to a measurement#

A calculation starts with physical assumptions and ends with an observer’s predicted reading. The intermediate steps have different jobs:

A component formula is usually a middle step, not the final observable. A clock calculation needs both the metric and the clock’s worldline.

24.2 A complete check with accelerating observers#

We will calculate the connection, curvature, accelerometer readings, and photon frequencies for the same family of accelerating observers.

Start in Minkowski spacetime, with inertial coordinates $(T,X,Y,Z)$:

Introduce accelerated coordinates $(t,x,y,z)$ by

where $a_0>0$ is a constant acceleration scale. Work in the wedge with

Differentiate the transformation. The mixed $dt\,dz$ terms from $-c^2dT^2$ and $dZ^2$ cancel, because the hyperbolic sine and cosine enter symmetrically. The remaining terms simplify using $\cosh^2\eta-\sinh^2\eta=1$:

This is the Rindler metric. We have changed coordinates, so spacetime must still be flat. But the metric now varies with position, and stationary clocks in these coordinates run at different rates relative to $t$:

These stationary coordinate positions belong to accelerating observers. Their relative clock rates depend on that choice of worldlines, even though the underlying spacetime is flat.

Calculate the connection#

Here the time coordinate is $t$ in seconds, so $g_{tt}=-c^2q^2$. The nonzero connection coefficients involving the $(t,z)$ sector are

where $q'=dq/dz=a_0/c^2$.

The first coefficient describes how the time basis changes as we move in $z$. The second is the coordinate acceleration term entering the $z$ geodesic equation.

An observer held at fixed $z$ has $u^t=dt/d\tau=1/q$ and $u^z=0$. Their four-acceleration is

Because $g_{zz}=1$ and the acceleration is spatial in this observer’s rest frame, the accelerometer magnitude is

The observers at different heights require different proper accelerations to maintain this stationary arrangement. They are not a collection of freely falling observers.

Calculate the curvature#

Compute a representative component directly:

The other terms vanish for this static diagonal metric. Substitute:

The cancellation matters. A derivative of a connection coefficient alone is not generally curvature. The quadratic connection terms are part of the definition. Here $q$ is linear in $z$, so $q''=0$, and every Riemann component vanishes.

The Einstein tensor is therefore zero. In the idealized test-apparatus limit, this metric is compatible with vacuum and $\Lambda=0$.

Calculate the frequency shift#

The metric is stationary, so the covariant photon momentum component $p_t$ is conserved along a null geodesic. A stationary observer measures

Since photon energy is proportional to frequency, a photon emitted at $z_e$ and received at $z_r$ satisfies

If the receiver is at larger $z$, the received frequency is lower. There is a frequency shift between accelerated observers even though spacetime curvature is identically zero.

The inertial description interprets the same experiment in terms of the observers’ motion during the light exchange. The accelerated description interprets it using a position-dependent lapse. Both predict the same detector readings.

What this calculation distinguishes. Nonconstant metric components, nonzero Christoffel symbols, acceleration readings, and frequency shifts between a specified family of observers do not individually prove nonzero spacetime curvature. Tidal curvature requires the appropriate invariant geometric test.

The chart covers the wedge $Z+c^2/a_0>|cT|$. Its boundaries $Z+c^2/a_0=\pm cT$ are null acceleration horizons for the stationary Rindler observers. They are neither curvature singularities nor black-hole horizons in Minkowski spacetime. Inertial coordinates extend across these boundaries in the same smooth Minkowski spacetime.

24.3 How to calculate a spacetime without getting lost#

Suppose someone hands you a metric and asks you to interpret it. Use this sequence.

1. Check the chart and domain. Which coordinate is time? Are angles dimensionless? Where is the matrix nondegenerate? Do any apparent singularities occur only at a chart boundary?
2. Invert the matrix. Verify $g^{\mu\alpha}g_{\alpha\nu}=\delta^\mu{}_{\nu}$. For a nondiagonal metric, componentwise reciprocals are incorrect.
3. Compute the determinant. This controls the volume element and often reveals where a coordinate chart fails.
4. Identify symmetries before differentiating. Independence of a coordinate can provide a Killing vector and conserved quantities. Symmetry can save pages of algebra.
5. Compute $\Gamma$ from $g$ and $\partial g$. Exploit its lower-index symmetry only for the Levi-Civita connection in a coordinate basis.
6. Compute Riemann with all its terms. Form the derivative terms and both quadratic terms with the chosen convention.
7. Contract carefully. Obtain Ricci, scalar curvature, and Einstein tensor; free indices must remain in the right places.
8. Compare with a physically admissible stress tensor. Conservation, matter equations, and an equation of state may rule out an apparently convenient interpretation.
9. Choose observers. Build four-velocities or a local orthonormal frame and project coordinate tensors into quantities those observers measure.
10. Check a known limit. Flat space, weak fields, small velocities, spherical symmetry, or an independently known invariant can reveal an error that elegant notation concealed.

Symbolic software can carry out the matrix operations and derivatives. Its output still needs the chart domain, matter assumptions, and observer definitions to become a physical prediction. Independent limits and invariants provide checks on the calculation.

WORKED EXAMPLE

Build a universe with matter, then observe it

Can you close the chain from metric to curvature, source, motion, and a photon measurement?

Builds onCurvature contractions Friedmann equations Cosmological observations

See the idea

Attempt this chain in your notebook before opening “Work it out.” Use c=1, zero cosmological constant, $t>0$, and the flat-slice FLRW metric $ds^2=-dt^2+a(t)^2d\mathbf x^2$, with dimensionless comoving coordinates and length-valued $a(t)=a_* (t/t_*)^{2/3}$. Find H, the independent orthonormal curvature entries, R, the source energy density and pressure, the comoving observers’ acceleration, and the redshift between two fixed emission/reception times. Flat spatial slices do not prejudge the four-dimensional answer.

Work it out

1. 01
   Differentiate the geometry

   The scale factor gives $H=2/(3t)$ and $\ddot a/a=-2/(9t^2)$. In an orthonormal frame, $R_{i0j0}=-(\ddot a/a)\delta_{ij}$ and $R_{ijkl}=H^2(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})$ for k=0. These formulas follow from the Christoffel calculation in Chapter 19. Contract them.

   $$R=6\left(\frac{\ddot a}{a}+H^2\right)=\frac4{3t^2},\qquad R_{abcd}R^{abcd}=12\left[\left(\frac{\ddot a}{a}\right)^2+H^4\right]=\frac{80}{27t^4}.$$

   Why this step works The scale factor’s time dependence creates spacetime curvature even with intrinsically flat spatial slices.

2. 02
   Infer the source

   The Einstein tensor gives $G_{00}=3H^2=4/(3t^2)$ and $G_{\hat i\hat j}=-(2\ddot a/a+H^2)\delta_{ij}=0$. With $G_{ab}=8\pi G_NT_{ab}$ in these units, the source is comoving pressureless matter.

   $$\epsilon=\frac1{6\pi G_Nt^2},\qquad p=0,\qquad \epsilon a^3=\text{constant}.$$

   Why this step works The conservation check uses a³ proportional to t² and independently confirms the inferred dust scaling.

3. 03
   Choose the observers and follow a ray

   Comoving observers have $u=\partial_t$, $u\cdot u=-1$, and $\Gamma^\mu{}_{tt}=0$, so their proper acceleration vanishes. A radial null ray satisfies $|d\chi/dt|=1/a(t)$. Its travel between emission $t_e$ and reception $t_o$ spans the comoving radial distance shown below.

   $$\Delta\chi=\frac{3t_*^{2/3}}{a_*}\left(t_o^{1/3}-t_e^{1/3}\right),\qquad 1+z=\frac{a(t_o)}{a(t_e)}=\left(\frac{t_o}{t_e}\right)^{2/3}.$$

   Why this step works The photon redshift compares comoving clock measurements at different events, not a coordinate light speed.

4. 04
   Make a new prediction before checking

   If $t_o=8t_e$, predict the redshift and received photon energy ratio. Also compare energy densities at t and 2t. Record which quantities change under coordinate relabeling and which are scalars or observer measurements.

   $$z=3,\qquad E_o/E_e=1/4,\qquad \epsilon(2t)/\epsilon(t)=1/4.$$

   Why this step works Both inverse-square-time density scaling and the observer-defined photon measurement are independently testable consequences.

Go deeper

A different source changes this chain. For $a\propto t^{1/2}$, the same contractions yield $R=0$ while the full curvature is nonzero; the Einstein equation gives $p=\epsilon/3$, appropriate to an isotropic radiation fluid. This provides a second counterexample to “zero scalar curvature means flat.” Derive that case independently: calculate H and $\ddot a/a$, infer $\epsilon$ and p, check $\epsilon a^4$ is constant, and find the redshift for $t_o=9t_e$. Expected checkpoints are $\epsilon=3/(32\pi G_Nt^2)$, $p=\epsilon/3$, and z=2. These classical ideal models have a singular boundary at t=0; the calculation does not establish how quantum gravity replaces it.

Further readingDavid Tong · General relativity

Test the idea

FIRST, PREDICT

The model has k = 0. Does that make its four-dimensional spacetime flat?

No. Its time-dependent scale factor gives nonzero Riemann curvature.Yes. Zero spatial curvature proves zero spacetime curvature.It is flat because every comoving observer is freely falling.

Compare the reasoning

No. Its time-dependent scale factor gives nonzero Riemann curvature.

The explicitly nonzero scalar and Kretschmann invariant verify that the spacetime is curved.

Yes. Zero spatial curvature proves zero spacetime curvature.

Spatial slices and the spacetime containing their evolution have different curvature tensors.

It is flat because every comoving observer is freely falling.

A whole family of freely falling observers can have nontrivial relative acceleration.

A hint

Calculate the orthonormal curvature entries before interpreting k.

NOW CHANGE THE EXAMPLE

For the dust capstone, evaluate $R\,t^2$.

A hint

Contract the supplied temporal and spatial curvature entries.

Work through the solution

$R=6[-2/(9t^2)+4/(9t^2)]=4/(3t^2)$, so $Rt^2=4/3$.

A complete GR prediction closes the chain from metric and source to observers and measured outcomes.

24.4 Four calibration geometries#

These four known geometries test different parts of a hand calculation or computer implementation. Here $A$ is the radius of a sphere; in the Schwarzschild row, $m=G_NM/c^2$; in the FLRW row, use $c=1$ and a spatially flat cosmology with $H=\dot a/a$.

The FLRW expression also shows that scalar curvature need not be positive even when the spatial slices are flat. “Spatially flat” is not “spacetime flat.”

24.5 Why substituting a solution into the action too early can destroy the question#

For a vacuum Ricci-flat solution with $\Lambda=0$, the Einstein–Hilbert bulk integrand $\sqrt{-g}R$ vanishes. Does this mean the action could not possibly determine that solution?

No. A function can vanish at a point without having zero derivative in every direction there. More specifically, a field configuration can satisfy $R=0$ while nearby off-shell configurations do not.

On shell means the fields satisfy their equations of motion. Off shell means we allow variations that need not satisfy those equations. A variational principle compares nearby off-shell configurations and asks whether the first change in the total action vanishes.

If you impose the equations before varying, you discard the directions that the variation is supposed to test. It is like replacing a function by its value at the minimum and then trying to recover the slope from that single number.

Boundary contributions add another reason not to identify a vanishing bulk integrand with a physically empty action. In gravitational problems, boundary conditions and boundary terms can carry decisive information.

24.6 Reconstructing the field equation#

Close the book for a moment and try to rebuild the logic.

A metric defines intervals and local causal structure. Its Levi-Civita connection supplies a derivative that respects the metric and has no torsion. The connection’s failure to return transported vectors unchanged around infinitesimal loops is curvature. Curvature’s contractions give the Ricci tensor and scalar.

A local gravitational action with the usual metric-only, two-derivative assumptions contains the scalar curvature and a constant term, integrated with the invariant volume measure. Matter has its own action and defines stress-energy through its response to a metric variation.

Vary the inverse metric. The curvature variation contributes $R_{\mu\nu}$ and a boundary divergence. The measure variation contributes $-\tfrac12Rg_{\mu\nu}$. The constant term contributes $\Lambda g_{\mu\nu}$. The matter variation contributes $-T_{\mu\nu}$ with the convention-dependent normalization already fixed.

Stationarity for arbitrary permitted interior variations gives

The Bianchi identity makes its left-hand side covariantly divergence-free. Diffeomorphism symmetry organizes the corresponding identity and matter balance law. The Newtonian limit fixes the coupling. Initial data and constraints select a physical history, up to gauge. Matter equations and the metric evolve together.

Then you ask an actual observer to make a measurement.

That final step is what keeps the entire geometric construction a theory of physics.

24.7 The highest-value change in intuition#

Before studying general relativity, you may ask, “What force pulls the body along that curve?”

After studying it, you have more precise questions available:

• Is that curve a geodesic of the physical metric?
• Is an accelerometer measuring nonzero proper acceleration?
• Does a neighboring geodesic reveal tidal curvature?
• Is the apparent effect a coordinate feature, an observer choice, or an invariant property?
• Which degrees of freedom are fixed by matter, and which arrive as gravitational initial or radiative data?
• Which approximation connects this mathematical model to the experiment?

You have not merely acquired a new answer about gravity. You have acquired a better set of questions.

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