Clocks, light, and Mercury

Two clocks start together. One stays on Earth; the other goes into orbit. When we compare their readings, how much time has each recorded? The field equation enters this question by determining the metric; the metric then determines the time accumulated along each clock’s path.

There is a repeatable answer. First solve, or approximate, the field equation for a metric. Then specify the worldlines of the source, detector, and light signals. Finally calculate quantities those observers can measure: elapsed proper time, frequency, angle, or separation. A coordinate component is an ingredient in that calculation; it is not automatically an observable.

16.1 How the metric changes clocks and rulers#

For a weak, approximately static field with negligible rotation, choose Cartesian spatial coordinates and write

Here $x^0=ct$, and $|\Phi|/c^2,|\Psi|/c^2\ll1$. Both potentials have units of velocity squared. The first alters the relation between coordinate time and clock time. The second alters the relation between coordinate distances and ruler lengths. We have omitted vector perturbations associated with mass currents, gravitational waves, and higher-order terms.

For an isolated, slowly moving, weakly gravitating source in GR, with pressure and directional stresses negligible compared with its rest-energy density and with the potentials vanishing far away,

outside a spherical body. In more general matter systems the equality needs justification; it is not part of the definition of a gravitational potential.

Section 12.5 derived this equality by keeping the temporal and spatial perturbations separate in the field equation. Here they are written as $\Phi=c^2\varphi$ and $\Psi=c^2\psi$, restoring velocity-squared units. We will use the same metric to calculate slow motion, clock readings, and light propagation.

For a slowly moving freely falling object, the leading spatial geodesic equation is

Staticity removes the time derivatives in the connection, giving

Consequently $d^2\mathbf x/dt^2=-\boldsymbol\nabla\Phi$: Newton emerges because the metric’s clock coefficient has the appropriate gradient. Terms involving spatial velocities enter at higher order for this slow particle. They cannot be discarded for light.

This is a coordinate acceleration, measured using the positions and time labels of the chosen chart. The falling object’s accelerometer reads zero. Its covariant four-acceleration $a^\mu=u^\nu\nabla_\nu u^\mu$ vanishes. A person standing on the floor has approximately zero coordinate acceleration in this chart but nonzero proper acceleration: the floor prevents a geodesic. A scale measures the supporting force that prevents this free fall.

16.2 What a clock actually accumulates#

Substitute $dx^i=v^i dt$ and $ds^2=-c^2d\tau^2$ into the weak-field metric:

Keep terms first order in $\Phi/c^2$ and $v^2/c^2$, dropping their product. Using $\sqrt{1+q}\simeq1+q/2$ gives

The gravitational and kinematic clock corrections now occupy the same line. A lower, more negative potential reduces accumulated proper time relative to this coordinate time. Motion also reduces it at this order. To compare clocks following different routes, integrate each expression along its own route and specify how their readings are compared.

Near Earth’s surface, two stationary clocks separated vertically by a small height $h$ have $\Delta\Phi\simeq gh$. For $h=1\,\mathrm m$,

Multiply the fractional rate difference by $86{,}400$ seconds to get the difference after a day: about $9.4$ picoseconds, where one picosecond is $10^{-12}$ seconds. The higher clock gains time.

This effect alone does not establish nonzero curvature. Accelerated observers in flat spacetime can also have systematically different clock rates. Curvature concerns the obstruction to removing gravitational effects throughout an extended region, especially tidal effects. It is stronger information than one pair of differently ticking clocks.

16.3 Gravitational redshift and observer energy#

In a stationary region, let $K^\mu$ be the timelike Killing field describing time-translation symmetry. Normalize it so that in an asymptotically flat static chart $K=\partial/\partial x^0$. Define its positive norm factor

An observer remaining on an orbit of this symmetry has four-velocity

Dividing by $N$ makes the norm of $K/N$ equal to $-1$; multiplying by $c$ then gives the required $u_\mu u^\mu=-c^2$.

Let $p^\mu$ be a photon’s four-momentum, transported along its null geodesic. The Killing equation is $\nabla_{(\mu}K_{\nu)}=0$. Therefore

The term differentiating $p_\mu$ vanished by the geodesic equation. The remaining contraction vanishes because $p^\alpha p^\mu$ is symmetric while the relevant part of $\nabla_\alpha K_\mu$ is antisymmetric. Thus

is conserved along the light ray. But the energy measured by a particular observer is

This is the energy convention used in Section 15.4: $p^0=E/c$, so the factor $c$ gives $E_K$ energy units. Observers at different values of $N$ measure different local energies even though the photon has the same conserved $E_K$ along its ray.

Since photon energy is proportional to measured frequency,

For the static weak-field metric, $N\simeq1+\Phi/c^2$. Thus

Light received higher in the potential has a lower measured frequency: it is redshifted. The conserved quantity is the symmetry energy $E_K$; the changing quantity is $E_{\rm local}=E_K/N$. In a time-dependent geometry without this timelike symmetry, the conserved quantity used in this derivation need not exist.

16.4 Deriving the bending of light#

For a null trajectory, $ds^2=0$. Let $d\ell^2=\delta_{ij}dx^i dx^j$ denote Euclidean coordinate path length. Then

In this static chart, the light path makes the travel-time functional $\int n\,d\ell$ stationary. This is the same mathematics as ray optics in an inhomogeneous refractive medium. No material ether has appeared: $n$ is a coordinate description of null geometry, and every local freely falling observer still measures light speed $c$.

To calculate the path, let the mass sit at the origin and describe the ray in the $xz$ plane by $x(z)$. A prime here means $d/dz$, and $d\ell=\sqrt{1+x'^2}\,dz$. Apply the Euler–Lagrange equation from Chapter 13 to $L=n(x,z)\sqrt{1+x'^2}$:

For weak bending, $n-1$ and the small slope $x'$ are both first-order quantities. Drop products of small quantities to obtain $x''\simeq\partial_xn$. The slope is the small angle the ray makes with its original direction, so integrating this equation gives its angle change.

The unperturbed ray has $x=b$, where $b$ is its impact parameter: its perpendicular distance from the mass if it continued straight. In vector form, $\boldsymbol\nabla_\perp$ means derivatives in the two directions perpendicular to the unperturbed ray. Thus

Why can we integrate along a straight path if the path bends? Because the bending is already first order in $G_NM$. Correcting the path inside this first-order integrand would produce a second-order correction.

Write $\Psi=\gamma\Phi$ for a constant comparison parameter. At transverse position $b>0$,

The transverse direction change is negative, toward the mass. Its magnitude is

The integral equals $2/b^2$; differentiating $z/[b^2\sqrt{b^2+z^2}]$ verifies it. GR gives $\gamma=1$, hence

For a ray grazing the Sun, set $b$ equal to the solar radius. The angle is about $1.75$ arcseconds; one arcsecond is $1/3600$ of a degree. Setting $\Psi=0$ while retaining the same $\Phi$ would halve the result.

The two contributions in this calculation belong to the chosen weak-field coordinates. Changing coordinates can change how we divide them between temporal and spatial metric terms. The predicted angle measured by the specified observer is independent of that division.

16.5 The extra travel time of a light signal#

The effective index also produces an additional travel time. Along a nearly straight path,

For a point mass, $\Phi+\Psi=-2G_NM/\sqrt{b^2+z^2}$ along the unperturbed ray. Its endpoints are at $z=-z_1$ and $z=z_2$, with $z_1,z_2>0$. The integral we need is

The inverse hyperbolic sine is $\operatorname{arsinh}q=\ln(q+\sqrt{1+q^2})$. Differentiating this logarithm gives $1/\sqrt{1+q^2}$ and verifies the antiderivative. It is an odd function, so evaluating the lower endpoint adds a second positive contribution:

For $q\gg1$, $\sqrt{1+q^2}\simeq q$, so the logarithm becomes $\operatorname{arsinh}q\simeq\ln(2q)$. When both endpoints are far from closest approach, $z_1,z_2\gg b$, this gives

This is a leading one-way coordinate delay relative to the corresponding flat path; $r_1,r_2$ are approximately the endpoint distances from the mass. An actual radar experiment models the return trip and converts the result into the tracking station’s proper time. Gravitational lensing more generally also involves different geometric path lengths. The observational model must keep both contributions.

The solar coefficient $2G_NM_\odot/c^3$ is about $9.85$ microseconds. For endpoints at about one astronomical unit on opposite sides of the Sun and a grazing ray, the logarithm is about 12.1. The one-way delay is then about $119$ microseconds in this approximation.

16.6 Mercury and the rotation of an orbit#

The preparation above derived the Newtonian orbit equation. We now use its relativistic extension; Section 17.5 derives that extension from the Schwarzschild metric. For a massive test particle in that geometry, define the conserved specific angular momentum $\ell=r^2d\phi/d\tau$ and let $u=1/r$. The exact equatorial orbit equation is

Here $u$ denotes reciprocal radius. The first term gives the Newtonian orbit equation already derived in the preparation; the term proportional to $u^2$ is the relativistic addition.

Set $p_{\rm orb}=\ell^2/(G_NM)$ and $m=G_NM/c^2$. The unperturbed orbit is

The dimensionless number $e$ is the eccentricity, describing the orbit’s shape. A circle has $e=0$; a bound ellipse has $0<e<1$. Choose $\phi=0$ at closest approach, called perihelion for an orbit around the Sun. Then

The semimajor axis $a_{\rm orb}$ is half the ellipse’s longest diameter, so $a_{\rm orb}=(r_{\min}+r_{\max})/2=p_{\rm orb}/(1-e^2)$. This gives the geometric meaning of the parameters we will use in the measured precession.

Insert $u_0$ into the small correction $3mu^2$. Its term proportional to $\cos\phi$ is $6me\cos\phi/p_{\rm orb}^2$. This drives the same angular frequency as the homogeneous operator $d^2/d\phi^2+1$. The resulting resonant particular solution is

because $(d^2/d\phi^2+1)(\phi\sin\phi)=2\cos\phi$. The other forcing terms produce bounded shape corrections, not the accumulated rotation we are seeking.

Now expand a slightly shifted oscillation:

Matching coefficients gives $\delta=3m/p_{\rm orb}$. One radial cycle therefore takes slightly more than $2\pi$ in azimuth:

per orbit, where $p_{\rm orb}=a_{\rm orb}(1-e^2)$ at the needed Newtonian order. For Mercury, using $a_{\rm orb}\simeq5.79\times10^{10}\,\mathrm m$ and $e\simeq0.206$, this gives about $0.104$ arcseconds per orbit. Mercury completes a revolution in about $87.97$ days, giving approximately $100(365.25)/87.97\simeq415$ orbits per century. Multiplying gives about $43$ arcseconds per century.

This is the relativistic contribution under the approximation of an isolated spherical Sun. Planetary perturbations, solar structure, and reference-frame modeling also affect the observed perihelion. The success lies in calculating the appropriate additional contribution, not declaring that every observed orbital change is relativistic.

16.7 GPS: calculate the competing clock effects#

Start with the clock formula already derived here:

Use a nonrotating, spherical Earth model. Compare a clock in a circular orbit of radius $r$ with a stationary surface clock at radius $R$. The potential is $\Phi=-G_NM/r$. Circular motion requires $v^2/r=G_NM/r^2$, so $v^2=G_NM/r$. Subtract the surface rate from the orbital rate:

Take $R=6371\,\mathrm{km}$ and $r=26{,}571\,\mathrm{km}$, corresponding to an approximate GPS altitude of $20{,}200\,\mathrm{km}$. With $c=299{,}792{,}458\,\mathrm{m/s}$ and the Earth parameter from Section 10.9, multiply each dimensionless rate by $86{,}400$ seconds per day:

A light signal travels about $11.5\,\mathrm{km}$ in $38.5\,\mu\mathrm s$. This converts a timing offset to a ranging scale; it is not a full prediction of an uncorrected receiver’s position error, which depends on how it estimates its own clock bias and uses satellite data.

Set the net rate to zero. The result is $1/R=3/(2r)$, or $r=3R/2$. In this model, a circular-orbit clock matches the surface rate at altitude $R/2\simeq3186\,\mathrm{km}$. Below that, it loses time; above that, it gains time. This crossover is not exact for the rotating, nonspherical real Earth. Operational GPS also includes eccentricity, Earth rotation, a chosen reference time and geoid, and propagation corrections. Ashby’s account of relativity in GPS.

16.8 A tossed clock can age more than the clock on the shelf#

Compare two ideal clocks that start together at height zero and reunite after coordinate time $T$. One remains supported at that height. The other is tossed vertically and falls freely between launch and catch. Ignore drag, recoil, and the brief launch and catch intervals; use a uniform $g$ and weak-field, slow-motion accuracy.

The free-fall path that returns after $T$ is

Choose $\Phi=gh$, so the shelf’s potential is zero. The tossed clock gains from height and loses from motion. Its net proper-time excess is

Do the two elementary integrals separately:

Thus $\Delta\tau=g^2T^3/(24c^2)>0$. With $g=9.81\,\mathrm{m/s^2}$ and $T=1\,\mathrm s$, the gain is approximately $4.46\times10^{-17}\,\mathrm s$, or 44.6 attoseconds, where one attosecond is $10^{-18}$ seconds. The height gain is twice the speed loss. In this short-path regime, the timelike free-fall path locally maximizes proper time between the endpoints. This does not make every geodesic a global maximum over arbitrary long journeys.

16.9 Gyroscopes and a compact experimental map#

A gyroscope supplies a direction that can be transported. Around a gravitating body, its orientation need not stay fixed relative to distant reference directions. Even a nonrotating source produces geodetic precession. A rotating source adds frame dragging. These are different contributions, not two names for the same effect.

The leading frame-dragging result below comes from parallel-transporting the spin in a weak rotating metric. We will identify that metric’s new time-angle component in Section 17.7; here we use the resulting rate to interpret the experiment. For source angular momentum $\mathbf J$,

Here $\hat{\mathbf r}$ is the radial unit vector. The expression is a vector: an orbital average must average its direction as well as its magnitude. For a circular polar orbit, choose $\mathbf J=J\hat{\mathbf z}$ and $\hat{\mathbf r}=(\sin\vartheta,0,\cos\vartheta)$. Over an orbit, the averages of $\sin\vartheta\cos\vartheta$ and $\cos^2\vartheta$ are 0 and $1/2$. Thus $\langle(\mathbf J\cdot\hat{\mathbf r})\hat{\mathbf r}\rangle=\mathbf J/2$, and the averaged precession vector is $G_N\mathbf J/(2c^2r^3)$. The measured projection also depends on the reference direction used by the experiment.

Gravity Probe B reported drift magnitudes of $6601.8\pm18.3$ milliarcseconds per year for the geodetic effect and $37.2\pm7.2$ for frame dragging, compared with predictions of $6606.1$ and $39.2$. A milliarcsecond is $10^{-3}$ arcseconds. The experiment’s signed drift convention is defined by its sky axes; magnitudes are quoted here to focus on scale. The collaboration’s 2011 final results.

This is a map of physical questions, not a ranking by a single “precision of GR.” Each measurement has its own model, observable, nuisance parameters, and uncertainty. MICROSCOPE does not establish exact equality or directly measure a metric coefficient. MICROSCOPE’s final analysis.