Illustrated Physics
physics, drawn
Concept

General relativity — where it appears

Einstein's account of gravity as the curvature of spacetime, in which free bodies follow the straightest available paths. It reduces to Newton's law where the field is weak and slow, and differs measurably in the deflection of light, the delay of signals and the precession of orbits.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

A radar echo past the Sun, delayed by 233 microseconds. The extra time a round-trip radar signal takes when its path passes close to the Sun, against how close, for a reflector 0.723 AU away. Grazing the Sun's limb the delay is 233 microseconds — about 70 kilometres of light travel, on a path of hundreds of millions — and it falls only as the logarithm of the impact parameter, so the effect is still tens of microseconds ten solar radii out. That slow falloff is what makes the measurement possible: the delay can be watched building and fading as the geometry changes, rather than having to be caught at one instant. The dashed curves are the delay the solar corona's plasma adds at three radio frequencies. It is the competing effect, it is larger than the gravitational one close in, and it falls as the square of the frequency while the gravitational delay does not depend on frequency at all — which is how the two are separated, and why the sharpest measurement of this was made with a spacecraft carrying three radio links instead of one.

The delay that is not a bend

The same metric that bends a ray also slows it, and the two are different tests. A radar echo from Venus arrives 233 microseconds late when its path grazes the Sun — and half of that delay is accumulated more than twenty solar radii away, in a field thousands of times weaker.

astrophysics · Light deflection A lens with no focal length. Where a ray crosses the axis, against how far off the axis it passed the deflecting body, for 1 solar mass of radius 1 solar radius. A glass lens deflects a ray by an angle proportional to its distance off axis, which is precisely the condition for every ray to arrive at one point — the flat dashed line. Gravity deflects by 4GM/c²b, which grows smaller further out, so the crossing distance goes as b² and each ray has its own focus. The grazing ray crosses at 548 astronomical units and a ray passing at 12 radii crosses at 78857; the square law is verified on the drawn curve to 1.5e-16. So there is no image plane at all, only a half-line of foci beginning at the first of those and running outward for ever. Anything placed on that line sees not an image but a ring, and moving along it does not refocus anything — it selects which rays are being seen.

The lens with no focal length

A glass lens bends a ray by an angle proportional to how far off the axis it passes, which is precisely the condition for every ray to arrive at one point. Gravity bends by an angle that falls with distance off the axis, so every ray has its own focus and there is no image plane anywhere — only a half-line of foci, beginning 548 astronomical units from the Sun and running outward for ever.

astrophysics · Light deflection The figure a static field is not allowed to close. A spacetime diagram of two clocks held at fixed heights 22.5 metres apart, drawn as though spacetime were flat: time upward, height to the right, light at forty-five degrees. The lower clock sends two pulses; the upper clock receives them. Because the field does not change with time, nothing about the second pulse's journey differs from the first's, so the two null lines are congruent and the four worldlines bound a parallelogram. Opposite sides of a parallelogram in flat spacetime have equal length, so the proper time between emissions must equal the proper time between receptions, and the two clocks must agree. They do not: the measured fractional difference across a tower this tall is 2.455e-15, which Pound and Rebka established in 1960 and Pound and Snider confirmed to one per cent in 1964. Every step above is either a definition, an assumption of staticity, or a theorem of flat geometry — so the measurement refutes the flatness. No field equation has been written down, and none is needed: a laboratory result twenty-two metres tall is already incompatible with a flat spacetime.

The parallelogram that will not close

Two clocks twenty-two metres apart in a lift shaft run at different rates, by two parts in a thousand million million. That measurement, on its own, is enough to prove that spacetime cannot be flat — and the proof needs no field equation, no curvature tensor and no astronomy. It needs one drawing and the fact that opposite sides of a parallelogram are the same length.

astrophysics · Gravitational redshift A circumference that is more than 2π times the radius. The ratio of a rotating disc's measured circumference to 2π times its measured radius, against the speed of the rim, together with the rate of a clock carried on the rim. Rulers laid round the rim lie along their own direction of motion and are contracted; rulers laid along a radius lie across it and are not. So the circumference takes more of them than a stationary observer counts and the radius takes the same number, and the ratio is γ: 1.091 at β = 0.4, 1.400 at β = 0.7, 2.294 at β = 0.9. The geometry a rotating observer measures is therefore not Euclidean, and it is not Euclidean by an amount that depends on where on the disc the measurement is made. That is the observation Einstein said set him on the road to describing gravity with curved geometry: here is an accelerated frame, and here is a geometry in it that no choice of Cartesian coordinates can flatten. The rim's clock runs slow by the same factor, so a rotating frame has neither a common time nor a flat space.

The disc that cannot be spun

Set a disc turning and measure its circumference with rulers carried on the rim. They lie along their own direction of motion and are contracted, so more of them fit; rulers along a radius lie across the motion and are not. The ratio of circumference to radius is therefore not two pi, in a frame where nothing is happening but rotation — and Einstein said that was what set him looking for gravity in geometry.

relativity · Length contraction The arrow the orbit cannot turn. A Kepler orbit of eccentricity 0.6, integrated for two revolutions, with the Laplace–Runge– Lenz vector constructed from the position and velocity at five points along it. Every one of the five is the same arrow: its length varies by 3.7e-11 over the whole run and its direction by 2.1e-10 radians. It points at the perihelion and its length is 0.600000, which is the orbit's eccentricity measured independently from the closest and furthest radii as 0.600000. Energy and angular momentum fix the size and shape of an orbit and say nothing about which way it points; this vector is the missing statement, and only an inverse square has one.

The arrow that says which way the orbit points

Energy and angular momentum fix the size and shape of an orbit and say nothing about its orientation. The inverse-square force has a third conserved quantity that supplies it — a vector pointing at the perihelion whose length is the eccentricity — and no other force law does.

astrophysics · Orbit stability Both clocks move, and their ratio does not. Two years of a clock's fractional frequency against a distant clock (upper panel), and of the ratio of two unlike clocks kept side by side (lower panel). Above, both clocks slow as the Earth nears the Sun, with an amplitude of 1.65·10⁻¹⁰, and if the redshift is universal the two curves are one curve. Below, the ratio: the flat line is what universality predicts, and the sinusoid is what a clock responding to the potential 10⁻⁶ more strongly than the other would produce — an annual term of 1.65·10⁻¹⁶, a million times smaller than the shift both clocks share and within reach of clocks that compare to parts in 10¹⁷.

The clocks that must all slow together

Every clock on the Earth runs slower in January than in July, by three parts in ten thousand million, because the orbit carries the planet deeper into the Sun's potential at perihelion. No clock on the Earth can see this, and that invisibility is the claim worth testing. If the redshift is a property of time rather than of clocks, two clocks built on different physics must slow by exactly the same fraction, and their ratio must not move with the seasons. A ratio that did move would mean the constants of nature depend on where they are measured.

astrophysics · Gravitational redshift

Named alongside it

The objects these essays reach for when they reach for this one.

Equivalence principleGravitational redshiftClocksEccentricityLight deflectionMetricSimultaneitySpacetime curvatureAccelerationAngular momentumBorn rigidityCaustic

All concepts