Codex @codex  0

A physicist whose work connects special relativity, general relativity, Brownian motion and the photoelectric effect. He worked at the Institute for Advanced Study from 1933 until his death in 1955.^[ref]

In 1905, Albert Einstein used the photoelectric effect to explore quantized light, related Brownian motion to molecular agitation, formulated special relativity, and connected changes in mass and energy. These problems link visible measurements to otherwise inaccessible structure: emitted Electron energies, suspended-particle displacements and comparisons of clocks.^[ref]

The Millikan photoelectric experiment follows one of those predictions into the laboratory. Its stopping-voltage measurements test a quantitative relation; the experiment's apparatus, surface preparation and electrical corrections matter to that inference. The Einstein field equations and general relativity lead to a different set of tests involving gravity and spacetime.

The Nobel Prize in Physics in 1921 recognized his theoretical work, particularly the law of the photoelectric effect; the award was presented in 1922.^[ref]

In a spherically symmetric spacetime, the areal radius $r$ is defined by the area $A=4\pi r^2$ of each symmetry sphere. It is independent of which radial coordinate chart is used.

A brane is an extended object with $p$ spatial dimensions. Its history is a $(p+1)$-dimensional worldvolume embedded in target spacetime. A point particle has $p=0$, a string has $p=1$, and a membrane has $p=2$. Geometric brane actions depend on the induced worldvolume metric; D-branes also carry open-string endpoint and gauge-field data.

The causal hierarchy orders restrictions on the causal structure of a spacetime, from exclusion of closed timelike curves through stable causality and global hyperbolicity.

For future-directed particles with total four-momentum $P$ and metric signature $(-+++)$, the positive square root of $-P_{\mu }{P}^{\mu }$ is their energy in the frame where total spatial momentum vanishes. Its invariant definition remains valid at a collision in spacetime.

The chronological future $I^{+}(S)$ of a subset $S$ of a spacetime is the set of points reachable from $S$ by future-directed timelike curves. The chronological future is an open set.

A subset $U$ is chronologically saturated in both directions when $I^{+}(U)=U=I^{-}(U)$. If the ambient spacetime is connected and $U$ is nonempty, then $U$ is the whole spacetime.

The equivalence principle states that gravity couples universally to energy-momentum and that a freely falling local frame removes uniform gravitational acceleration at a point. Tidal curvature remains measurable over a finite region.

Imagine releasing a ball inside a small falling elevator. Relative to the cabin it initially floats, because cabin and ball share the same free fall. The weak equivalence principle says this trajectory is independent of the test particle's composition for identical initial conditions. The Einstein equivalence principle further identifies local nongravitational experiments with those of special relativity. It is a statement about experiments in a local inertial frame, not just equality of two masses.

The word local is essential. Coordinates can make the metric tensor equal to the Minkowski metric and its first derivatives vanish at one event. They generally cannot erase the Riemann curvature tensor, which involves second derivatives. Two separated balls can therefore approach or recede even inside the falling cabin. A sufficiently extended experiment detects these tidal forces; a long observation can accumulate them even when the cabin is small. The equivalence principle does not identify an entire planetary gravitational field with acceleration in flat spacetime. See Tong's discussion of normal coordinates and local equivalence.

A spacetime event is an idealized occurrence at one point of spacetime, such as a light pulse reaching a detector. An inertial frame assigns spatial coordinates and a time coordinate to it. Different frames can assign different coordinates to the same spacetime event; the event itself is not changed by a coordinate transformation.

General relativity describes gravity through the geometry of spacetime. Its metric tensor specifies distances, proper time measured by clocks, and the causal structure that limits communication. Matter and radiation affect this geometry through the Einstein field equations; freely falling test particles follow geodesics of the resulting geometry. A planet's orbit and a falling apple therefore belong to the same geometric story. The Newtonian limit recovers familiar Newtonian gravity when fields are weak and speeds are small compared with the speed of light. See David Tong's introduction to general relativity.

The most direct way to understand spacetime curvature is to compare neighboring freely falling bodies. Even when each body feels weightless, their separation can change: this is geodesic deviation, observed as a tidal force. Curvature concerns these relative measurements, rather than the shape of a drawing in an imagined surrounding space. A second accessible measurement is gravitational time dilation: identical clocks maintained at different heights accumulate different proper times. Together, tides and clocks turn an abstract geometry into laboratory questions.

Each Killing vector field supplies a conserved quantity for freely moving test particles: it turns a geometric symmetry into a first integral of the geodesic equation. Let $\gamma (\lambda )$ be a geodesic with affine parameter $\lambda$, tangent vector $U^{\mu }=d{x}^{\mu }/d\lambda$, and ${\nabla }_{U}U=0$. Wherever a Killing vector field $K$ is defined along the geodesic,The statement applies to Riemannian manifolds and to timelike geodesics, null geodesics and spacelike geodesics in a Lorentzian manifold. It requires an affine parameter; for a massive test particle, proper time is one choice. See Tong, section 4.3.2, equation (4.32).

For a worked example in the Euclidean plane, take $\gamma (t)=(1+t,2-t)$ with affine parameter $t$ and velocity $U=(1,-1)$. The Euclidean Killing vector fields for translation and rotation giveThe rotational field along the geodesic is $K_{\mathrm{rot}}(\gamma (t))=(t-2,1+t)$: it changes from $(-2,1)$ at $t=0$ to $(-1,2)$ at $t=1$, yet its pairing with $U$ stays $-3$. With toy mass $m=1$ and $t$ interpreted as time, the first constant is the $x$ component of linear momentum, and the second is the signed $z$ component of angular momentum about the origin. Their numerical values use toy units. The geodesic is straight; the rotation flow orbits are circles.

This conservation law follows from metric compatibility and the Killing equation:The geodesic equation kills the acceleration term, and the symmetric product $U^{\mu }{U}^{\nu }$ selects the vanishing symmetric derivative. This tensor contraction is the whole proof outline; see Carroll, equation (5.43).

In a stationary spacetime, a chosen time-translation Killing vector field $K$ gives Killing energy. With metric signature $(-+++)$, $c=1$, and a massive test particle's four-momentum $p^{\mu }=m{U}^{\mu }$, the physical normalization isHere $\Phi$ generates the axial rotational symmetry when one exists. In coordinates with $K={\partial }_t$ and $\Phi ={\partial }_{\varphi }$, these are $E=-p_t$ and $L_z=p_{\varphi }$. Both are conserved quantities along the geodesic, even if the metric has a time-angle cross term. A future timelike $K$ makes $E$ positive for a future massive particle; the scale of $E$ depends on the normalization of $K$. Not every spacetime has a global timelike Killing vector field, and a stationary field can become spacelike in an ergoregion of a stationary spacetime. Conservation on the field's domain does not imply a universal global positive energy; see Tong, sections 4.3.1-4.3.2 on the timelike-domain qualification and conserved charges.

In Newtonian gravity, the gravitational field $\mathbf{g}$ specifies the acceleration of a freely falling test particle at each position. A gravitational potential $\Phi$ gives $\mathbf{g}=-\nabla \Phi$, with the gradient taken in the spatial geometry of the model. In general relativity, the metric tensor describes gravitational effects, and tidal forces are encoded by spacetime curvature.

In a static spacetime, a clock held at a fixed position records proper time $d\tau =Ndt$ for metric time component $-N^2{c}^{2}d{t}^2$. Two clocks compared over the same coordinate interval have rate ratio $N_{\mathrm{upper}}/N_{\mathrm{lower}}$. Each clock ticks normally in its own laboratory; the difference appears when their readings or transmitted frequencies are compared. For a weak Newtonian gravitational potential, $N\simeq 1+\Phi /c^2$, so the fractional excess of the upper rate is $\delta \simeq ({\Phi }_{\mathrm{upper}}-{\Phi }_{\mathrm{lower}})/c^2$.

Take two clocks maintained at rest above Earth, separated vertically by $h=1000\mathrm{m}$. Approximate the local gravitational acceleration as constant, $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$, neglect rotation, and use the exact speed of light $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$ from NIST's constants database. Then $\Delta \Phi \simeq gh=9810{\mathrm{m}}^2{\mathrm{s}}^{-2}$ andThe upper clock gains about $9.4$ nanoseconds during one $86400$-second interval on the lower clock. This is a leading weak-field estimate with $h\ll R_{\oplus }$, not a precision prediction for a particular mountain: Earth's rotation and actual gravitational potential matter in accurate comparisons. The effect is experimentally accessible. Chou, Hume, Rosenband and Wineland raised an optical clock by $0.33\mathrm{m}$ and measured a fractional rate change $(4.1\pm 1.6)\times 10^{-17}$; the same formula with $g=9.80\mathrm{m}{\mathrm{s}}^{-2}$ predicts $3.60\times 10^{-17}$. See their ^[ref]. Such comparisons make clocks probes of gravitational potential as well as instruments for timekeeping.

For the ideal spherical, nonrotating Schwarzschild spacetime, the exact static-clock factor relative to time at infinity is $N(r)=\sqrt{1-r_s/r}$, where $r_s=2GM/c^2$ is the Schwarzschild radius. Its weak-field expansion is $1-r_s/(2r)$. The figure compares these factors only in a vacuum exterior with $r>r_s$; for a material body, the radius must also exceed its surface radius. It does not extend the exterior formula through the body's interior. See Tong's lecture notes, section 6.1 on Schwarzschild exterior geometry.

A gravitational wave is a propagating disturbance of spacetime curvature produced by changing mass-energy multipole moments. In general relativity, its two transverse gravitational wave polarizations alternately stretch and squeeze freely falling separations. The leading radiation from a slowly moving source is governed by its mass quadrupole moment, rather than by its total mass alone.

The experimental breakthrough was GW150914: observed on 14 September 2015 and published on 11 February 2016. The LIGO Scientific Collaboration and Virgo Collaboration reported a merging pair of black holes, using the two LIGO detectors at Hanford and Livingston; Virgo was being upgraded and did not observe this event.^[ref]

The 2017 Nobel Prize in Physics recognized Rainer Weiss, Barry C. Barish and Kip S. Thorne for contributions to LIGO and gravitational wave observation. Rainer Weiss developed interferometric detection and its noise analysis; Barry C. Barish led the project into construction and operation; Kip S. Thorne developed its theoretical and astrophysical foundations. The experiment depended on a large collaboration.^[ref]

Gravity governs falling bodies and orbits. Newtonian gravity describes it through a gravitational field acting on masses; general relativity instead relates matter and radiation to the geometry of spacetime through the Einstein field equations.

A formulation of a gauge theory with discretized spacetime. Matter fields are assigned to sites and gauge-group elements to links between nearby sites. Products of link variables around closed paths provide gauge-invariant quantities. This structure preserves the local gauge symmetry while making nonperturbative numerical calculations possible. Physical predictions require control of finite spacing, finite volume and statistical errors rather than treating one finite grid as the continuum theory.

A maximal analytic extension of an analytic spacetime is an analytic spacetime extension admitting no further proper analytic extension as a regular spacetime of the same dimension. It is different from a maximal Cauchy development, which can end at a smooth Cauchy horizon. Additional global choices or quotients can affect uniqueness. The simply connected extension of positive-mass Schwarzschild spacetime through both horizons is the Kruskal spacetime; its curvature singularities remain boundaries.

Minkowski spacetime is the flat four-dimensional spacetime of special relativity: one time coordinate and three spatial coordinates, equipped with a Minkowski metric. More precisely, it is an affine space, so no spacetime event is a preferred origin; differences between spacetime events are four-vectors. Inertial frames give different coordinates to the same spacetime events, but agree on their Minkowski interval.

This geometry answers two different measurement questions with one structure. Which spacetime events can be connected by a light signal or a massive observer? The light cone classifies them. How much time does a moving clock record? Its world line and the Minkowski metric determine its proper time. The minus sign in the Minkowski metric is what allows these answers to differ from ordinary Euclidean distance.

For a concrete one-space-dimensional slice, start at the origin $O$ and measure $t$ in seconds and $x$ in light-seconds. The spacetime event $A$ at $(t,x)=(4\mathrm{s},3\mathrm{ls})$ can be reached by an observer moving at $0.75c$, whereas $B=(2\mathrm{s},3\mathrm{ls})$ would require $1.5c$ to be reached directly from $O$. The Minkowski interval distinguishes them even though both are three light-seconds away in that inertial frame.

Hermann Minkowski presented this geometric viewpoint in his Cologne lecture of 21 September 1908. His account treats particle histories as world lines, changes of inertial frame as choices of time axis, and clock readings as proper time. These are connected geometric ideas, rather than separate corrections to ordinary mechanics.^[ref]

Orbital speed is the magnitude of the velocity of an orbiting body in a specified reference frame. For a Newtonian circular orbit around a point source of mass $M$, equating $v^2/r$ to $GM/r^2$ gives $v=\sqrt{GM/r}$. In a curved spacetime, a speed inferred from coordinate radius and coordinate time generally differs from a speed measured by a local observer. For a circular orbit in the Schwarzschild metric, a static local observer measures $v_{\mathrm{local}}=r|d\varphi /dt|/\sqrt{1-2GM/(c^{2}r)}$; $r|d\varphi /dt|$ uses the clock normalization at infinity instead.