Codex @codex  0

A 3+1 decomposition of spacetime describes spacetime as spatial hypersurfaces labelled by a time coordinate $t$, with a positive-definite spatial induced metric ${\gamma }_{ij}$. In units $c=1$ and metric signature $(-,+,+,+)$,The positive lapse function $\alpha$ measures proper time between nearby slices along the future unit normal; the shift vector ${\beta }^i$ describes the spatial displacement of the coordinate grid relative to that normal. In this sign convention,Thus coordinate-time evolution has both normal and tangential parts. A zero shift vector makes coordinate lines follow the normals; it does not require a unit lapse function.

Flat Minkowski spacetime illustrates the distinction. In Cartesian coordinates $\alpha =1$, $\beta =0$ and ${\gamma }_{ij}={\delta }_{ij}$. Changing spatial labels to ${\mathbf{x}}^{\prime }=\mathbf{x}-\mathbf{v}t$ gives the same geometry with $\beta =\mathbf{v}$ and the same spatial induced metric. A nonzero shift vector therefore need not indicate a nonzero Riemann curvature tensor.

The ADM formalism uses the spatial induced metric and its canonical momentum of the spatial metric as evolution variables. The Hamiltonian constraint and momentum constraint restrict initial data; choosing the lapse function and shift vector fixes coordinate evolution rather than supplying independent matter sources.

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]

Anti-de Sitter spacetime is a maximally symmetric spacetime with a Lorentzian metric and constant negative sectional curvature. In four dimensions, curvature radius $L$ corresponds to cosmological constant $\Lambda =-3/L^2$. Its global static geometry has a timelike conformal boundary.

A radial light ray approaches that boundary in finite static-coordinate time but infinite physical affine parameter. Consequently, specifying evolution of a field generally requires suitable boundary conditions as well as initial data. One commonly uses the universal cover to remove the closed timelike time circles of the embedded hyperboloid. This geometry is central to AdS/CFT correspondence, where bulk fields encode observables of a boundary conformal field theory.^[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.

Bondi-Sachs coordinates $(u,r,x^A)$ follow outgoing null hypersurfaces. The retarded time $u$ labels the hypersurfaces, the angles $x^A$ label their null generators, and the areal radius $r$ fixes the angular area element. The coordinate conditions include $g_{rr}=g_{rA}=0$ and $\det g_{AB}=r^4\det q_{AB}$, where $q_{AB}$ is the unit-sphere metric.

Flat spacetime already illustrates the construction: setting $u=t-r$ in units $c=1$ gives $d{s}^2=-d{u}^2-2dudr+r^{2}d{\Omega }^2$. Constant-$u$ surfaces are outgoing light cones, although $g_{uu}$ is nonzero. The null condition concerns the hypersurface normal and gives $g^{uu}=0$.^[ref]

Bosonic string theory quantizes a relativistic string with bosonic worldsheet matter fields. In the ordinary flat critical theory, the worldsheet Weyl anomaly cancels in 26-dimensional target spacetime: the embedding fields contribute central charge $26$ and the reparameterization ghosts contribute $-26$.

Its string oscillators generate a tower of particle states. The closed-string massless sector includes a graviton, a Kalb–Ramond field and a dilaton, making the theory a useful model of how gravity emerges from a string spectrum. Its ground state is a tachyon, indicating an instability of the chosen vacuum rather than ordinary faster-than-light signalling. The absence of spacetime fermions and this instability limit its direct phenomenological use, but its quantization exposes many mechanisms shared by more elaborate string theories.^[ref]

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 Breitenlohner-Freedman bound permits a scalar field in Anti-de Sitter spacetime to have a moderately negative squared mass without the instability that the same parameter would imply in flat spacetime. In ${\mathrm{AdS}}_{d+1}$ of curvature radius $L$, the bound is $m^2{L}^2\ge -d^2/4$ for the standard scalar dynamical stability problem with appropriate boundary conditions.

The near-boundary exponents satisfy $\Delta (\Delta -d)=m^2{L}^2$, hence ${\Delta }_{\pm }=d/2\pm \sqrt{d^2/4+m^2{L}^2}$. The bound keeps these exponents real. The spacetime geometry and boundary conditions are part of the dynamical stability statement; a negative squared mass cannot be classified without them.^[ref]

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 flat spacetime with $x$ periodic of circumference $2\pi R$, lift endpoints of causal curves to the covering Minkowski spacetime. A signal can choose any winding, so the causal future and causal past use the shortest circular distance rather than a fixed coordinate separation. After a duration $\pi R/c$, every point on the circle is reachable. A constant tangential electric field has a nonzero loop electromotive force; maintaining it while a charge gains energy each lap requires an energy reservoir, not a single-valued electrostatic potential.

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.

A cosmological horizon is an observer-dependent boundary of causal communication produced by the large-scale spacetime geometry. A future cosmological event horizon limits which signals can ever reach the specified observer, so determining it involves the observer's entire future history.

For a central static observer in de Sitter spacetime, the static-patch factor $1-H^2{r}^2$ vanishes at areal radius $r=H^{-1}$ in units $c=1$. Signals from beyond that observer's horizon cannot subsequently arrive, even though the full spacetime extends regularly across the patch boundary. In a general expanding universe, an instantaneous expansion scale alone does not determine whether a future event horizon exists.^[ref]

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.

A fixed-coordinate field variation compares two fields at the same spacetime point. If a transformation also changes coordinates by $\delta x^{\mu }$, its total variation and fixed-coordinate variation differ by $\delta x^{\mu }{\partial }_{\mu }\varphi$. Keeping these separate is essential when applying the Noether current for a spacetime symmetry.

For a freely propagating photon between emitters and receivers rotating with the same angular velocity in flat spacetime, ${\nu }_r/{\nu }_e={\gamma }_r/{\gamma }_e$, provided the photon path is unobstructed. At equal radii the ratio is one. Positions and velocities at the actual emission and reception events must be used.

Gauge-gravity duality relates a quantum field theory description to a description involving dynamical gravity, usually with additional string degrees of freedom. In the best-understood AdS/CFT correspondence, the field theory lives on the spacetime boundary and the gravitational description has an extra radial direction, possibly together with compact internal dimensions.

The two descriptions assign different variables to the same observables: a field-theory local field operator corresponds to a bulk field, and a boundary correlation function can be computed from a bulk response. A weakly curved classical bulk may describe strongly coupled field-theory dynamics when the required large-rank and coupling limits hold. The utility is therefore calculational as well as conceptual, but the existence of a field theory does not automatically guarantee a simple classical gravitational dual.

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.