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Use geometrized units $G=c=1$ and metric signature $(-,+,+,+)$. Put $f=1-2M/r$ in the Schwarzschild metric. The Schwarzschild tortoise coordinate satisfies $d{r}_{\ast }/dr=f^{-1}$, soSubstituting $dt=dv-f^{-1}dr$ gives the ingoing Eddington-Finkelstein coordinates:The radial metric tensor has determinant $-1$ and inverse components $g^{vv}=0$, $g^{vr}=1$, $g^{rr}=f$. Thus it is nondegenerate and analytic at $r=2M$. The same expression defines a Lorentzian metric for every $r>0$, extending the exterior across the future Schwarzschild event horizon into the black hole. It does not include the other exterior or the white hole of the full Kruskal spacetime. At $r=0$, the Kretschmann scalar $R_{abcd}{R}^{abcd}=48M^2/r^6$ diverges, so this is a curvature singularity, not a removable coordinate singularity.

For the P(X, phi) scalar field theory specialized to $F(X)$, vary the inverse metric tensor while holding $\varphi$ fixed. With $q_a={\partial }_a\varphi$,Using $T_{ab}=-2(-g{)}^{-1/2}\delta S/\delta g^{ab}$ yieldsIn any local Lorentz frame, set $A=F^{\prime }(X)\ge 0$, $Y=X{F}^{\prime }(X)-F(X)\ge 0$, $s={\sum }_i{q}_i^2$, and $X=(q_0^2-s)/2$. ThenA direct calculation givesHence $\rho \ge \sqrt{T_{0i}{T}_{0i}}$ in every local Lorentz frame, proving the dominant energy condition. This proof covers timelike, null, spacelike, and zero $\nabla \varphi$ alike; it does not assume $X>0$ or use a perfect fluid in general relativity rest frame. The hypotheses need only hold at every attained value of $X$; the stated all-$X$ assumptions are sufficient. For example, $F=X-V_0$, $V_0\ge 0$, satisfies them.

Write $h_{ij}$ for the spatial metric tensor and lower the shift vector with $h_{ij}$. For the negative-shift convention, the four-dimensional metric and its inverse have componentsThe normal has covariant components $n_{\mu }=(-N,0,0,0)$, so $n_{\mu }{n}^{\mu }=-1$ and it annihilates every tangent vector to the spatial slice. Since $n_i=0$, its covariant derivative gives $n_{i;j}=N{\Gamma }^0{}_{ij}$ and therefore $K_{ij}=-N{\Gamma }^0{}_{ij}$.

Compute the required Christoffel symbol using the symmetric connection formula. Its time-index term and spatial-index term combine to giveThe spatial expression in parentheses is ${\dot{h}}_{ij}+D_i{N}_j+D_j{N}_i$, where $D$ is the covariant derivative of $h$. Thus the extrinsic curvature with a negative shift isThis derivation fixes both signs from the actual 3+1 decomposition of spacetime, rather than transferring a formula using the opposite shift vector convention. The connection formula used here has $g_{\lambda \kappa ,\nu }$ as its second differentiated term; the printed repeated-index version is a typographical error.

For the spherical coordinates in the question, the coordinate tangent vectors are mutually orthogonal, with lengths $1,r,r\sin \theta$. Their metric tensor is therefore $\mathrm{diag}(1,r^2,r^2{\sin }^2\theta )$ and the Jacobian determinant is $r^2\sin \theta$. Applying the divergence formula to the gradient givesFor example, the formula follows directly from $\Delta u=|\det h{|}^{-1/2}{\partial }_i(|\det h{|}^{1/2}{h}^{ij}{\partial }_{j}u)$ for this diagonal metric. These coordinates are valid away from the polar axis and angular seam; the Laplacian is a smooth geometric operator there as well, described by other charts. The coordinate volume form integrates over $1/2<r<2$, $0<\theta <\pi$, $0\le \varphi <2\pi$, with the usual periodic identification.

Put $q=r_0^2$ and $\eta =d\psi +\frac{1}{2}\cos \theta d\varphi$. Expanding the metric tensor in the stationary direction gives the useful exact identitiesTaking the exterior derivative givesThe fibered form of the metric tensor has determinant $\det g=-r^6{\sin }^2\theta /16$. Its inverse components needed for the flux areThereforesince $h(1-\alpha \Omega )=1$. This calculation includes the rotational mixed term; dropping it prematurely would miss the exact cancellation.

With the stipulated negative coordinate orientation, ${ϵ}_{tr\psi \theta \varphi }=-r^3\sin \theta /4$. The pullback of the Hodge star operator to the constant-$t,r$ three-dimensional spacelike submanifold is consequentlyThe coordinate ranges and standard sphere identifications giveThus the five-dimensional Komar mass, in the normalization given and $G_5=1$, isThe flux is already independent of $r$ in the vacuum region, so its limit at infinity has the same value. The negative orientation in the question is crucial for the positive sign; reversing that orientation reverses the flux.

For the zero-angular-momentum curves, the fiber term in the induced metric tensor vanishes along the tangent, leavingfor a null geodesic. In the exterior, a future geodesic has positive conserved energy $E=(f/h)\dot{t}$. Its equations areDefine the tortoise coordinate, separately on intervals bounded by horizon radii, byFor an outgoing curve $\dot{r}=+E\sqrt{h}$,for an ingoing curve $\dot{r}=-E\sqrt{h}$, $d(t+r_{\ast })/d\lambda =0$. Thus $u=t-r_{\ast }$ and $v=t+r_{\ast }$ are the respective constant labels. The derivative $\sqrt{h}/f$, with its sign, is required for continuation; taking $\sqrt{h}/|f|$ would give the wrong labels between the horizons.

For a nonextremal horizon $r=r_H$, the tortoise coordinate has leading behaviorAt the double root of the extremal black hole it instead has a pole. These singularities are coordinate effects at a regular Killing horizon; they are not the curvature singularity at $r=0$. To cross the future horizon, the ingoing coordinatesgive the regular induced metric tensorThis also fixes the future branch used in the causal argument below.

Relative to a chosen asymptotically flat end, define the black hole region bywhere ${\mathcal{I}}^{+}$ is that end's future null infinity in a conformal completion. It consists of events unable to send a future causal signal to that infinity; its boundary is the corresponding future event horizon.

For $0<\alpha <r_0/2$, $h>0$ and $f<0$ on $r_{-}<r<r_{+}$. The gradient of $r$ has norm $g^{ab}{\nabla }_{a}r{\nabla }_{b}r=f$, so it is timelike there. Use the future extension regular in the ingoing coordinates above, reached by an exterior future ingoing null geodesic with $\dot{r}=-E\sqrt{h}<0$. This determines $\nabla r$ to be future timelike in the inter-horizon block. Every nonzero future causal tangent $V$ therefore obeysAt the future outer horizon the gradient becomes future null and gives the corresponding one-way inequality $dr/d\lambda \le 0$. No future signal can cross that horizon outward to the selected exterior. The future inter-horizon block is consequently within $\mathcal{B}$; a future-directed path can leave it only through the inner horizon, not return through the outer horizon to the selected infinity.

This statement requires a branch and an end. The maximal extension also contains time-reversed inter-horizon blocks, where future-directed curves have increasing $r$ and emerge into an exterior: they are white holes, not the selected black hole region. Thus the unqualified claim for every copy of $r_{-}<r<r_{+}$ in an arbitrary extension is false. Analytic continuation past a Cauchy horizon can also introduce other asymptotic ends; the displayed definition explicitly refers to the chosen component of infinity.

To sketch the causal structure, suppress the spacelike $\psi$ circle by its orbit-space projection. The induced three-dimensional metric tensor isThe causal projection along a spacelike circular fiber is exact: any causal curve projects to a $q$-causal curve, and every base curve has a lift $d\psi =\Omega dt$ with exactly its base norm. The diagrams therefore represent this two-dimensional orbit space, not a constant-$\psi$ slice of the three-dimensional submanifold.

There are three nonextremal block types. For $r>r_{+}$, $r_{\ast }$ spans the full real line from the outer horizon to infinity, producing an exterior diamond. For $r_{-}<r<r_{+}$, $r_{\ast }$ spans the full line with the time and space roles reversed; these diamonds contain the future black-hole or past white-hole branches. For $0<r<r_{-}$, $r_{\ast }$ is finite at zero and diverges at the inner horizon, producing a static half-diamond with a timelike singular edge. The supplied Kretschmann scalar diverges as $384{\alpha }^4{r}_0^4/r^{12}$ at $r=0$, and $f>0$ on the inner side, so that edge is a genuine timelike curvature singularity. In contrast, both horizon radii are regular in horizon-adapted coordinates.

The finite strip shows two exterior levels, a future trapped block, an inner static level with timelike singularities, and the next emerging block. Continuing through the inner Cauchy horizons repeats this pattern upward and downward in the maximal analytic extension. The inner boundary is not a spacelike Schwarzschild-type singularity. Such an ideal extension need not describe a physical collapse past its unstable inner horizon.

At $\alpha =r_0/2$, put $R=r_0/\sqrt{2}$. Then $r_{+}=r_{-}=R$, $f=(r^2-R^2{)}^2/r^4$, and the interval $r_{-}<r<r_{+}$ is empty. The lapse is positive on both sides, so there is no inter-horizon trapped diamond. The horizon is a degenerate Killing horizon, with $\kappa =0$. In the exterior $r_{\ast }\to -\infty$ as $r\downarrow R$, while inside $r_{\ast }\to +\infty$ as $r\uparrow R$; more precisely its leading pole is $-\sqrt{2}{R}^2/[4(r-R)]$. Static proper distance to the horizon is infinite. Future ingoing rays still cross it at finite affine parameter, as the regular ingoing metric tensor shows.

The extremal sketch gives the exterior diamond and singular interior half-diamond, with the complete gluing prescription: the exterior future horizon attaches to the interior past horizon; the interior future horizon attaches to the past horizon of another exterior. Repeating these attachments gives the maximal extension. The marked throat endpoints are conformal ideal endpoints, not bifurcation points of the spacetime. This block representation avoids incorrectly treating the extremal geometry as two transverse horizons with a collapsed trapped region.

For $k=1$, put $r=\sin \chi$. The spatial metric tensor becomesThe global closed spatial slice is a 3-sphere with $0\le \chi \le \pi$; the original $r$ chart covers only a hemisphere, so two charts are needed. A circle at fixed $\chi$ in an equatorial angular plane has circumference $\boxed{2\pi a\sin \chi }$. The full spatial volume isIntegrating a single $r\in [0,1]$ chart without its second copy would give only half this volume.

At $p$, choose local normal coordinates with $g_{\alpha \beta }(p)=\mathrm{diag}(-1,1)$. Let $e_0,e_1$ be their coordinate basis. The vectors $V=e_0+e_1$ and $W=e_0-e_1$ have zero metric tensor square, and their component matrix has determinant $-2\ne 0$. Thus two independent null vectors form a basis of the tangent space. Normal coordinates are only needed at the chosen point, not on a whole neighbourhood with flat metric.

In Minkowski spacetime with metric $-d{t}^2+d{x}^2+d{y}^2+d{z}^2$, takeIts tangent vector is $(1,-\sin t,\cos t,0)$, with metric tensor square $-1+{\sin }^{2}t+{\cos }^{2}t=0$. Its acceleration is $(0,-\cos t,-\sin t,0)$, nonzero and not proportional to the tangent because its time component is zero. Thus it is a regular null curve which is not a geodesic under any reparametrization.

In conformal gauge, ${\gamma }_{\mu \nu }=e^{2\omega }\mathrm{diag}(-1,1)$. The Weyl invariance of the Polyakov action removes $\omega$, leavingIts complete variation isFix the string embedding map on the initial and final time slices. The interior Euler-Lagrange field equation is the wave equationAt each spatial endpoint the remaining variation vanishes under either Dirichlet boundary conditions, $\delta X^m=0$, or Neumann boundary conditions, $X^{m\prime }=0$, independently for each target-space component. Thus a fixed endpoint or a free endpoint makes the boundary contribution vanish. Fixing a spatial point throughout time imposes Dirichlet boundary conditions on its spatial coordinates, while the time coordinate may retain Neumann boundary conditions.

The independent metric tensor equation must also be retained after gauge fixing. Its Virasoro constraints areTogether with the wave equation, these ensure equivalence to the Nambu–Goto action; varying only the already fixed metric would lose these Virasoro constraints.

The canonical momentum equation is $\dot{X}=eP+u{X}^{\prime }$. Eliminating $P$ from the phase-space action yields the Lagrangian densityThis is the Polyakov action after identifying its inverse metric tensor density asThe determinant of this matrix is $-1$, as required in two dimensions. The Weyl transformation leaves the matrix unchanged, so $e,u$ encode the metric modulo its conformal factor.

Define the induced worldsheet metric by $g_{\mu \nu }={\eta }_{mn}{\partial }_{\mu }{X}^m{\partial }_{\nu }{X}^n$. Varying the independent inverse metric tensor in the Polyakov action givesConsequently $g_{\mu \nu }=f{\gamma }_{\mu \nu }$, where $f=\frac{1}{2}{\gamma }^{\rho \sigma }{g}_{\rho \sigma }$. For a nondegenerate Lorentzian string worldsheet with the usual compatible time orientation, take $f>0$, or ${\gamma }_{\mu \nu }=e^{2\omega }{g}_{\mu \nu }$. The undetermined function is precisely Weyl invariance. Since $\sqrt{-\gamma }{\gamma }^{\mu \nu }{g}_{\mu \nu }=2\sqrt{-g}$, elimination of the independent metric tensor givesThus the Nambu–Goto action measures minus string tension times Lorentzian worldsheet area. The metric elimination holds in the nondegenerate interior; null endpoints are understood as boundary limits.

A perfect fluid in general relativity has stress-energy tensorwhere $u^{\mu }{u}_{\mu }=-1$. Besides the metric tensor, one needs the energy density $\rho$, the pressure $P$, and three independent components of the normalized four-velocity $u^{\mu }$: five scalar functions of spacetime in total.

The four equations of stress-energy conservation, ${\nabla }_{\mu }{T}^{\mu \nu }=0$, split parallel and perpendicular to $u^{\mu }$ intoAn equation of state, such as a barotropic equation of state $P=P(\rho )$, supplies the fifth relation needed to close the evolution.

For any vector field $Y$, apply the Leibniz rule to the scalar $\omega (Y)$:Using ${\mathcal{L}}_{V}Y=[V,Y]$ and expanding the Lie bracket in a coordinate chart leavesThe connection terms cancel because the Levi-Civita connection is torsion-free. Applying this formula to each slot of the metric tensor and using metric compatibility givesEquivalently, one may prove both identities at a point in normal coordinates; since both sides are tensors, the result then holds in every coordinate system.

Matching spacetime symmetries to dynamical symmetries is a powerful demand for a coherent interpretation, but an unconditional equality theorem does not follow from logic or from general covariance. A spacetime symmetry preserves the structure declared physically significant: for Minkowski spacetime, the metric tensor and relevant orientations. A dynamical symmetry takes solutions to solutions with the laws' fixed backgrounds held fixed. The appropriate comparison concerns external transformations on the same physical quantities; internal gauge symmetries have no reason to be isometries of spacetime.

One direction becomes a theorem after important premises are supplied. Suppose all fixed background structure is geometrical and the matter equations are constructed covariantly from it. An automorphism of that full structure then takes solutions of those equations to solutions. For example, the Klein-Gordon equation $({\eta }^{ab}{\partial }_a{\partial }_b+m^2)\varphi =0$, in units $c=\hslash =1$, has the Poincaré symmetry of the fixed Minkowski metric. This is an explanation of symmetry inheritance conditional on a coupling prescription; it does not derive that prescription from the bare existence of a metric tensor.

A counterexample exposes the missing premise. On the same Minkowski spacetime, add a fixed timelike vector field $n^a$ and a matter termFor generic nonzero $\alpha$, a boost which preserves $\eta$ but changes $n$ is not a symmetry with that background held fixed. The full background has fewer symmetries than the metric tensor alone. Writing the equations in arbitrary coordinates and transforming $n$ as a vector field gives general covariance, not Lorentz invariance. Conversely, redundant unobservable geometrical structure could make the declared spacetime symmetry group artificially too small. Accidental symmetries of a restricted matter sector could make its dynamical symmetry group too large.

There are deeper motivations for avoiding such mismatches. If all measuring devices obey Lorentz-invariant dynamics, their behaviour supports a common Minkowski metric interpretation; adding an undetectable preferred rest structure sacrifices explanatory economy. If a preferred structure affects some device, then the original metric-only geometry was incomplete as a description of physical background structure. These observations support either revising the geometry or explaining the universal matter symmetry, but they are methodological and interpretive arguments, not an automatic metaphysical identity.^[ref]

Two qualifications matter. Boundary conditions and initial conditions can select a rest frame in an individual solution while the laws remain boost symmetric. Also, a symmetry of the complete theory need not be detectable by transforming only a subsystem relative to its environment. The best principle therefore matches the physically relevant geometry to the external symmetries of the complete laws, after all backgrounds and couplings are included. Calling it merely a bookkeeping preference understates its explanatory value; calling it an unavoidable consequence of covariance overstates its logical force.

The ideal slow clock transport synchronization procedure reproduces Einstein synchronisation in an inertial frame, as a limiting result about proper time. It does not determine distant simultaneity without assumptions about clocks and their dynamics. Begin with identical ideal clocks together at $x=0$, with matching rates and readings. Leave one there and transport the other to a laboratory position $L$ at small constant speed $u$, neglecting gravity. Use Einstein-synchronised laboratory coordinates to calculate the result, rather than assuming that the transported clock already displays coordinate time.

The journey takes laboratory time $T=L/u$. The transported clock's proper time is $\tau =T\sqrt{1-u^2/c^2}$, giving the lagConsequentlyThe small time dilation accumulates over a long journey, but its quadratic dependence on $u$ dominates the journey's inverse dependence on $u$. Saying that the journey is arbitrarily long is therefore not an objection to this limit. For a nonuniform journey with $|v|\le u$, $\delta =\int (1-\sqrt{1-v^2/c^2})dt$; a simple monotonic path of fixed length has the same vanishing bound. Acceleration stages can be arranged with vanishing displacement and proper time deficit, assuming the clock hypothesis.^[ref]

This procedure establishes synchronization operationally for stationary laboratory clocks, once ideal-clock behaviour has empirical support. It does not show that every actual device is insensitive to transport: thermal shifts, acceleration-dependent mechanisms or differing gravitational potentials can spoil the idealization. Nor does the same construction define global synchronization for rotating laboratories, where path dependence is significant.

The conventionality question is separate. Resynchronize the same laboratory with $t_{\kappa }=t+\kappa x/c$, where $|\kappa |<1$. At the arrival event the transported clock approaches the reading $t$, whereas a stationary clock displaying $t_{\kappa }$ reads $t+\kappa L/c$. Slow transport has not eliminated that difference. The transformed metric tensor has an off-diagonal term, and the proper time calculation still gives the same physical clock reading: changing labels cannot change its ticks. This is the invariance of reunion ages under clock resynchronization, applied to transport comparisons.

Slow clock transport synchronization is thus a useful physical realization of a preferred conventional coordinate choice given Lorentz-invariant dynamics and ideal clocks. It does not independently prove that the one-way speed of light must be isotropic in every admissible synchronization. It supplies a reproducible convention with a strong physical rationale, rather than a convention-free observation of an absolute present.

Spacetime geometry gives an acceptable structural explanation of length contraction, provided the relation between material rods, measurement and the metric tensor is included. It is not a force that squeezes a rod, and the mere existence of a quadratic form cannot guarantee how arbitrary matter behaves. The explanatory issue is whether a common geometrical constraint is sufficient for the phenomenon at issue, or whether a constructive theory of the rod is also required.

For an equilibrium rod with proper length $L_0$, use its rest frame $S^{\prime }$, where its endpoint histories are $x_A^{\prime }=0$ and $x_B^{\prime }=L_0$. In a laboratory frame useA laboratory measurement compares endpoints at the same laboratory time. Subtract the two endpoint equations at $\Delta t=0$ to obtainThe two frames measure different pairs of spacetime events. There is no contradiction in each assigning the other's moving rods smaller lengths; their simultaneity relations differ. This is neither an optical distortion nor an invariant compression diagnosed by an attached strain gauge. A photograph also involves unequal emission times and propagation of light, so it is not this simultaneous endpoint measurement.

The geometrical explanation has genuine unifying content. Once all suitable equilibrium rods measure the spatial distances induced by the same Minkowski metric, the result is independent of chemical composition. It relates length contraction to time dilation and relativity of simultaneity, rather than postulating separate mechanisms for every device. A structural explanation need not be a temporal causal story: the result compares descriptions of an existing world tube, not successive states of a rod being accelerated.

The main objection asks what warrants the italicized measurement premise. Matter on a mathematical Minkowski spacetime could obey equations involving an additional preferred vector field, as in 3. Such matter need not form devices with the usual transformation behaviour. Harvey R. Brown and Oliver Pooley argue that a constructive understanding must attend to the Lorentz-invariant dynamics of matter.^[ref] This objection has force against treating a bare metric tensor as an independently acting agent. It does not invalidate the endpoint derivation once the physical metric interpretation is established.

Acceleration introduces a different explanatory task. A rod pushed at one end can undergo stresses, heating and oscillations; its final proper length can change. A dynamical calculation is needed to determine whether it settles into an equivalent equilibrium state. The comparison $L=L_0/\gamma$ assumes a specified equilibrium proper length, and does not assert that every acceleration history preserves it. Geometry constrains the measurement of that state; material dynamics explain its formation and stability. Those explanatory roles can coexist without making either empty.

The phrase equivalence principle covers several related claims. The weak equivalence principle concerns the universality of free fall for test particles of negligible self-gravity, with the same initial conditions and no relevant nongravitational forces. In a Newtonian notation,Composition-independent motion requires the ratio of passive gravitational mass to inertial mass to be universal. Its common value can then be absorbed into the normalization of $\mathbf{g}$; the empirical assertion is universality, not the numerical convention used to express the ratio.

The Einstein equivalence principle includes that universality and extends the claim to local nongravitational experiments through local Lorentz invariance and local position invariance. A sufficiently small freely falling laboratory should recover the nongravitational laws of special relativity. The strong equivalence principle extends the scope to local gravitational experiments and bodies with appreciable gravitational self-energy. An experiment establishing one part of the weak statement does not automatically establish these stronger claims.

The motivating elevator comparison makes both the power and the limitation clear. Release a small object in a freely falling cabin. The cabin and object share the local free fall, so the object initially floats relative to the cabin. In a cabin supported against gravity, a released object instead accelerates relative to the floor, as it does in a suitably accelerated cabin in flat spacetime. This suggested replacing a universal gravitational force by geometric inertial motion. It does not identify a finite planetary gravitational field with an accelerated frame everywhere: separated falling objects can have a measurable relative acceleration.

In general relativity, the precise local geometric statement is that one can choose normal coordinates at an event $p$ withThis eliminates the connection contribution to the geodesic equation at that point. It does not generally set the Riemann curvature tensor to zero. In normal coordinates the first metric corrections away from $p$ are quadratic in the coordinate separation, and geodesic deviation measures the associated tidal forces. Equivalence is a local approximation for finite experiments whose extent and duration make these corrections negligible; it is not an assertion that curvature has been transformed away.^[ref]

There are direct experimental consequences beyond the elevator thought experiment. An Eötvös experiment compares bodies of different composition on a torsion balance, using a differential signal to test the universality of gravitational acceleration. MICROSCOPE performed an analogous test in orbit using electrostatic control of titanium-alloy and platinum-alloy test masses. Its first published result in 2017 constrained their differential free fall; this was already available before the examination. It tested a specific weak-equivalence prediction, not every possible gravitational law.^[ref] Comparisons of different atomic clocks can instead test local position invariance, including the expected gravitational redshift. These measurements constrain related but distinct parts of the framework.

The principle remains important as a restriction on how matter couples to gravity and as an organizing guide for experimental tests. It does not, on its own, derive the Einstein field equations. Universally coupling matter to a metric tensor still leaves choices for that metric's action; the equivalence principle does not uniquely fix field equations. For example, a metric gravitational action containing a scalar curvature squared term can retain the same minimal matter coupling while changing the gravitational dynamics. Selecting the Einstein-Hilbert action requires further dynamical assumptions and empirical success. The present role of the principle is consequently precise and limited: it supports local relativistic physics and universal matter coupling, while curvature, gravitational field equations and their quantitative predictions remain additional content.

There are two distinct explanatory questions. One asks why a proposed system of equations has a particular dynamical symmetry. The other asks why the common spacetime geometry is the right geometry for the behavior of physical rods, clocks and fields. A geometric account and a dynamical account give different directions of explanation, and their merits cannot be decided by a change of notation alone.

For the geometric account, Minkowski spacetime has a fixed Lorentzian metric; its spacetime isometries form the Poincaré group. If the matter equations use that metric tensor and no additional symmetry-breaking background structure, with couplings and boundary assumptions appropriate to the transformation, these spacetime isometries take solutions into solutions. For example, the Klein-Gordon equationin metric signature $(-,+,+,+)$ is invariant under the Poincaré group. This provides one common reason for the same transformation rules appearing in many laws. An individual solution, such as a particular plane wave, need not possess every symmetry of the equation.

Geometry alone is insufficient unless the assumed coupling to geometry is included. In the covariant equation with a preferred timelike vector,the fixed timelike $n$ selects a preferred rest frame. All the quantities can be transformed consistently under a coordinate transformation, but boosts that change $n$ are not symmetries with the physical background fixed. Thus general covariance, tensor notation and the mere presence of the Minkowski metric do not by themselves establish Einstein's principle of relativity.

A dynamical account instead starts with the common invariance of the actual interactions and asks why their composite rods and clocks measure a common interval. If their equations and suitable equilibrium solutions transform consistently under Lorentz transformations, their moving configurations can exhibit length contraction and their periodic processes time dilation. This is the explanatory strategy defended by Harvey R. Brown and Oliver Pooley, who deny that merely redescribing these effects geometrically supplies their microscopic explanation.^[ref] Such an account must still explain why independent interactions have the same relevant symmetry and why a particular clock is a good clock; it cannot simply infer those facts from the word “dynamics”.

The positions therefore need not disagree about experimental predictions. The geometric presentation makes a shared constraint explicit and permits economical deductions. The constructive presentation investigates why material systems realize that constraint. Defining geometry from universal dynamical behavior is a substantive proposal; deriving behavior from geometry also requires substantive matter-coupling assumptions. Neither direction is made compulsory by the formal equivalence of the descriptions.

The geodesics of the product-weighted diagonal plane metric follow from the geodesic Lagrangian $L=(y^2{\dot{x}}^2+x^2{\dot{y}}^2)/2$. With dots denoting an affine parameter derivative, its Euler-Lagrange equations giveSubstituting $y=mx$ with constant nonzero $m$ reduces both equations to $x\ddot{x}+{\dot{x}}^2=0$. Thus $x\dot{x}$ is constant, and a suitable affine parametrization exists; explicitly, $x^2=C_1\lambda +C_2$ on a segment of fixed sign.

These straight rays are geodesics only in the regular domain $xy\ne 0$. The metric tensor determinant is $x^2{y}^2$. On the axes, including the origin, this is a degenerate metric tensor: the calculation does not establish a regular geodesic continuing through the origin. The slope-zero ray is outside the regular domain. Geometrically, on each quadrant the change $\rho =|xy|/\sqrt{2}$, $\vartheta =\log |y/x|$ gives $d{s}^2=d{\rho }^2+{\rho }^{2}d{\vartheta }^2$. A fixed slope has constant $\vartheta$ and is therefore locally a radial geodesic of a flat metric tensor.