We can now read the geometric parts of the opening physics articles. This chapter gives the remaining short translations and records where a different subject, rather than more differential geometry, is needed.
7.1. Non-relativistic mechanics
Read Framework, quantities and geometry. Chapters 1–3 and §5.1 supply its regularity, adapted coordinates, and regular constraints. For \(N\) labelled particles a configuration is a tuple \(q=(q_1,\ldots,q_N)\in(\mathbb R^3)^N\). A configuration space is the specified set of admissible tuples, with its manifold structure. Removing coincidences \(q_i=q_j\) leaves an open set: each difference map is continuous, and \(\mathbb R^3\setminus\{0\}\) is open. Imposing \(r\) regular scalar constraints gives dimension \(3N-r\), by Proposition 2.7.
Definition 7.1. Mass metric
For positive masses \(m_1,\ldots,m_N\), the mass metric on a regular configuration submanifold \(Q\subset(\mathbb R^3)^N\) is the restriction to its included tangent vectors of
\[g_m(v,w)=\sum_{a=1}^Nm_a\,v_a\cdot w_a.\]The associated mass operator on the ambient displacement space is \(Mv=(m_1v_1,\ldots,m_Nv_N)\).
Proposition 7.2. The mass metric is Riemannian
Proof.
The ambient components are constant. Pulling back by the inclusion gives smooth components. Symmetry follows from the Euclidean dot product. For a nonzero included vector, at least one \(v_a\ne0\), so \(g_m(v,v)=\sum_am_a|v_a|^2>0\). The inclusion differential is injective by Proposition 3.10, so this tests every nonzero vector of \(T_qQ\).
□Example. For one unit-length pendulum of mass \(m\), \(q(\theta)=(\cos\theta,\sin\theta)\). The tangent derivative is \((-\sin\theta,\cos\theta)\), whose squared Euclidean length is one. Therefore the mass metric is \(m\,d\theta^2\). “One degree of freedom” means one local configuration coordinate; velocities then need one additional coordinate.
The Galilean maps form a group of bijections. Indeed if \(g_i(t,x)=(t+b_i,R_ix+v_it+a_i)\), direct substitution gives parameters
\[(b_1+b_2,\ R_2a_1+v_2b_1+a_2,\ R_2v_1+v_2,\ R_2R_1)\]for \(g_2\circ g_1\). The identity parameters are \((0,0,0,I)\); the inverse parameters are \((-b,-R^T(a-vb),-R^Tv,R^T)\). Orthogonal matrices satisfy \(R^{-1}=R^T\); transpose and determinant identities show that products and inverses stay in \(SO(3)\). This proves closure and inverses; associativity is associativity of function composition. For a pure boost, the inverse simply reverses the boost velocity. See the worldline and frame-change theorem.
7.2. Special relativity
Read Spacetime Primitives. Chapters 2–5 supply its smooth atlas, tangent identification, orientation, volume form, and causal vectors. Its affine Minkowski space is an affine four-space with a fixed Lorentz form \(\eta\) on its displacement space, a spatial orientation convention, and a chosen future cone. The corresponding metric is constant in affine coordinates, by the canonical identification \(T_pA\cong V\).
A future inertial frame chooses an origin and a positive Lorentz-orthonormal tetrad with future timelike first vector. Such frames exist: choose a future timelike vector, divide by \(\sqrt{-\eta(v,v)}\), and complete it with Proposition 5.11; change one spatial sign if needed to match the chosen orientation. Its affine chart uses \(x^0=ct\). Transitions preserve \(\eta\), time orientation, and the chosen orientation. For example the one-direction boost
\[x'^0=\gamma(x^0-\beta x^1),\qquad x'^1=\gamma(x^1-\beta x^0),\qquad \gamma=(1-\beta^2)^{-1/2},\quad |\beta|<1\]with the other two coordinates unchanged preserves \(-(x^0)^2+(x^1)^2\): expansion gives \(\gamma^2(1-\beta^2)(-(x^0)^2+(x^1)^2)\). Its determinant is \(\gamma^2(1-\beta^2)=1\), and its image of the future time unit vector has positive time component, so the cone criterion proves it preserves the future component.
7.3. General relativity
Read Framework And Geometry after chapters 1–6. Its geometric data are a smooth time-oriented Lorentzian four-manifold. Unlike affine Minkowski space, it need not have global affine coordinates or constant metric components. The inverse metric, Levi-Civita connection, curvature, Ricci contraction, and proper time are exactly the constructions proved above. A nonempty example is Minkowski space with its constant metric and \(N=\partial_0\).
Definition 7.3. Stress-energy tensor and pointwise energy conditions
A model’s stress-energy tensor \(T\) is a specified smooth symmetric covariant 2-tensor. Its physical construction is part of the matter model. In this paragraph all observers \(n\) have \(g(n,n)=-1\). The following are optional pointwise inequalities:
- Weak energy condition (WEC): \(T(v,v)\ge0\) for every timelike \(v\).
- Null energy condition (NEC): \(T(k,k)\ge0\) for every null \(k\).
- Dominant energy condition (DEC): for every future timelike \(v\), the vector \(J_v=-(T(v,\cdot))^\sharp\) is future causal or zero.
- Strong energy condition (SEC), in four dimensions: \(\big(T-\tfrac12(\operatorname{tr}_gT)g\big)(v,v)\ge0\) for every timelike \(v\).
The observer’s energy-density reading is \(T(n,n)\), in the unit conventions of the model. The inequalities are definitions of classes of tensors, not consequences of Lorentzian geometry.
Worked example for Definition 7.3
Counterexample: The energy conditions are different requirements
Example. For \(T=\rho\,n^\flat\otimes n^\flat\), \(\rho\ge0\), the WEC and NEC follow from \(T(v,v)=\rho g(n,v)^2\ge0\). Also \(J_v=-\rho g(n,v)n\) is future timelike or zero for future timelike \(v\), proving DEC. Finally \(\operatorname{tr}_gT=-\rho\), so the SEC expression is \(\rho(g(n,v)^2+\tfrac12g(v,v))\). Write \(v=an+w\), \(w\perp n\), using Proposition 5.11. It becomes \(\tfrac\rho2(a^2+g(w,w))\ge0\), proving SEC. In contrast \(T=g\) is a smooth symmetric tensor with \(T(n,n)=-1\), so it violates WEC. A tensor’s type alone imposes no positive-energy inequality.
7.4. Classical electromagnetism
Read Primitives and postulates. Chapters 1, 3–5, and the flat-space case of chapter 6 supply the geometry. A restricted inertial chart identifies the laboratory region with \(I\times\Omega\), where \(I\subset\mathbb R\) is an open time interval and \(\Omega\subset\mathbb R^3\) is open. A fixed-time spatial field is a map \(\Omega\to\mathbb R^m\); a time-dependent field is a map \(I\times\Omega\to\mathbb R^m\). Joint \(C^1\) regularity means all first time and spatial partials are continuous together.
Definition 7.4. Gradient, divergence, curl, and cross product
In an oriented Cartesian spatial chart, for a scalar \(f\) and vector fields \(E,B\), define
\[(\operatorname{grad}f)_i=\partial_i f,\qquad \operatorname{div}E=\sum_i\partial_i E_i,\qquad (\operatorname{curl}E)_i=\sum_{j,k}\epsilon_{ijk}\partial_jE_k,\qquad (E\times B)_i=\sum_{j,k}\epsilon_{ijk}E_jB_k.\]The symbol \(\epsilon_{ijk}\) was defined in chapter 4. These formulas use the Euclidean spatial metric and its chosen orientation; in general coordinates the Cartesian formulas cannot simply be copied unchanged. Intrinsically the gradient is \((df)^\sharp\).
Proposition 7.5. The elementary differential identities
For \(C^2\) fields, \(\operatorname{curl}(\operatorname{grad}f)=0\) and \(\operatorname{div}(\operatorname{curl}E)=0\). Divergence and curl are linear.
Proof.
For the first identity, \(\sum_{j,k}\epsilon_{ijk}\partial_j\partial_k f=0\): the coefficient changes sign under \(j\leftrightarrow k\), while the mixed partial does not. For the second, \(\sum_{i,j,k}\epsilon_{ijk}\partial_i\partial_jE_k=0\) by the same pairing of \(i,j\). Partial derivatives are linear, so their displayed finite sums are linear.
□Example. For \(E=(-y,x,0)\), \(\operatorname{div}E=0\) and \(\operatorname{curl}E=(0,0,2)\), so the divergence of that curl is zero. For \(f=x^2+y^2+z^2\), \(\operatorname{grad}f=(2x,2y,2z)\) has zero curl. Finite linear combinations of smooth Maxwell solutions with their sources are solutions because every term in the adopted equations is a linear derivative or a fixed scalar multiple. Distributing a finite sum through each term proves this superposition theorem; it does not prove Maxwell’s equations themselves.
An oriented regular surface patch in Euclidean three-space is a smooth parametrization \(\sigma:O\subset\mathbb R^2\to\mathbb R^3\) with linearly independent derivatives \(\sigma_u,\sigma_v\), using the order \((u,v)\) for its orientation. Its unit normal and area density are
\[n=\frac{\sigma_u\times\sigma_v}{|\sigma_u\times\sigma_v|},\qquad dS=|\sigma_u\times\sigma_v|\,du\,dv.\]The denominator is nonzero because expansion of the cross-product formula gives \(|a\times b|^2=|a|^2|b|^2-(a\cdot b)^2\), positive for independent vectors by the equality case of Cauchy–Schwarz. The same expansion gives \((a\times b)\cdot a=(a\times b)\cdot b=0\), proving that \(n\) is a unit normal. Under a parameter change, the chain rule and alternating bilinearity multiply \(\sigma_u\times\sigma_v\) by the parameter Jacobian determinant. Thus orientation-preserving changes leave \(n\) unchanged and scale the area density by the absolute Jacobian, as required for integration. The flux density of a vector field \(J\) is \(J\cdot n\,dS\). For \(\sigma(u,v)=(u,v,0)\), the normal is \((0,0,1)\), \(dS=du\,dv\), and the flux density is \(J_3\,du\,dv\). See oriented normals and flux. This supplies the geometry; evaluating general surface integrals needs further integration theory.
The two-form \(F\) in chapter 4 defines an endomorphism \(\Phi\) by \(g(v,\Phi(w))=F(v,w)\), namely \(\Phi^a{}_b=\sum_cg^{ac}F_{cb}\). Nondegeneracy proves uniqueness, and the inverse-metric formula proves existence. Antisymmetry gives \(g(w,\Phi(w))=F(w,w)=0\), explaining why the covariant Lorentz-force term is orthogonal to the four-velocity.
There is also a simple observer split of any 2-form. For a unit observer \(n\), let \(e=F(n,\cdot)\). Then \(e(n)=0\), and
\[B=F+n^\flat\wedge e,\qquad B(n,\cdot)=0,\qquad F=-n^\flat\wedge e+B.\]Indeed \((n^\flat\wedge e)(n,v)=-e(v)\). Thus \(e\) and \(B\) restrict to the observer’s rest space and determine \(F\). For the laboratory observer \(n=\partial_0\), \(e_i=F_{0i}=-E_i/c\); the spatial part has \(B_{ij}=\epsilon_{ijk}B_k\). This proves the geometric split; signs and factors of \(c\) follow the declared convention.
7.5. Non-relativistic quantum mechanics
Read Framework And Quantities. Its classical event arena uses the Newtonian affine structure of §5.1. Its apparatus trajectories are curves with that smooth structure. A pure quantum state is not a tangent vector to spacetime, and the framework does not assign it a classical worldline between detections.
The Hilbert spaces, unbounded operators, spectral measures, and probability postulates in that page require functional analysis and measure theory. This series supplies its spacetime geometry, not those subjects. Likewise electromagnetism’s weak distributional branch and volume or surface source integrals need integration and distribution theory beyond the one-dimensional integral introduced here. Those are additional analysis prerequisites, not hidden differential-geometric assumptions.
7.6. Where to stop
| Opening physics page | Geometry to read here |
|---|---|
| Non-relativistic classical mechanics | 1–3, §5.1, §7.1 |
| Special relativity | 1–5, §7.2 |
| General relativity | 1–6, §7.3 |
| Classical electromagnetism | 1, 3–5 (using the chart definitions in 2), flat connection in 6, §7.4 |
| Non-relativistic quantum mechanics | 1–2, §5.1, §7.5 |
The physics library’s separate mathematics companion categories provide further prerequisites, not additional physical models. Its categories marked “in preparation” had no opening articles to target on the date of this audit. Later topics such as geodesic existence, integration on manifolds, global causal theory, and field-equation existence are outside this series.
Worked examples
Each example checks the hypotheses and the calculation for the linked statements. The links in the statements return here.
Worked example 1. A constrained two-particle configuration
Applies to: Definition 7.1, Proposition 7.2.
For two particles of positive masses \(m_1,m_2\) joined by a rod of length \(\ell>0\), let \(Q=\{(q_1,q_2):|q_2-q_1|^2=\ell^2\}\subset\mathbb R^6\). The constraint derivative is \(2(q_2-q_1)\cdot(v_2-v_1)\), a nonzero linear functional on \(Q\), so \(Q\) is a smooth five-dimensional configuration space. Tangent vectors obey \((q_2-q_1)\cdot(v_2-v_1)=0\). The mass metric gives \(g_m(v,v)=m_1|v_1|^2+m_2|v_2|^2>0\) for nonzero included tangent vectors; restriction of a constant bilinear form is smooth and symmetric. A rigid translation \(v_1=v_2=w\) has squared length \((m_1+m_2)|w|^2\), and the mass operator sends it to \((m_1w,m_2w)\). These checks show why both positivity of masses and regularity of the constraint are used.
Further proof details
The configuration’s local coordinates can be exhibited explicitly. Put \(d=q_2-q_1\). At a rod configuration some component \(d_j\ne0\). Keep \(q_1\in\mathbb R^3\) and the other two components of \(d\) as five coordinates; recover the remaining one by \(d_j=\epsilon\sqrt{\ell^2-\sum_{i\ne j}d_i^2}\), with the fixed local sign \(\epsilon=\operatorname{sgn}d_j\). The radicand is positive on a neighborhood of the chosen point. The resulting smooth map and its coordinate projection are inverse, so these are genuine five-dimensional local charts. In any such chart \(u\), the mass metric entries are \(\sum_a m_a\sum_i(\partial q_a^i/\partial u^j)(\partial q_a^i/\partial u^k)\), smooth products of chart derivatives. The differential of the parametrization is injective because its five-coordinate projection is the identity. Hence every nonzero coordinate tangent gives a nonzero included velocity, and positive masses make its squared length strictly positive. This proves smoothness and positive definiteness on the actual constrained manifold.
Worked example 2. A stress tensor with all four energy conditions
Applies to: Definition 7.3.
In Minkowski space take the constant unit future observer \(n=e_0\) and \(T=\rho\,n^\flat\otimes n^\flat\) with a constant \(\rho>0\). This is smooth and symmetric. For any timelike or null \(v\), \(T(v,v)=\rho g(n,v)^2\ge0\), proving WEC and NEC. For future timelike \(v\), \(J_v=-\rho g(n,v)n\) is a positive multiple of \(n\), so DEC holds. Write \(v=an+w\), \(w\perp n\). Since \(\operatorname{tr}_gT=-\rho\), the SEC expression is \(\rho(a^2+g(w,w))/2\ge0\). The observer reading is \(T(n,n)=\rho\). In contrast \(T=g\) is smooth and symmetric but gives \(T(n,n)=-1\); symmetry alone therefore does not imply the inequalities.
Worked example 3. Cartesian differential operators and an oriented flux
Applies to: Definition 7.4, Proposition 7.5.
In the usual oriented Euclidean frame let \(E=(-y,x,0)\), \(f=x^2+y^2+z^2\). Direct derivatives give \(\operatorname{div}E=0\), \(\operatorname{curl}E=(0,0,2)\), \(\operatorname{grad}f=(2x,2y,2z)\), \(\operatorname{curl}\operatorname{grad}f=0\), and \(\operatorname{div}\operatorname{curl}E=0\). For \(B=e_z\), \(E\times B=(x,y,0)\); scaling and adding these fields commutes with each derivative, checking linearity.
The regular patch \(\sigma(u,v)=(u,v,u)\), \(0\le u,v\le1\), has \(\sigma_u\times\sigma_v=(-1,0,1)\), unit normal \((-1,0,1)/\sqrt2\), and area density \(\sqrt2\). A constant current \(J=e_z\) thus has flux density 1 and total flux 1. Reversing the parameter order reverses the normal and the signed flux. As a separate Maxwell example, constant fields \((E,B)=(e_x,0)\) and \((0,e_z)\) with zero sources solve the smooth vacuum equations because every derivative vanishes. Their real linear combinations do too, verifying the prose superposition result without imposing a wave relation on constant fields.
Worked example 4. An electromagnetic two-form and a Lorentz boost
Use \(c=1\), \(g=\operatorname{diag}(-1,1,1,1)\), and a constant electric field \(E=e_x,B=0\). The declared convention gives \(F=-dx^0\wedge dx^1\), so \(F_{01}=-1,F_{10}=1\). Raising the first index gives \(\Phi^0{}_1=1,\Phi^1{}_0=1\), all other components zero. Thus \(\Phi(w)=(w^1,w^0,0,0)\) and \(g(w,\Phi(w))=-w^0w^1+w^1w^0=0\). With \(n=e_0\), \(e=F(n,\cdot)=-dx^1\), \(n^\flat=-dx^0\); hence \(F+n^\flat\wedge e=0\), exactly the observer’s zero magnetic part.
For the boost parameter \(\beta=3/5\), \(\gamma=5/4\), the time-space block is \(\begin{pmatrix}5/4&-3/4\\-3/4&5/4\end{pmatrix}\). Its determinant is 1, it sends \(e_0\) to the future unit vector \((5/4,-3/4,0,0)\), and direct expansion preserves \(-(x^0)^2+(x^1)^2\). This verifies the unboxed frame-change example and supplies a nontrivial orthonormal observer frame.
Counterexamples and limits of the hypotheses
Each construction below refutes the explicitly stated stronger claim. It does not refute the related theorem with its original hypotheses.
Counterexample 1. The energy conditions are different requirements
Related statements: Definition 7.3.
Claim refuted. The weak energy condition forces the dominant or strong energy condition.
Proof.
In a Minkowski orthonormal frame let \(T=\operatorname{diag}(1,2,2,2)\). It is a constant smooth symmetric covariant tensor, and \(T(v,v)=(v^0)^2+2|\mathbf v|^2\ge0\) for every vector, proving WEC and NEC. For the future unit timelike vector \(v=(\sqrt2,1,0,0)\), raising the covector \(T(v,\cdot)=(\sqrt2,2,0,0)\) and negating gives \(J_v=(\sqrt2,-2,0,0)\). Its squared Lorentz length is \(-2+4=2>0\), so it is spacelike and DEC fails.
For a separate failure, choose a constant \(\rho>0\) and \(\widetilde T=-\rho g\). Its value on timelike \(v\) is positive, on null \(k\) is zero, and \(J_v=\rho v\) is future timelike for every future timelike \(v\); thus WEC, NEC, and DEC hold. But \(\operatorname{tr}_g\widetilde T=-4\rho\), so \(\widetilde T-\tfrac12(\operatorname{tr}_g\widetilde T)g=\rho g\), negative on every timelike vector. SEC fails. These are algebraic examples of the definitions; assigning either tensor to physical matter is a separate modeling step.
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