A coordinate chart, a frame, and an observer are different objects. We now define the geometric data that let the physics articles distinguish them.
5.1. Events and displacements
Definition 5.1. Affine space
An affine space modelled on a real vector space \(V\) is a nonempty set \(A\) with an operation \((p,v)\mapsto p+v\) satisfying \(p+0=p\), \((p+v)+w=p+(v+w)\), and: for every \(p,q\in A\), exactly one \(v\in V\) satisfies \(q=p+v\). Write this vector as \(q-p\). An affine coordinate chart chooses an origin \(o\in A\) and a basis \((e_1,\ldots,e_n)\) of \(V\), and assigns the unique coordinates \(x^i\) in \(p=o+\sum_i x^ie_i\).
Proposition 5.2. Affine smooth structure
Affine charts give a canonical smooth structure on a finite-dimensional affine space. Its tangent spaces identify canonically with \(V\).
Proof.
Two charts have transition \(y=Bx+b\), with \(B\) invertible, obtained by changing the displacement basis and the origin. Give \(A\) the topology transported from any one chart; the continuous affine transitions make it independent of the choice. A chart identifies it with \(\mathbb R^n\), so it is Hausdorff and second-countable. Affine transitions and their inverses are smooth and hence form an atlas. The map \(V\to T_pA\) sending \(v\) to the velocity of \(t\mapsto p+tv\) is linear and, in any affine chart, has the vector’s basis coordinates. It is an isomorphism by Proposition 3.3 and uses no chosen origin.
□Example. Points in a plane are affine points; the arrow from one point to another is a displacement. Picking a different origin translates all position coordinates but leaves every displacement unchanged. In spacetime, events play the role of points.
Definition 5.3. Newtonian affine time space
Take a four-dimensional \(V\), a surjective linear functional \(\tau:V\to\mathbb R\), a positive-definite symmetric bilinear form \(h\) on \(K=\ker\tau\), an orientation of \(K\), and an affine space \(A\) modelled on \(V\). Events \(p,q\) are simultaneous if \(\tau(q-p)=0\). An adapted frame chooses an origin \(o\), a vector \(u\) with \(\tau(u)=1\), and a positive \(h\)-orthonormal basis \(e_i\) of \(K\). Its chart is
\[p=o+tu+\sum_{i=1}^3x^ie_i.\]Proposition 5.4. Slices and adapted coordinates
Adapted coordinates exist uniquely for each frame. Simultaneity is an equivalence relation; its classes are affine three-spaces modelled on \(K\). Adapted chart transitions have the form
\[t'=t+b,\qquad x'=Rx+vt+a,\qquad R\in SO(3),\]where \(SO(3)=\{R:R^TR=I,\det R=1\}\). For a time-parametrized \(C^2\) worldline, velocity and acceleration transform as \(\dot x'=R\dot x+v\), \(\ddot x'=R\ddot x\).
Proof.
Surjectivity supplies \(u\); rank–nullity gives \(\dim K=3\). Gram–Schmidt in the positive-definite form supplies an orthonormal basis, with one sign corrected for orientation. Explicitly \(t=\tau(p-o)\), and \(p-o-tu\in K\) has unique coordinates in that basis. The relation is reflexive, symmetric, and transitive because \(\tau\) is linear; the slice through \(p\) is \(p+K\). Changing origins adds constants. Both time directions have \(\tau=1\), so their difference lies in \(K\); expressing it in the new spatial basis gives the term \(vt\). The change between two positive orthonormal spatial bases is an orthogonal matrix of positive determinant. Differentiating the affine transition, with \(dt'/dt=1\), gives both derivative laws.
□Example. For \(t'=t\), \(x'=x-v_0t\), a particle at rest in the original frame has velocity \(-v_0\) in the new one and zero acceleration in both. This arena defines spatial lengths only within a simultaneous slice: \(d(p,q)=\sqrt{h(q-p,q-p)}\). There is no spacetime inner product among the specified data.
5.2. Metrics and index raising
Definition 5.5. Metric tensor
A pseudo-Riemannian metric \(g\) on \(M\) is a smooth symmetric covariant 2-tensor that is nondegenerate at every point: \(g_p(v,w)=0\) for every \(w\) forces \(v=0\). It is Riemannian if each \(g_p\) is positive definite. It is Lorentzian with our signature convention if, at every point, some basis gives matrix \(\operatorname{diag}(-1,1,\ldots,1)\). In four dimensions write the signature as \((-+++ )\).
This “metric” is a bilinear form on tangent vectors. In Lorentzian geometry it is not a metric-space distance: a nonzero vector can have zero or negative squared norm. The signature convention fixes one negative direction, not a preferred time coordinate.
Proposition 5.6. Inverse metric and musical maps
The map \(\flat:T_pM\to T_p^*M\), \(v^\flat=g_p(v,\cdot)\), is an isomorphism. Write \(\sharp\) for its inverse. If \(G=(g_{ij})\) and \(G^{-1}=(g^{ij})\), then
\[(v^\flat)_i=\sum_jg_{ij}v^j,\qquad (\alpha^\sharp)^i=\sum_jg^{ij}\alpha_j.\]The inverse components are smooth and transform contravariantly. The metric trace of a covariant 2-tensor is \(\operatorname{tr}_g A=\sum_{i,j}g^{ij}A_{ij}\), independent of the chart.
Proof.
Nondegeneracy says \(\flat\) is injective; equal finite dimensions make it bijective. The coordinate equations follow by inserting basis vectors and solving the resulting linear system. Cofactors divided by the nonzero determinant prove smoothness of \(G^{-1}\). From \(G'=J^TGJ\), where \(J=\partial x/\partial y\), inversion gives \((G')^{-1}=J^{-1}G^{-1}(J^{-1})^T\). Contracting this with \(A'=J^TAJ\) cancels both inverse pairs. Equivalently the metric trace is the ordinary trace of the linear map defined by \(g(Lv,w)=A(v,w)\), which explains its basis independence.
□Example. For \(g=dx^2+dy^2\), with \(dx^2\) abbreviating \(dx\otimes dx\), use \(x=r\cos\theta,y=r\sin\theta\) on a chart with \(r>0\) and an angle interval of length less than \(2\pi\). Differentiating gives \(g=dr^2+r^2d\theta^2\). Thus \((d\theta)^\sharp=r^{-2}\partial_\theta\), not \(\partial_\theta\). The inverse coordinate chart is smooth locally by the inverse function theorem, since the Jacobian determinant is \(r\ne0\).
5.3. Lorentzian vectors and observers
Definition 5.7. Causal type and time orientation
A nonzero tangent vector is timelike if \(g(v,v)<0\), null if \(g(v,v)=0\), and spacelike if \(g(v,v)>0\). “Causal” means timelike or null. Zero is excluded from the causal vectors in this series.
A time orientation is a smooth everywhere timelike vector field \(N\). A causal \(v\) is future-directed when \(g(v,N)<0\), and past-directed when \(g(v,N)>0\). A spacetime here is a smooth boundaryless four-manifold with a Lorentzian metric and a time orientation.
Proposition 5.8. The two timelike cones
In a Lorentz-orthonormal basis write \(v=(v^0,\mathbf v)\). The timelike vectors have exactly two connected components, \(v^0>|\mathbf v|\) and \(v^0<-|\mathbf v|\); each is convex. The nonzero closure of either consists of its timelike vectors and the adjoining null vectors. Pairing with any timelike \(N\) selects the component containing \(N\) by \(g(v,N)<0\), and never vanishes on a causal vector.
Proof.
The inequality \(-(v^0)^2+|\mathbf v|^2<0\) is exactly \(|v^0|>|\mathbf v|\). The triangle inequality proves convexity of each signed cone. A segment is a continuous path, so each is connected; a connected subset cannot meet both because the continuous function \(v^0\) would cross zero. Their closures are the corresponding weak inequalities: boundary points are approached by increasing \(|v^0|\). If \(N^0>|\mathbf N|\) and \(v^0\ge|\mathbf v|\), \(v\ne0\), then \(v^0>0\) and
\[g(v,N)=-v^0N^0+\mathbf v\cdot\mathbf N\le-v^0(N^0-|\mathbf N|)<0.\]For the opposite causal cone the sign reverses. Replacing \(N\) by \(-N\) treats the other case. Thus the sign criterion is intrinsic and selects precisely the claimed cone.
□Example. In \(\mathbb R^4\) with \(\eta=-(dx^0)^2+(dx^1)^2+(dx^2)^2+(dx^3)^2\) and \(N=\partial_0\), the vectors \((1,0,0,0)\), \((1,1,0,0)\), and \((0,1,0,0)\) are respectively future timelike, future null, and spacelike. These are tangent vectors at an event, not other events.
Proposition 5.9. Comparison of time orientations
Two time orientations choose the same future cone or opposite future cones on each connected subset of spacetime.
Proof.
For timelike fields \(N,\widetilde N\), the function \(g(N,\widetilde N)\) is smooth and never zero by Proposition 5.8. Its sign cannot change on a connected subset: positive and negative inverse images would separate that subset. The sign criterion then proves the assertion.
□Definition 5.10. Point observer and rest projection
A unit point observer is a future vector \(n\) with \(g(n,n)=-1\). Its rest space is \(n^\perp=\{w:g(w,n)=0\}\). Define its rest projection by \(h_n(v)=v+g(v,n)n\). A smooth field of unit future vectors is an observer congruence in the geometric sense used here. A tetrad is a basis of four tangent vectors; an orthonormal tetrad has metric matrix \(\operatorname{diag}(-1,1,1,1)\). A tetrad field is a smooth such choice at every point of its domain.
Worked example for Definition 5.10
Counterexample: The rest-projection formula needs a unit observer
Proposition 5.11. The observer’s rest geometry is Euclidean
\(h_n(v)\in n^\perp\), \(h_n^2=h_n\), and \(v=-g(v,n)n+h_n(v)\). The metric on \(n^\perp\) is positive definite. In particular \(n\) can be completed to a Lorentz-orthonormal tetrad.
Worked example for Proposition 5.11
Counterexample: The rest-projection formula needs a unit observer
Proof.
The projection equations follow by inserting \(g(n,n)=-1\). In any Lorentz-orthonormal basis, orthogonality gives \(w^0=\mathbf n\cdot\mathbf w/n^0\). Since \((n^0)^2=1+|\mathbf n|^2\),
\[g(w,w)=|\mathbf w|^2-\frac{(\mathbf n\cdot\mathbf w)^2}{(n^0)^2}\ge\frac{|\mathbf w|^2}{(n^0)^2}.\]This is positive for nonzero \(w\): if \(\mathbf w=0\), orthogonality also gives \(w^0=0\). The kernel of the nonzero functional \(g(\cdot,n)\) is three-dimensional. Gram–Schmidt there supplies three unit spatial basis vectors, completing \(n\) to the required tetrad.
□Example. For \(n=(1,0,0,0)\), \(h_n(v^0,\mathbf v)=(0,\mathbf v)\). A frame in affine Minkowski space chooses a fixed tetrad and an origin, giving a chart; a point observer chooses only \(n\) at one event. A general tetrad field need not be a coordinate basis.
5.4. Worldlines and clocks
Definition 5.12. Proper time and four-velocity
A future timelike worldline is a \(C^1\) curve \(\gamma\) with future timelike velocity. For a positive model constant \(c\), its elapsed proper time is
\[\tau(s)-\tau(s_0)=\frac1c\int_{s_0}^s\sqrt{-g(\dot\gamma(u),\dot\gamma(u))}\,du.\]Its four-velocity is \(U=d\gamma/d\tau\). Interpreting this parameter as an ideal clock reading is a physical postulate, separate from its mathematical definition.
Proposition 5.13. Parameter independence and normalization
Elapsed proper time is unchanged by an increasing \(C^1\) reparametrization with positive derivative. It gives a local \(C^1\) parameter, and \(g(U,U)=-c^2\).
Proof.
For \(s=\phi(r)\), the chain rule multiplies the velocity by \(\phi'(r)>0\); the square root in the integral is multiplied by the same factor. Substitution, proved in chapter 1, gives invariance. The fundamental theorem gives \(d\tau/ds=\sqrt{-g(\dot\gamma,\dot\gamma)}/c>0\). The one-dimensional inverse theorem supplies a local inverse, with derivative \(ds/d\tau=c/\sqrt{-g(\dot\gamma,\dot\gamma)}\). Multiplying \(\dot\gamma\) by this factor proves the norm identity.
□Example. In Minkowski coordinates with \(x^0=ct\), a worldline \(\gamma(t)=(ct,\mathbf vt)\), \(|\mathbf v|<c\), has \(d\tau/dt=\sqrt{1-|\mathbf v|^2/c^2}\). A null curve has zero integrand, so proper time cannot be used as its parameter. This is why the timelike hypothesis matters.
Worked examples
Each example checks the hypotheses and the calculation for the linked statements. The links in the statements return here.
Worked example 1. Events without a preferred origin
Applies to: Definition 5.1, Proposition 5.2.
Take events \(A=\mathbb R^4\) with the translation action of \(V=\mathbb R^4\). For \(p=(1,2,0,0)\), \(q=(3,5,0,0)\), the unique displacement is \(q-p=(2,3,0,0)\); adding it to \(p\) gives \(q\). Changing the chosen origin to \(o=(1,1,0,0)\) changes both event coordinates but not their difference. Two affine charts differ by \(x'=Lx+a\), with invertible \(L\); the derivative is the constant \(L\) and the inverse is affine and smooth. A displacement \(v\) is canonically the velocity of \(s\mapsto p+sv\), independent of the chosen origin. This checks the affine axioms, smooth structure, and tangent identification in this concrete arena.
Worked example 2. A Galilean change of frame
Applies to: Definition 5.3, Proposition 5.4.
On this affine space set \(\tau(t,x)=t\) and give \(K=\{0\}\times\mathbb R^3\) its ordinary dot product and orientation. The standard time vector has \(\tau(u)=1\), so the standard frame is adapted. The change \(t'=t+1,\ x'=x+(2,0,0)t+(3,0,0)\) has \(R=I\in SO(3)\) and is an adapted change; its inverse is \(t=t'-1,\ x=x'-(2,0,0)t'-(1,0,0)\). Thus both maps are bijections of the required form. Events with equal \(t\) still have equal \(t'\), and each simultaneity class is a translated spatial three-space. For \(x(t)=(t^2,0,0)\), differentiation with respect to \(t'=t+1\) gives \(\dot x'=(2t+2,0,0)\), \(\ddot x'=(2,0,0)\). These are exactly \(\dot x+v\) and \(\ddot x\). Composing two pure boosts adds their velocities, and reversing a boost gives its inverse, illustrating the group calculation used in chapter 7.
Worked example 3. Raising and lowering with a Lorentz metric
Applies to: Definition 5.5, Proposition 5.6.
On \(\mathbb R^4\), let \(g=\operatorname{diag}(-1,1,1,1)\). It is constant, smooth, symmetric, and invertible, hence Lorentzian. For \(v=(2,1,0,0)\), \(v^\flat=(-2,1,0,0)\). Raising this covector with the inverse matrix \(g^{-1}=g\) recovers \(v\). A covector \((a,b,c,d)\) raises to \((-a,b,c,d)\), showing bijectivity directly. For \(A=\operatorname{diag}(3,2,0,0)\), \(\operatorname{tr}_gA=-3+2=-1\). By contrast the identity matrix defines a Riemannian metric: its value on a nonzero vector is a positive sum of squares. Thus nondegeneracy, signature, and the musical maps are distinct, explicitly checked requirements.
Worked example 4. Selecting a future cone
Applies to: Definition 5.7, Proposition 5.8, Proposition 5.9.
For the same \(g\), choose \(N=(1,0,0,0)\). The vectors \((2,1,0,0)\), \((1,1,0,0)\), and \((0,1,0,0)\) have squared lengths \(-3,0,1\): timelike, null, and spacelike. The first two are future since their pairing with \(N\) is negative. If \(v^0>|\mathbf v|\), \(w^0>|\mathbf w|\), then for \(0\le s\le1\), \(sv^0+(1-s)w^0>|s\mathbf v+(1-s)\mathbf w|\), using the triangle inequality. This verifies convexity of the future timelike cone; negation gives the past cone. Null \((1,1,0,0)\) is approached by \((1+1/n,1,0,0)\).
The smooth field \(N'=(\cosh x^1,\sinh x^1,0,0)\) has length \(-1\). For future causal \(v\), \(g(v,N')\le-v^0(\cosh x^1-|\sinh x^1|)<0\), so it selects the same future cone everywhere. The field \(-N'\) selects the opposite cone. This checks the comparison of time orientations on a connected spacetime.
Worked example 5. The rest projection in an observer tetrad
Applies to: Definition 5.10, Proposition 5.11.
With \(n=(1,0,0,0)\), a future unit observer, \(n^\perp=\{(0,a,b,c)\}\). For \(v=(t,x,y,z)\), \(h_n(v)=v-tn=(0,x,y,z)\), because \(g(v,n)=-t\). Thus \(h_n^2=h_n\) and \(v=tn+h_n(v)\). The restricted metric is \(x^2+y^2+z^2\), positive on every nonzero rest vector. Completing \(n\) by the three coordinate spatial vectors gives a Lorentz-orthonormal tetrad. These constant vectors also define a smooth tetrad field and observer congruence. This verifies the projection and the Euclidean rest geometry in coordinates.
Worked example 6. Proper time for constant velocity
Applies to: Definition 5.12, Proposition 5.13.
In coordinates \(x^0=ct\), let \(\gamma(t)=(ct,vt,0,0)\) with constants \(c>0\), \(|v|<c\). Its tangent has length \(-c^2+v^2<0\) and positive time component. Thus \(\tau(t)-\tau(0)=t\sqrt{1-v^2/c^2}\) for \(t\ge0\). The four-velocity is \(U=(c,v,0,0)/\sqrt{1-v^2/c^2}\), whose squared length is \(-c^2\). Under \(t=2s\), the square root in the integrand gains a factor 2 while \(ds=dt/2\), leaving the elapsed proper time unchanged. This checks an increasing reparametrization with positive derivative and the exact normalization, separately from its physical interpretation as a clock reading.
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 rest-projection formula needs a unit observer
Related statements: Definition 5.10, Proposition 5.11.
Claim refuted. The formula h(v)=v+g(v,N)N is a rest projection for any timelike N without normalization.
Proof.
In Minkowski space with metric \(\operatorname{diag}(-1,1,1,1)\), take the smooth future timelike field \(N=2e_0\), of squared length \(-4\). The proposed map sends \(e_0\) to \(e_0+(-2)2e_0=-3e_0\), whose pairing with \(N\) is 6, not zero. Applying the map again gives \(9e_0\), not \(-3e_0\), so it is not idempotent. The correct projection for a nonunit timelike \(N\) is \(v-g(v,N)N/g(N,N)\): pairing with \(N\) cancels exactly, it fixes vectors in \(N^\perp\), and it kills \(N\), proving idempotence by the resulting decomposition. Only when \(g(N,N)=-1\) does this reduce to the chapter’s formula.
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