Curves, regular causal curves and future timelike worldlines in affine Minkowski space

Let $(A,V,\eta ,C^{+})$ be an oriented time-oriented affine Minkowski space (Oriented time-oriented affine Minkowski space (the spacetime primitives)) and let $I$ be an order-convex subset of $\mathbb{R}$ with at least two elements (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length); a curve of the model is a map $x:I\to A$.

Derivative and regularity of the parametrization

For $s\in I$ and $h\mathrm{\ne }0$ with $s+h\in I$ the difference quotient $(x(s+h)-x(s))\mathrm{/}h$ is an element of $V$, the displacement from $x(s)$ to $x(s+h)$ divided by the real number $h$ (Oriented time-oriented affine Minkowski space (the spacetime primitives)). Convergence in $V$ is understood through an inertial frame: for the frame isomorphism $E:{\mathbb{R}}^4\to V$ of an inertial frame (Inertial frames and inertial charts of the affine Minkowski model) put $\mathrm{\mid }v{\mathrm{\mid }}_E:=\mathrm{\mid }{E}^{-1}v{\mathrm{\mid }}_2$, the Euclidean norm transported by $E$. If $E^{\mathrm{\prime }}$ is the isomorphism of another inertial frame, then $E^{\mathrm{\prime }-1}E$ and its inverse are linear maps ${\mathbb{R}}^4\to {\mathbb{R}}^4$, so they are bounded with constants $K,K^{\mathrm{\prime }}\ge 0$ (Every Euclidean linear map has a unique matrix and satisfies $\mathrm{\parallel }Lh{\mathrm{\parallel }}_2\le K\mathrm{\parallel }h{\mathrm{\parallel }}_2$ for some $K\ge 0$): $\mathrm{\mid }v{\mathrm{\mid }}_{E^{\mathrm{\prime }}}\le K^{\mathrm{\prime }}\mathrm{\mid }v{\mathrm{\mid }}_E$ and $\mathrm{\mid }v{\mathrm{\mid }}_E\le K\mathrm{\mid }v{\mathrm{\mid }}_{E^{\mathrm{\prime }}}$ for all $v\in V$. The two transported norms are therefore equivalent, give the same limits, and give the same notion of differentiability. The curve is differentiable at $s$ when the limit of the difference quotients as $h\to 0$ (over $h\mathrm{\ne }0$, $s+h\in I$; one-sided at the endpoints of $I$) exists in $V$; then this limit, written $x^{\mathrm{\prime }}(s)\in V$, is the tangent vector of $x$ at $s$, and it does not depend on the inertial frame used to describe the convergence.

If $x_A:A\to {\mathbb{R}}^4$ is the global chart of an inertial frame (Inertial frames and inertial charts of the affine Minkowski model, The affine atlas makes the Minkowski model a smooth 4-manifold diffeomorphic to R^4), then $x$ is differentiable at $s$ exactly when the ${\mathbb{R}}^4$-valued map $x_A\circ x$ is differentiable at $s$ in the sense of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, because $E^{-1}$ is linear and carries the difference quotients of $x$ to those of $x_A\circ x$; and then

$x^{\mathrm{\prime }}(s)=E((x_A\circ x{)}^{\mathrm{\prime }}(s)),$

where $E:{\mathbb{R}}^4\to V$ is the frame isomorphism $E(u)={\sum }_a{u}^a{e}_a$. This value does not depend on the inertial chart used: two inertial charts are related by an affine bijection with an invertible linear part (The affine atlas makes the Minkowski model a smooth 4-manifold diffeomorphic to R^4), the chain rule transports the coordinate derivative accordingly, and all norms on the finite-dimensional space $V$ give the same limit.

The curve is $C^1$ when it is differentiable at every point of $I$ and the derivative map $s\mapsto x^{\mathrm{\prime }}(s)$, $I\to V$, is continuous, meaning that the ${\mathbb{R}}^4$-valued map $s\mapsto (x_A\circ x{)}^{\mathrm{\prime }}(s)$ is continuous (Vector-valued functions $f:A\to {\mathbb{R}}^m$, their limits and continuity, with the dictionary to the metric notions); by the equivalence of transported norms this condition is independent of the inertial frame. It is $C^k$ for $k\ge 1$ when the derivative map is $C^{k-1}$, with the same transported reading. A curve on a compact interval $[a,b]$ contained in $I$ is piecewise $C^1$ when there is a finite partition $a=s_0<s_1<\cdots <s_m=b$ such that $x$ is $C^1$ on every open subinterval $(s_{i-1},s_i)$ and the finite one-sided difference-quotient derivatives at both ends of every piece exist in $V$. These are limits of $(x(t+h)-x(t))\mathrm{/}h$ over the relevant side of the piece, not assertions that $x^{\mathrm{\prime }}(s)$ has a limit as $s$ approaches a corner; their finiteness implies continuity of $x$ at the corners (for a one-sided quotient derivative $v$ at $t$ one has $x(t+h)-x(t)=h(v+o(1))$ as $h\to 0$ on that side). No continuous extension of the open-piece derivative is required. Derivatives at the endpoints of $I$ are the one-sided difference-quotient limits over $I$.

Causal character

Let $x$ be differentiable at $s\in I$. Then $x$ is regular at $s$ if $x^{\mathrm{\prime }}(s)\mathrm{\ne }0$; $x$ is timelike (respectively null, spacelike, causal) at $s$ if the vector $x^{\mathrm{\prime }}(s)$ is timelike (respectively null, spacelike, causal) in the sense of Timelike, null, spacelike and causal vectors of a Lorentz vector space, and $x$ is future-directed at $s$ if $x^{\mathrm{\prime }}(s)$ is future timelike or future null in the sense of Time orientation, future and past timelike/null/causal vectors, and the closed cones. The zero vector has no causal character, so only a regular curve has a causal character at $s$.

Worldlines

A future timelike worldline (equivalently, a massive worldline) is a $C^1$ curve $x:I\to A$ with $x^{\mathrm{\prime }}(s)$ future timelike for every $s\in I$; a future causal curve (respectively a future null curve) is a $C^1$ curve with $x^{\mathrm{\prime }}(s)$ future causal (respectively future null) for every $s\in I$. A piecewise $C^1$ worldline is a piecewise $C^1$ curve whose tangents at every point of each open smooth piece and whose finite one-sided difference-quotient tangents at both endpoints of every piece are future timelike; a $C^k$ worldline is a worldline whose parametrization is $C^k$. $C^2$ regularity is what makes the four-acceleration of Four-velocity and four-acceleration of a future timelike worldline available.

The parametrization is part of the curve: the image of a future timelike worldline is also the image of the past-directed curve $s\mapsto x(-s)$, which is not a worldline of this model, and no worldline or clock claim is attached to it. The definition asserts the existence of no worldline beyond those given by an explicit formula.

• Well-definedness. The difference quotient is an element of $V$ because the affine action assigns a displacement to every ordered pair of events. The limit in $V$ is taken in the transported Euclidean norm $\mathrm{\mid }v{\mathrm{\mid }}_E=\mathrm{\mid }{E}^{-1}v{\mathrm{\mid }}_2$ of an inertial frame isomorphism $E$, and the choice of frame does not matter: for inertial frames $E$ and $E^{\mathrm{\prime }}$ the maps $E^{\mathrm{\prime }-1}E$ and its inverse are Euclidean linear maps, so by Every Euclidean linear map has a unique matrix and satisfies $\mathrm{\parallel }Lh{\mathrm{\parallel }}_2\le K\mathrm{\parallel }h{\mathrm{\parallel }}_2$ for some $K\ge 0$ they are bounded with constants $K,K^{\mathrm{\prime }}\ge 0$, which gives $\mathrm{\mid }v{\mathrm{\mid }}_{E^{\mathrm{\prime }}}\le K^{\mathrm{\prime }}\mathrm{\mid }v{\mathrm{\mid }}_E$ and $\mathrm{\mid }v{\mathrm{\mid }}_E\le K\mathrm{\mid }v{\mathrm{\mid }}_{E^{\mathrm{\prime }}}$ on $V$; equivalent norms have the same convergent sequences and the same limits, and hence the same derivative $x^{\mathrm{\prime }}(s)$. The identification with the coordinate calculus, the value $x^{\mathrm{\prime }}(s)=E((x_A\circ x{)}^{\mathrm{\prime }}(s))$, the independence of the class $C^k$ and the continuity of $x^{\mathrm{\prime }}$ are then read through the same linear isomorphism, with the published vector-valued derivative and continuity of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral and Vector-valued functions $f:A\to {\mathbb{R}}^m$, their limits and continuity, with the dictionary to the metric notions as the coordinate calculus, and the one-sided endpoint derivatives are the one-sided limits over the domain $I$ used by that same definition. The affine chart transitions of The affine atlas makes the Minkowski model a smooth 4-manifold diffeomorphic to R^4 are consistent with this dictionary: they are affine with an invertible linear part, so they compose the coordinate calculus with a bounded linear change of coordinates.
• Regularity is implicit in every causal character. Because the zero vector is excluded from timelike, null and spacelike alike (Timelike, null, spacelike and causal vectors of a Lorentz vector space), the future timelike condition $x^{\mathrm{\prime }}(s)\in C^{+}$ already contains $x^{\mathrm{\prime }}(s)\mathrm{\ne }0$, and each side of a corner in the piecewise clause is required to carry a nonzero future timelike one-sided tangent. Null and spacelike curves are distinguished from massive worldlines and are used later only to delimit the proper-time construction, which is stated for future timelike worldlines alone.
• No choice principle is used. All objects are finite-dimensional or explicitly parametrized; the definition selects nothing from a family.