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\begin{document}

\begin{titlepage} \centering \vspace*{2.4cm}

{\Huge\bfseries From Riemannian to Lorentzian Geometry\par} \vspace{0.4cm} {\LARGE\bfseries Connections and Covariant Derivatives\par}

\vspace{1.2cm}

\begin{tcolorbox}[ width=0.88\textwidth, colback=CommonBlue!5, colframe=CommonBlue, boxrule=1pt, arc=2mm ] \centering

$$\boxed{ \text{Riemannian geometry} \longrightarrow \text{semi-Riemannian geometry} \longrightarrow \text{Lorentzian geometry} }$$

\end{tcolorbox}

\vspace{1cm}

\begin{minipage}{0.84\textwidth} \large The central theme of these notes is that the passage from Riemannian to Lorentzian geometry does \emph{not} require inventing a new theory of connections. Covariant differentiation, torsion, parallel transport, geodesics, and curvature belong to a common differential-geometric framework.

What changes dramatically is the linear algebra of the metric: positive definiteness is replaced by indefinite signature. The result is causal geometry---timelike, null, and spacelike vectors; light cones; time orientation; proper time; and null hypersurfaces. \end{minipage}

\vfill

$$\boxed{ \begin{array}{c} \textbf{same differential-geometric machinery}\\[1mm] +\\[1mm] \textbf{different metric algebra}\\[1mm] =\\[1mm] \textbf{Lorentzian geometry} \end{array}}$$

\vfill

{\large Signature convention throughout: $\;(-,+,\ldots,+)$.\par}

\end{titlepage}

\tableofcontents \newpage

% ============================================================ \section{Orientation: what changes, and what does not?} % ============================================================

\subsection{Why ordinary differentiation stops making sense}

On $\R^n$, if $Y\colon \R^n\to\R^n$ is a vector field, then

$$\frac{Y(x+h)-Y(x)}{\|h\|}$$

makes sense because every value $Y(x)$ belongs to the \emph{same} vector space $\R^n$.

On a manifold $M$, however,

$$Y(p)\in T_pM, \qquad Y(q)\in T_qM,$$

and $T_pM$ and $T_qM$ are different vector spaces. The expression $Y(q)-Y(p)$ is therefore meaningless until we supply a rule for comparing tangent vectors based at different points.

\begin{figure}[htbp] \centering \begin{tikzpicture}[scale=1] \draw[thick] (-3,0) .. controls (-2,1.4) and (-0.8,1.0) .. (0,0.5) .. controls (1.3,-0.2) and (2.3,-1.1) .. (3,-0.1) .. controls (2.4,-1.8) and (-2.1,-1.8) .. (-3,0);

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\draw[-{Latex[length=3mm]},very thick,CommonBlue] (p) -- ++(0.65,0.65) node[above] {$Y(p)$}; \draw[-{Latex[length=3mm]},very thick,SemiGreen] (q) -- ++(0.2,0.85) node[above right] {$Y(q)$};

\draw[dashed,<->,ChangeRed] (-0.8,1.4) .. controls (0,2.0) .. (0.9,1.2) node[midway,above] {$\text{no canonical identification}$}; \end{tikzpicture} \caption{Vectors at different points belong to different tangent spaces. A connection supplies a rule for differentiating despite this.} \label{fig:different-tangent-spaces} \end{figure}

A \emph{connection} is precisely the additional structure that solves this problem infinitesimally.

\biginsight{The theory of connections is not intrinsically Lorentzian.}

A connection may be defined on a vector bundle without any metric at all. It therefore makes no sense to speak of a Riemannian covariant derivative'' and a fundamentally different Lorentzian covariant derivative.'' The same abstract notion is used in both settings.

What \emph{does} change when the metric becomes Lorentzian is the geometry determined by that metric.

\biginsight{The geometry induced by a Lorentzian metric is qualitatively different from Riemannian geometry.}

The decisive issue is \emph{indefiniteness}. In Riemannian geometry,

$$g(v,v)>0 \qquad\text{for every }v\neq0.$$

In Lorentzian geometry, a nonzero vector may satisfy

$$g(v,v)<0,\qquad g(v,v)=0,\qquad\text{or}\qquad g(v,v)>0.$$

The middle possibility---a nonzero vector with zero quadratic value---is responsible for light cones, null directions, degenerate induced metrics on null hypersurfaces, and much of causal geometry.

\begin{keybox} Throughout the handout we repeatedly separate three logical levels:

$$\boxed{\text{arbitrary connection}} \qquad \boxed{\text{nondegenerate metric}} \qquad \boxed{\text{Lorentzian signature}}.$$

When a construction does not require a metric, we will say so. When it requires only nondegeneracy, we will say so. When it uses index one and causal cones, we will mark it as genuinely Lorentzian. \end{keybox}

\subsection{Our signature convention}

A semi-Riemannian metric is a smooth nondegenerate symmetric bilinear form on each tangent space.

Authors disagree about the ordering of signature pairs. In these notes, if we write

$$\operatorname{sig}(g)=(p,q),$$

then $p$ is the number of positive directions and $q$ the number of negative directions. Thus:

$$\text{Riemannian: }(n,0), \qquad \text{Lorentzian: }(n-1,1).$$

Many authors reverse the order and call the same Lorentzian signature $(1,n-1)$.

Our matrix convention is

$$\boxed{\operatorname{diag}(-1,+1,\ldots,+1)}.$$

% ============================================================ \section{Connections on vector bundles} % ============================================================

\subsection{The general vector-bundle setting}

Let

$$\pi:E\longrightarrow M$$

be a smooth real vector bundle, and let $\Gamma(E)$ denote its space of smooth sections.

\begin{definition}[Koszul connection] A \emph{Koszul connection} on $E$ is a map

$$\nabla:\X(M)\times\Gamma(E)\longrightarrow\Gamma(E), \qquad (X,s)\longmapsto\nabla_Xs,$$

such that for all $X,Y\in\X(M)$, $s,t\in\Gamma(E)$, $f\in C^\infty(M)$, and $a,b\in\R$, \begin{align*} \nabla_{aX+bY}s &=a\nabla_Xs+b\nabla_Ys,\ \nabla_X(as+bt) &=a\nabla_Xs+b\nabla_Xt,\ \nabla_{fX}s &=f\nabla_Xs,\ \nabla_X(fs) &=X(f)s+f\nabla_Xs. \end{align*} \end{definition}

The last two identities encode a crucial asymmetry.

\begin{keybox} The first argument is $C^\infty(M)$-linear:

$$\nabla_{fX}s=f\nabla_Xs.$$

The second argument behaves like differentiation:

$$\nabla_X(fs)=X(f)s+f\nabla_Xs.$$

Thus $\nabla_X$ differentiates the section $s$, while the direction $X$ is used only through its value at the point. \end{keybox}

This asymmetry is exactly what one expects from directional differentiation. For an ordinary directional derivative on $\R^n$,

$$D_{fX}Y=fD_XY,$$

but

$$D_X(fY)=X(f)Y+fD_XY.$$

\begin{proposition}[Locality and first-order behavior] Let $\nabla$ be a connection on $E$. Then: \begin{enumerate} \item $\nabla_Xs(p)$ depends on $X$ only through $X_p$. \item $\nabla_Xs(p)$ depends on $s$ only through its germ near $p$. \item In a local frame $e_1,\ldots,e_r$, if

$$s=s^ae_a,$$

then

$$\nabla_Xs=X(s^a)e_a+s^a\nabla_Xe_a.$$

\end{enumerate} \end{proposition}

\begin{proof} The third statement follows immediately from the Leibniz rule:

$$\nabla_X(s^ae_a) = X(s^a)e_a+s^a\nabla_Xe_a.$$

The first follows from $C^\infty(M)$-linearity in $X$: locally, $X=X^i\partial_i$, so

$$\nabla_Xs=X^i\nabla_{\partial_i}s,$$

and at $p$ only the numbers $X^i(p)$ remain.

For locality in $s$, suppose $s$ vanishes on a neighborhood of $p$. Choose a smooth function $\chi$ supported in that neighborhood with $\chi=1$ near $p$. Then $\chi s=0$, hence

$$0=\nabla_X(\chi s) =X(\chi)s+\chi\nabla_Xs.$$

At $p$, $\chi(p)=1$ and $s(p)=0$, so $\nabla_Xs(p)=0$. Thus changing $s$ away from $p$ does not change $\nabla_Xs(p)$. \end{proof}

\begin{commonbox} Nothing in the definition above involves a metric, positive definite or otherwise. Connections belong to differential geometry before one chooses Riemannian or Lorentzian structure. \end{commonbox}

\subsection{Connections on the tangent bundle}

Now specialize to

$$E=TM.$$

Then

$$\nabla:\X(M)\times\X(M)\longrightarrow\X(M).$$

\begin{definition}[Linear or affine connection] A connection on $TM$ is called a \emph{linear connection} or \emph{affine connection}. It provides the covariant derivative

$$\nabla_XY$$

of a vector field $Y$ in the direction $X$. \end{definition}

Terminology varies between authors. A useful dictionary is

$$\boxed{ \begin{array}{ccl} \text{Koszul connection} &\longleftrightarrow& \text{connection on a vector bundle}, \\[1mm] \text{linear/affine connection} &\longleftrightarrow& \text{connection on }TM, \\[1mm] \text{covariant derivative} &\longleftrightarrow& \text{the differentiation operation supplied by }\nabla. \end{array}}$$

Some authors use covariant derivative'' as a synonym for connection.'' Others reserve it for the operation $\nabla_X$.

\begin{warningbox} \textbf{Koszul connection} and \textbf{Koszul formula} are different notions.

A Koszul connection is an arbitrary connection on a vector bundle.

The Koszul formula, introduced later, is a formula characterizing the Levi-Civita connection of a nondegenerate metric.

A Koszul connection is therefore \emph{not} automatically a Levi-Civita connection. \end{warningbox}

\subsection{Local connection coefficients}

Let $e_1,\ldots,e_r$ be a local frame of $E$, and $\partial_i$ a coordinate frame on $M$. There exist functions $A^a{}_{ib}$ such that

$$\nabla_{\partial_i}e_b=A^a{}_{ib}e_a.$$

These are the local coefficients of the connection in the chosen frame.

If

$$s=s^be_b, \qquad X=X^i\partial_i,$$

then

$$\nabla_Xs = X^i \left( \partial_is^a+A^a{}_{ib}s^b \right)e_a.$$

The appearance of derivatives of the frame-change matrix in the transformation law is already a warning: connection coefficients are not tensor components.

% ============================================================ \section{The connection-theoretic core shared by Riemannian and Lorentzian geometry} % ============================================================

This section establishes the common baseline. Everything here makes sense for an arbitrary affine connection unless explicitly stated otherwise.

\subsection{Torsion}

\begin{definition}[Torsion] For an affine connection $\nabla$, its torsion is

$$T(X,Y) = \nabla_XY-\nabla_YX-[X,Y].$$

\end{definition}

\begin{proposition} The torsion $T$ is $C^\infty(M)$-linear in both arguments. Hence

$$T\in\Gamma(\Lambda^2T^*M\otimes TM).$$

\end{proposition}

\begin{proof} For example, \begin{align*} T(fX,Y) &= \nabla_{fX}Y-\nabla_Y(fX)-[fX,Y]\ &= f\nabla_XY-\bigl(Y(f)X+f\nabla_YX\bigr) -\bigl(f[X,Y]-Y(f)X\bigr)\ &= fT(X,Y). \end{align*} The other slot is analogous. Antisymmetry follows directly from the definition. \end{proof}

Thus torsion is a genuine tensor even though $\nabla_XY$ itself is not $C^\infty$-linear in $Y$.

\subsection{Curvature}

\begin{definition}[Curvature] The curvature of an affine connection is

$$R(X,Y)Z = \nabla_X\nabla_YZ -\nabla_Y\nabla_XZ -\nabla_{[X,Y]}Z.$$

\end{definition}

\begin{proposition} The curvature is $C^\infty(M)$-linear in $X,Y,Z$, and therefore

$$R\in\Gamma\!\left(\Lambda^2T^*M\otimes T^*M\otimes TM\right).$$

\end{proposition}

\begin{proof}[Proof sketch] Linearity in $X$ and $Y$ follows from the connection axioms together with the Leibniz rule for the Lie bracket. The key cancellation in the third slot is

$$\nabla_X\nabla_Y(fZ) - \nabla_Y\nabla_X(fZ) - \nabla_{[X,Y]}(fZ),$$

where all first- and second-derivative terms involving $f$ cancel, leaving

$$fR(X,Y)Z.$$

\end{proof}

\subsection{What an arbitrary connection gives you}

Once a connection is chosen, one may construct:

\begin{itemize} \item covariant derivatives of vector fields; \item covariant derivatives of covectors and general tensors; \item torsion; \item differentiation along curves; \item parallel vector fields and parallel transport; \item geodesics; \item curvature. \end{itemize}

\begin{commonbox} Every item in the preceding list exists before choosing a metric.

The same definitions are available on a Riemannian manifold, a Lorentzian manifold, or a smooth manifold with no metric whatsoever. \end{commonbox}

\subsection{A schematic view of parallel transport}

A connection gives a rule for carrying vectors along a curve while declaring them to remain ``parallel.''

\begin{figure}[htbp] \centering \begin{tikzpicture}[scale=1] \draw[very thick,CommonBlue] (-3,-0.4) .. controls (-1.8,1.3) and (-0.4,1.1) .. (0.5,0.2) .. controls (1.2,-0.45) and (2.0,-0.8) .. (3,0.25); \node[CommonBlue] at (3.15,0.5) {$\gamma$};

\foreach \x/\y/\ang in {-2.5/0.1/62,-1.4/0.8/49,-0.2/0.75/36,1.0/-0.15/28,2.25/-0.35/20}{ \fill (\x,\y) circle (1.5pt); \draw[-{Latex[length=2.5mm]},very thick,SemiGreen] (\x,\y) -- ++({0.75cos(\ang)},{0.75sin(\ang)}); }

\node[SemiGreen] at (0,-1.2) {$\displaystyle \frac{DV}{dt}=0$}; \end{tikzpicture} \caption{Parallel transport along a curve. The vectors are not being compared by an ambient Euclidean translation; the connection itself defines what ``parallel'' means.} \label{fig:parallel-transport} \end{figure}

% ============================================================ \section{Adding a nondegenerate metric} % ============================================================

\subsection{Semi-Riemannian metrics}

\begin{definition}[Semi-Riemannian metric] A \emph{semi-Riemannian metric} on $M$ is a smooth symmetric $(0,2)$-tensor field $g$ such that

$$g_p:T_pM\times T_pM\to\R$$

is nondegenerate for every $p\in M$. \end{definition}

Nondegeneracy means:

$$g_p(v,w)=0\ \text{for all }w\in T_pM \quad\Longrightarrow\quad v=0.$$

It does \emph{not} mean positive definiteness.

By Sylvester's law of inertia, at each point there exists a basis in which

$$[g_p] = \operatorname{diag} (\underbrace{1,\ldots,1}_{p}, \underbrace{-1,\ldots,-1}_{q}).$$

For a semi-Riemannian metric the pair $(p,q)$ is locally constant and, on a connected manifold, constant.

\begin{semibox} The essential hypothesis for most metric connection theory is \emph{nondegeneracy}, not positive definiteness. \end{semibox}

\subsection{Metric compatibility}

The covariant derivative of the metric is defined by

$$(\nabla_Xg)(Y,Z) = X[g(Y,Z)] - g(\nabla_XY,Z) - g(Y,\nabla_XZ).$$

\begin{definition}[Metric compatibility] A connection $\nabla$ is \emph{metric-compatible} with $g$ if

$$\nabla g=0.$$

Equivalently,

$$X[g(Y,Z)] = g(\nabla_XY,Z) + g(Y,\nabla_XZ).$$

\end{definition}

Geometrically, metric compatibility says that the connection differentiates inner products by the usual product rule.

\subsection{Torsion-free connections}

\begin{definition} An affine connection is \emph{torsion-free} if

$$T(X,Y)=0$$

for all $X,Y$, equivalently

$$\nabla_XY-\nabla_YX=[X,Y].$$

\end{definition}

\subsection{The Levi-Civita theorem}

\begin{theorem}[Fundamental theorem of semi-Riemannian geometry] Let $g$ be a smooth nondegenerate semi-Riemannian metric on $M$. There exists a unique affine connection $\nabla$ satisfying

$$T=0, \qquad \nabla g=0.$$

It is called the \emph{Levi-Civita connection} of $g$. \end{theorem}

\begin{commonbox} The theorem is formally identical for Riemannian and Lorentzian metrics.

Positive definiteness is not used. Nondegeneracy is the relevant hypothesis. \end{commonbox}

Thus:

$$\boxed{ \begin{array}{ccl} \text{Riemannian} &:& \operatorname{sig}(g)=(n,0), \\[1mm] \text{Lorentzian} &:& \operatorname{sig}(g)=(n-1,1) \quad \text{in our ordering}. \end{array}}$$

The same Levi-Civita construction applies to both.

% ============================================================ \section{The Koszul formula: the common metric theory} % ============================================================

\begin{theorem}[Koszul formula] Let $\nabla$ be the Levi-Civita connection of a nondegenerate metric $g$. Then \begin{align} 2g(\nabla_XY,Z) ={}& X[g(Y,Z)] + Y[g(Z,X)]

Z[g(X,Y)] \nonumber\ &- g(X,[Y,Z]) + g(Y,[Z,X]) + g(Z,[X,Y]). \label{eq:koszul} \end{align} \end{theorem}

\subsection{Derivation}

Metric compatibility gives \begin{align*} Xg(Y,Z) &= g(\nabla_XY,Z)+g(Y,\nabla_XZ),\ Yg(Z,X) &= g(\nabla_YZ,X)+g(Z,\nabla_YX),\ Zg(X,Y) &= g(\nabla_ZX,Y)+g(X,\nabla_ZY). \end{align*} Take the first two equations and subtract the third. Then use torsion-freeness:

$$\nabla_XY-\nabla_YX=[X,Y],$$

and its cyclic permutations. After cancellation, one obtains \eqref{eq:koszul}.

\subsection{Where nondegeneracy enters}

The right-hand side of the Koszul formula, for fixed $X,Y$, is a $C^\infty(M)$-linear function of $Z$. Hence at each $p$ it defines a covector

$$\alpha_p\in T_p^*M.$$

To recover $\nabla_XY$, one must solve

$$g_p(\nabla_XY,\cdot)=\frac12\alpha_p.$$

Nondegeneracy says that the map

$$g^\flat_p:T_pM\longrightarrow T_p^*M, \qquad v\longmapsto g_p(v,\cdot),$$

is an isomorphism. Therefore there is a unique vector $\nabla_XY$ realizing the right-hand side.

\begin{semibox} Nondegeneracy is used to turn a covector

$$Z\longmapsto g(\nabla_XY,Z)$$

into a unique vector $\nabla_XY$.

Positive definiteness is unnecessary. \end{semibox}

\subsection{Uniqueness of the Levi-Civita connection}

\begin{proof}[Proof of uniqueness] If $\nabla$ is torsion-free and metric-compatible, the preceding derivation forces it to satisfy the Koszul formula. The right-hand side depends only on $g,X,Y,Z$, not on any further choice.

If $\nabla$ and $\widetilde\nabla$ were two such connections, then for every $Z$,

$$g(\nabla_XY-\widetilde\nabla_XY,Z)=0.$$

By nondegeneracy,

$$\nabla_XY=\widetilde\nabla_XY.$$

\end{proof}

\subsection{Existence}

\begin{proof}[Proof sketch of existence] Use the right-hand side of \eqref{eq:koszul} to \emph{define} $\nabla_XY$ by

$$2g(\nabla_XY,Z)=\text{right-hand side of \eqref{eq:koszul}}.$$

One checks that the right-hand side is $C^\infty$-linear in $Z$, so nondegeneracy produces a unique smooth vector field $\nabla_XY$.

A direct calculation then verifies

$$\nabla_{fX}Y=f\nabla_XY, \qquad \nabla_X(fY)=X(f)Y+f\nabla_XY.$$

Subtracting the formula with $X,Y$ interchanged yields

$$\nabla_XY-\nabla_YX=[X,Y],$$

so the connection is torsion-free. Combining the Koszul formula for appropriate permutations of $X,Y,Z$ yields

$$Xg(Y,Z) = g(\nabla_XY,Z)+g(Y,\nabla_XZ),$$

so $\nabla g=0$. \end{proof}

\biginsight{The Koszul formula is identical in Riemannian and Lorentzian geometry.}

\begin{changebox} The formula does not change when the metric becomes Lorentzian.

What changes is the metric algebra encoded in $g$: signs, null directions, causal cones, and the behavior of orthogonality. \end{changebox}

% ============================================================ \section{Local coordinates and Christoffel symbols} % ============================================================

Let

$$(x^1,\ldots,x^n)$$

be local coordinates and write

$$\partial_i=\frac{\partial}{\partial x^i}.$$

\begin{definition}[Connection coefficients] The local coefficients of an affine connection are the functions $\Gamma^k{}_{ij}$ defined by

$$\nabla_{\partial_i}\partial_j = \Gamma^k{}_{ij}\partial_k.$$

\end{definition}

If

$$X=X^i\partial_i, \qquad Y=Y^j\partial_j,$$

then \begin{align*} \nabla_XY &= X^i\nabla_{\partial_i}(Y^j\partial_j)\ &= X^i \left[ (\partial_iY^k)\partial_k + Y^j\Gamma^k{}_{ij}\partial_k \right]. \end{align*} Hence

$$\boxed{ \nabla_XY = X^i \left( \partial_iY^k+\Gamma^k{}_{ij}Y^j \right)\partial_k. }$$

\subsection{Torsion in coordinates}

Because

$$[\partial_i,\partial_j]=0,$$

we have

$$T(\partial_i,\partial_j) = (\Gamma^k{}_{ij}-\Gamma^k{}_{ji})\partial_k.$$

Thus a connection is torsion-free iff

$$\Gamma^k{}_{ij}=\Gamma^k{}_{ji}$$

in every coordinate chart.

\subsection{Christoffel symbols of the Levi-Civita connection}

Write

$$g_{ij}=g(\partial_i,\partial_j)$$

and let $g^{ij}$ denote the inverse matrix:

$$g^{ik}g_{kj}=\delta^i_j.$$

Since coordinate vector fields commute, the Koszul formula gives

$$2g(\nabla_{\partial_i}\partial_j,\partial_\ell) = \partial_i g_{j\ell} + \partial_j g_{i\ell} - \partial_\ell g_{ij}.$$

But

$$g(\nabla_{\partial_i}\partial_j,\partial_\ell) = \Gamma^k{}_{ij}g_{k\ell}.$$

Multiplying by $g^{m\ell}$,

$$\boxed{ \Gamma^m{}_{ij} = \frac12g^{m\ell} \left( \partial_i g_{j\ell} + \partial_j g_{i\ell} - \partial_\ell g_{ij} \right). }$$

\begin{commonbox} The Christoffel-symbol formula is unchanged in Lorentzian signature.

The only algebraic difference is that the inverse matrix $g^{ij}$ comes from an indefinite rather than positive-definite matrix. \end{commonbox}

\subsection{Christoffel symbols are not tensor components}

Under a coordinate change $x\mapsto \widetilde x$, \begin{align*} \widetilde\Gamma^a{}{bc} ={}& \frac{\partial\widetilde x^a}{\partial x^k} \frac{\partial x^i}{\partial\widetilde x^b} \frac{\partial x^j}{\partial\widetilde x^c} \Gamma^k{}{ij} \ &+ \frac{\partial\widetilde x^a}{\partial x^k} \frac{\partial^2x^k} {\partial\widetilde x^b\partial\widetilde x^c}. \end{align*} The second-derivative term is the obstruction to tensoriality.

\begin{warningbox}

$$\boxed{\Gamma^k{}_{ij}\text{ are not tensor components}.}$$

A tensor transformation law is homogeneous. The transformation law for Christoffel symbols contains an inhomogeneous second-derivative term. \end{warningbox}

This fact is completely independent of signature.

\begin{tcolorbox}[ breakable, colback=SoftGray, colframe=DeepGray, title=\textbf{Coordinate comparison} ] \textbf{Same as Riemannian geometry:}

$$\text{connection coefficients}, \qquad \text{Christoffel transformation law}, \qquad \text{Levi-Civita formula}.$$

\textbf{What becomes Lorentzian:}

$$\text{the metric contractions and the causal interpretation of vectors}.$$

\end{tcolorbox}

\subsection{A coordinate frame on a curved surface}

\begin{figure}[htbp] \centering \begin{tikzpicture}[scale=0.95] \draw[thick] (-3,-1.2) .. controls (-2.2,1.5) and (1.7,1.5) .. (3,-0.7); \draw[thick] (-3,-1.2) .. controls (-2.0,-2.0) and (2.0,-1.8) .. (3,-0.7);

\foreach \s in {-2.2,-1.3,-0.4,0.5,1.4,2.2}{ \draw[CommonBlue!60] (\s,-1.35) .. controls (\s-0.25,-0.3) and (\s+0.25,0.7) .. (\s,1.0); }

\foreach \h in {-0.8,-0.25,0.3,0.78}{ \draw[SemiGreen!65] (-2.65,\h) .. controls (-1.1,\h+0.4) and (1.5,\h-0.2) .. (2.6,\h); }

\coordinate (p) at (0.35,0.2); \fill (p) circle (2pt) node[below right] {$p$}; \draw[-{Latex[length=2.5mm]},very thick,CommonBlue] (p)--++(0.95,0.18) node[right] {$\partial_1$}; \draw[-{Latex[length=2.5mm]},very thick,SemiGreen] (p)--++(-0.2,0.9) node[above] {$\partial_2$}; \end{tikzpicture} \caption{A coordinate frame on a curved surface. Even when the coordinate basis varies from point to point, a connection differentiates the basis itself through the coefficients $\Gamma^k{}_{ij}$.} \label{fig:coordinate-frame} \end{figure}

% ============================================================ \section{Covariant differentiation of tensors} % ============================================================

A connection on $TM$ canonically induces connections on the dual bundle and all tensor bundles.

\subsection{Covectors}

Let $\alpha\in\Omega^1(M)$. Define $\nabla_X\alpha$ by requiring the usual product rule:

$$X[\alpha(Y)] = (\nabla_X\alpha)(Y) + \alpha(\nabla_XY).$$

Thus

$$\boxed{ (\nabla_X\alpha)(Y) = X[\alpha(Y)] - \alpha(\nabla_XY). }$$

In coordinates,

$$\alpha=\alpha_j\,dx^j$$

gives

$$(\nabla_i\alpha)_j = \partial_i\alpha_j - \Gamma^k{}_{ij}\alpha_k.$$

\subsection{General tensors}

If

$$A\in\Gamma(T^r_sM),$$

then $\nabla_XA$ is characterized by: \begin{enumerate} \item $\nabla_X$ is $\R$-linear; \item it satisfies a Leibniz rule for tensor products; \item it commutes with contraction; \item it agrees with the given connection on vector fields and the induced connection on covectors. \end{enumerate}

For a $(1,1)$-tensor $A=A^i{}_j\partial_i\otimes dx^j$,

$$(\nabla_kA)^i{}_j = \partial_kA^i{}_j + \Gamma^i{}_{k\ell}A^\ell{}_j - \Gamma^\ell{}_{kj}A^i{}_\ell.$$

More generally, for

$$T^{i_1\cdots i_r}{}_{j_1\cdots j_s},$$

one adds one $+\Gamma$-term for every contravariant index and one $-\Gamma$-term for every covariant index: \begin{align*} \nabla_k T^{i_1\cdots i_r}{}{j_1\cdots j_s} ={}& \partial_kT^{i_1\cdots i_r}{}{j_1\cdots j_s} \ &+ \sum_{a=1}^r \Gamma^{i_a}{}{k\ell} T^{i_1\cdots \ell\cdots i_r}{}{j_1\cdots j_s} \ &- \sum_{b=1}^s \Gamma^\ell{}{kj_b} T^{i_1\cdots i_r}{}{j_1\cdots \ell\cdots j_s}. \end{align*}

\begin{commonbox} Induced connections on $T^*M$, tensor bundles, and exterior-power bundles work exactly the same way for Riemannian and Lorentzian metrics. The construction depends on the connection, not on positive definiteness. \end{commonbox}

\subsection{Differential forms: $d$ versus $\nabla$}

For a $k$-form $\omega$,

$$d:\Omega^k(M)\longrightarrow\Omega^{k+1}(M)$$

is the canonical exterior derivative, requiring no connection.

By contrast,

$$\nabla\omega \in \Gamma(T^*M\otimes\Lambda^kT^*M).$$

The first new slot records the direction in which $\omega$ is covariantly differentiated.

Thus:

$$d\omega \quad\text{is a }(k+1)\text{-form},$$

whereas

$$\nabla\omega \quad\text{is generally not fully alternating}.$$

If the connection is torsion-free, then

$$d\omega = \operatorname{Alt}(\nabla\omega),$$

more explicitly,

$$d\omega(X_0,\ldots,X_k) = \sum_{i=0}^k (-1)^i (\nabla_{X_i}\omega) (X_0,\ldots,\widehat X_i,\ldots,X_k).$$

% ============================================================ \section{Covariant derivative along curves} % ============================================================

Let

$$\gamma:I\to M$$

be a smooth curve. A \emph{vector field along $\gamma$} is a smooth assignment

$$t\longmapsto V(t)\in T_{\gamma(t)}M.$$

\begin{definition}[Covariant derivative along a curve] The covariant derivative of $V$ along $\gamma$ is

$$\frac{DV}{dt} = \nabla_{\dot\gamma}V.$$

\end{definition}

If

$$V(t)=V^k(t)\partial_k|_{\gamma(t)},$$

then

$$\boxed{ \frac{DV^k}{dt} = \frac{dV^k}{dt} + \Gamma^k{}_{ij}(\gamma(t)) \dot\gamma^iV^j. }$$

\begin{definition}[Parallel vector field] A vector field $V$ along $\gamma$ is \emph{parallel} if

$$\frac{DV}{dt}=0.$$

\end{definition}

The resulting linear ODE has a unique solution for each initial vector $V(t_0)$. Hence the connection defines a linear isomorphism

$$P_{\gamma,t_0\to t_1}: T_{\gamma(t_0)}M \longrightarrow T_{\gamma(t_1)}M,$$

called \emph{parallel transport}.

\begin{definition}[Geodesic] A curve $\gamma$ is an \emph{affinely parametrized geodesic} if

$$\boxed{ \frac{D\dot\gamma}{dt}=0. }$$

\end{definition}

In local coordinates:

$$\boxed{ \ddot x^k + \Gamma^k{}_{ij}\dot x^i\dot x^j = 0. }$$

\begin{proposition}[Affine reparametrization] If $\gamma(t)$ is an affinely parametrized geodesic and

$$t=as+b, \qquad a\neq0,$$

then $\widetilde\gamma(s)=\gamma(as+b)$ is also affinely parametrized.

A general nonlinear reparametrization need not preserve the affine geodesic equation. \end{proposition}

\begin{proof} Since

$$\frac{d\widetilde\gamma}{ds} = a\dot\gamma,$$

and $a$ is constant,

$$\nabla_{\widetilde\gamma'}\widetilde\gamma' = a^2\nabla_{\dot\gamma}\dot\gamma = 0.$$

\end{proof}

\begin{commonbox} Parallel transport, covariant differentiation along curves, and the geodesic equation are formally identical in Riemannian and Lorentzian geometry. \end{commonbox}

% ============================================================ \section{The decisive transition: Lorentzian signature} % ============================================================

We now specialize to a Lorentzian metric $g$ of signature

$$(-,+,\ldots,+).$$

\subsection{The causal trichotomy}

For a nonzero tangent vector $v$,

$$\boxed{ \begin{array}{ccl} g(v,v)<0 &:& \text{timelike},\\[1mm] g(v,v)=0 &:& \text{null or lightlike},\\[1mm] g(v,v)>0 &:& \text{spacelike}. \end{array}}$$

Under the usual convention adopted here,

$$v=0$$

is neither timelike, null, nor spacelike.

A nonzero vector is called \emph{causal} if it is timelike or null.

\begin{warningbox} The scalar $g(v,v)$ is not the square of a norm in Lorentzian geometry.

In particular,

$$v\neq0 \quad\text{and}\quad g(v,v)=0$$

can occur. Hence $v\mapsto\sqrt{|g(v,v)|}$ is not a norm on the vector space. \end{warningbox}

\subsection{Riemannian versus Lorentzian tangent spaces}

In a Riemannian tangent space, the set

$$\{v:g(v,v)=1\}$$

is an ellipsoid, and no nonzero vector has zero squared length.

\begin{figure}[htbp] \centering \begin{tikzpicture}[scale=1.05] \draw[->] (-2.3,0)--(2.3,0) node[right] {$e_1$}; \draw[->] (0,-2.0)--(0,2.0) node[above] {$e_2$}; \draw[CommonBlue,very thick] (0,0) circle (1.3); \draw[-{Latex[length=2.5mm]},very thick,SemiGreen] (0,0)--(0.95,0.62) node[right] {$v$}; \draw[-{Latex[length=2.5mm]},very thick,LorentzPurple] (0,0)--(-0.62,0.95) node[above left] {$v^\perp$}; \draw[dashed,DeepGray] (-1.6,2.45) -- (1.6,-2.45); \node at (1.65,1.55) {$g(v,v)>0$ for $v\neq0$}; \end{tikzpicture} \caption{A Riemannian tangent-space picture: there are no nonzero null vectors, and the orthogonal complement of a nonzero vector is transverse to its span.} \label{fig:riemannian-tangent} \end{figure}

In a Lorentzian tangent space, the equation

$$g(v,v)=0$$

defines the \emph{light cone}.

\begin{figure}[htbp] \centering \begin{tikzpicture}[scale=1.05] \fill[LorentzPurple!8] (0,0)--(-1.7,2.4)--(1.7,2.4)--cycle; \fill[LorentzPurple!8] (0,0)--(-1.7,-2.4)--(1.7,-2.4)--cycle;

\draw[->] (-3,0)--(3,0) node[right] {$x$}; \draw[->] (0,-2.8)--(0,2.8) node[above] {$t$};

\draw[very thick,LorentzPurple] (-1.9,-2.7)--(1.9,2.7); \draw[very thick,LorentzPurple] (1.9,-2.7)--(-1.9,2.7);

\node at (0.55,1.65) {$\timelike$}; \node at (0.65,-1.65) {$\timelike$}; \node at (2.15,1.15) {$\spacelike$}; \node at (-2.2,-1.2) {$\spacelike$}; \node[LorentzPurple] at (1.65,2.25) {$\nulltype$}; \end{tikzpicture} \caption{The light cone in a $1+1$-dimensional Lorentzian tangent space. Timelike vectors lie inside the cone, null vectors on it, and spacelike vectors outside it.} \label{fig:light-cone} \end{figure}

\begin{indefbox} The new phenomenon is not merely that one diagonal entry of the metric has a minus sign. The crucial fact is that the quadratic form is \emph{indefinite}. This produces nonzero null vectors and divides tangent directions into causal types. \end{indefbox}

% ============================================================ \section{Causal cones and time orientation} % ============================================================

At $p$, define the timelike cone

$$\mathcal T_p = \{v\in T_pM:g(v,v)<0\}.$$

For Lorentzian signature, $\mathcal T_p$ has two connected components.

\begin{figure}[htbp] \centering \begin{tikzpicture}[scale=1] \fill[CommonBlue!10] (0,0)--(-1.6,2.4)--(1.6,2.4)--cycle; \fill[ChangeRed!10] (0,0)--(-1.6,-2.4)--(1.6,-2.4)--cycle;

\draw[->] (-2.7,0)--(2.7,0) node[right] {$x$}; \draw[->] (0,-2.8)--(0,2.8) node[above] {$t$}; \draw[very thick,LorentzPurple] (-1.8,-2.7)--(1.8,2.7); \draw[very thick,LorentzPurple] (1.8,-2.7)--(-1.8,2.7);

\draw[-{Latex[length=2.8mm]},very thick,CommonBlue] (0,0)--(0.25,1.7); \draw[-{Latex[length=2.8mm]},very thick,ChangeRed] (0,0)--(-0.25,-1.7);

\node[CommonBlue] at (0.9,1.7) {future}; \node[ChangeRed] at (-0.9,-1.7) {past}; \end{tikzpicture} \caption{After a time orientation has been chosen, one timelike component is called future and the other past.} \label{fig:future-past} \end{figure}

\begin{definition}[Time orientation] A \emph{time orientation} is a continuous choice, at every point $p$, of one of the two connected components of $\mathcal T_p$.

Vectors in the chosen component are \emph{future-directed timelike}; those in the other are \emph{past-directed timelike}. Future-directed null vectors lie in the boundary of the chosen future cone. \end{definition}

\begin{definition}[Time-orientable] A Lorentzian manifold is \emph{time-orientable} if it admits a global time orientation. \end{definition}

Equivalently, a Lorentzian manifold is time-orientable iff it admits a smooth timelike vector field $T$.

\begin{proposition} A Lorentzian manifold is time-orientable if and only if it admits a global smooth timelike vector field. \end{proposition}

\begin{proof}[Proof sketch] If $T$ is a global timelike vector field, declare the component containing $T_p$ to be future.

Conversely, assume a continuous choice of future cone is given. Choose local smooth future timelike fields $T_\alpha$. A partition of unity $\{\varphi_\alpha\}$ produces

$$T=\sum_\alpha\varphi_\alpha T_\alpha.$$

At each point all vectors being averaged lie in the same future cone, and a Lorentzian future timelike cone is convex. Therefore $T$ is again future timelike. \end{proof}

\begin{lorentzbox} A Lorentzian metric does \emph{not} automatically determine which cone is future.

Thus

$$\boxed{\text{Lorentzian metric}}$$

and

$$\boxed{\text{time-oriented Lorentzian manifold}}$$

are different structures. \end{lorentzbox}

% ============================================================ \section{Lorentzian orthogonality} % ============================================================

For any nonzero $v$,

$$v^\perp = \{w:g(v,w)=0\}.$$

In positive-definite geometry,

$$\Span(v)\cap v^\perp=\{0\}$$

for every nonzero $v$, because if $v\in v^\perp$, then

$$g(v,v)=0,$$

forcing $v=0$.

That argument fails in Lorentzian geometry.

\begin{proposition}[Null self-orthogonality] If $k\neq0$ is null, then

$$k\in k^\perp.$$

Consequently,

$$\Span(k)\subset k^\perp.$$

\end{proposition}

\begin{proof} Since $k$ is null,

$$g(k,k)=0.$$

By the definition of $k^\perp$, this says exactly that $k\in k^\perp$. \end{proof}

\begin{indefbox} A nonzero null vector is orthogonal to itself.

This is impossible in positive-definite geometry and is one of the clearest algebraic signals that Lorentzian orthogonality behaves differently. \end{indefbox}

\subsection{Three types of orthogonal complement}

In a Lorentzian vector space $V$:

\begin{itemize} \item if $v$ is timelike, $v^\perp$ is positive definite; \item if $v$ is spacelike, $v^\perp$ is Lorentzian; \item if $v$ is null, $v^\perp$ is degenerate and contains $v$. \end{itemize}

\begin{proposition} Let $V$ have Lorentzian signature and let $k\neq0$ be null. Then the restriction of $g$ to $k^\perp$ has radical

$$\rad(k^\perp)=\Span(k).$$

\end{proposition}

\begin{proof} Because $k\in k^\perp$ and

$$g(k,w)=0 \qquad \forall\,w\in k^\perp,$$

we have

$$\Span(k)\subset\rad(k^\perp).$$

Since $g$ is nondegenerate on $V$,

$$\dim k^\perp=\dim V-1.$$

For Lorentzian index one, a null hyperplane cannot have a radical of dimension larger than one: two independent radical vectors would span a totally null two-plane, impossible for index one. Hence the radical is exactly $\Span(k)$. \end{proof}

% ============================================================ \section{Lorentzian linear algebra} % ============================================================

\subsection{Sylvester's law of inertia}

\begin{theorem}[Sylvester's law of inertia] Let $B$ be a real symmetric nondegenerate bilinear form on a finite dimensional real vector space $V$. There exists a basis in which

$$[B] = \operatorname{diag} (\underbrace{1,\ldots,1}_{p}, \underbrace{-1,\ldots,-1}_{q}),$$

and the pair $(p,q)$ is independent of the chosen diagonalizing basis. \end{theorem}

Thus the signature is an invariant of the form.

A Lorentzian vector space is one with exactly one negative direction under our sign convention.

\subsection{Subspaces}

Let $W\subset V$.

\begin{definition} A subspace $W$ is: \begin{itemize} \item \emph{spacelike} if $g|_W$ is positive definite; \item \emph{timelike} if $g|_W$ is nondegenerate and Lorentzian; \item \emph{null} if $g|_W$ is degenerate. \end{itemize} \end{definition}

For a hyperplane $H\subset V$:

\begin{itemize} \item $H$ is spacelike iff its normal line is timelike; \item $H$ is timelike iff its normal line is spacelike; \item $H$ is null iff its normal line is null and lies inside $H$. \end{itemize}

\subsection{A null hyperplane}

\begin{figure}[htbp] \centering \begin{tikzpicture}[scale=1] \coordinate (O) at (0,0); \draw[->] (O)--(0,3) node[above] {$t$}; \draw[->] (O)--(3,0) node[right] {$x$}; \draw[->] (O)--(-1.7,-1.3) node[below left] {$y$};

\fill[LorentzPurple!10,opacity=.9] (0,0)--(2.4,2.4)--(1.1,1.4)--(-1.3,-1.0)--cycle; \draw[LorentzPurple,thick] (0,0)--(2.4,2.4)--(1.1,1.4)--(-1.3,-1.0)--cycle;

\draw[-{Latex[length=3mm]},very thick,ChangeRed] (O)--(1.55,1.55) node[above right] {$k$};

\node[LorentzPurple] at (1.75,0.55) {$k^\perp$}; \node at (1.7,-0.65) {$k\in k^\perp$}; \end{tikzpicture} \caption{A schematic null hyperplane in $1+2$ dimensions. Its null normal $k$ is also tangent to the hyperplane.} \label{fig:null-hyperplane} \end{figure}

\subsection{Reverse Cauchy--Schwarz}

Ordinary Cauchy--Schwarz says, for a positive-definite inner product,

$$|g(u,v)| \le \sqrt{g(u,u)}\sqrt{g(v,v)}.$$

For future-directed timelike vectors in Lorentzian geometry, the inequality reverses.

\begin{proposition}[Reverse Cauchy--Schwarz] Let $u,v$ be future-directed timelike vectors in a Lorentzian vector space with signature $(-+\cdots+)$. Then

$$\boxed{ -g(u,v) \ge \sqrt{-g(u,u)}\sqrt{-g(v,v)}. }$$

Equality holds iff $u$ and $v$ are positive scalar multiples of one another. \end{proposition}

\begin{proof} Choose a Lorentz-orthonormal basis in which

$$u=(a,0,\ldots,0), \qquad a=\sqrt{-g(u,u)}>0.$$

Write

$$v=(v^0,\vec v), \qquad v^0>0.$$

Then

$$-g(u,v)=av^0.$$

Since $v$ is timelike,

$$-(v^0)^2+|\vec v|^2<0,$$

so

$$(v^0)^2 = -g(v,v)+|\vec v|^2 \ge -g(v,v).$$

Hence

$$-g(u,v) = av^0 \ge a\sqrt{-g(v,v)}.$$

Equality occurs precisely when $\vec v=0$, i.e.\ when $v$ is a positive multiple of $u$. \end{proof}

\begin{lorentzbox} The reverse Cauchy--Schwarz inequality is a statement about vectors in the same time cone. Without the common time-orientation hypothesis, the sign of $g(u,v)$ changes and the displayed inequality is not the correct formulation. \end{lorentzbox}

\subsection{Timelike, null, and spacelike vectors in one picture}

\begin{figure}[htbp] \centering \begin{tikzpicture}[scale=1.05] \draw[->] (-3,0)--(3,0) node[right] {$x$}; \draw[->] (0,-2.5)--(0,2.7) node[above] {$t$}; \draw[very thick,LorentzPurple] (-1.7,-2.4)--(1.7,2.4); \draw[very thick,LorentzPurple] (1.7,-2.4)--(-1.7,2.4);

\draw[-{Latex[length=3mm]},very thick,CommonBlue] (0,0)--(0.45,1.75) node[right] {timelike}; \draw[-{Latex[length=3mm]},very thick,LorentzPurple] (0,0)--(1.3,1.84) node[right] {null}; \draw[-{Latex[length=3mm]},very thick,IndefOrange] (0,0)--(2.15,0.8) node[right] {spacelike}; \end{tikzpicture} \caption{The three causal types are determined by the position of a vector relative to the light cone.} \label{fig:three-causal-types} \end{figure}

% ============================================================ \section{Proper time} % ============================================================

Let

$$\gamma:I\to M$$

be a future-directed timelike curve.

\begin{definition}[Proper time] Its proper time between parameters $a$ and $b$ is

$$\boxed{ \tau(\gamma) = \int_a^b \sqrt{-g(\dot\gamma,\dot\gamma)}\,dt. }$$

\end{definition}

The integrand is real and strictly positive because

$$g(\dot\gamma,\dot\gamma)<0.$$

If $\gamma$ is parametrized so that

$$g(\dot\gamma,\dot\gamma)=-1,$$

then the parameter itself measures proper time:

$$\tau=b-a.$$

\begin{lorentzbox} Proper time is defined for timelike curves and has a causal interpretation: it measures the elapsed time recorded along the worldline.

A Riemannian curve has an arc length, but there is no Riemannian future/past causal interpretation corresponding to proper time. \end{lorentzbox}

\subsection{Three notions of parameter}

It is essential to distinguish:

\begin{description} \item[Parameter time.] An arbitrary parameter $t$ used to describe the curve.

\item[Proper time.] For a timelike curve, a parameter $\tau$ satisfying

$$g\!\left(\frac{d\gamma}{d\tau}, \frac{d\gamma}{d\tau}\right)=-1.$$

\item[Affine parameter.] A parameter for which a geodesic satisfies

$$\nabla_{\dot\gamma}\dot\gamma=0.$$

\end{description}

A timelike geodesic can be affinely parametrized by proper time. A null geodesic cannot: for every parametrization,

$$g(\dot\gamma,\dot\gamma)=0.$$

Null geodesics therefore require affine parameters not normalized by proper time.

% ============================================================ \section{Lorentzian geodesics: same equation, different geometry} % ============================================================

The equation is unchanged:

$$\boxed{ \nabla_{\dot\gamma}\dot\gamma=0. }$$

In coordinates,

$$\boxed{ \ddot x^k+\Gamma^k{}_{ij}\dot x^i\dot x^j=0. }$$

What changes is the possible causal character of $\dot\gamma$.

\begin{proposition}[Causal type is constant along an affine geodesic] Let $\gamma$ be an affinely parametrized geodesic for the Levi-Civita connection. Then

$$g(\dot\gamma,\dot\gamma)$$

is constant. \end{proposition}

\begin{proof} Metric compatibility gives

$$\frac{d}{dt}g(\dot\gamma,\dot\gamma) = 2g(\nabla_{\dot\gamma}\dot\gamma,\dot\gamma) = 0.$$

\end{proof}

Thus a nonconstant geodesic is everywhere of one type:

$$\boxed{ \text{timelike}, \qquad \text{null}, \qquad \text{or spacelike}. }$$

\begin{commonbox} The differential equation defining a geodesic is the same.

The Lorentzian distinction arises because the conserved quantity $g(\dot\gamma,\dot\gamma)$ can be negative, zero, or positive. \end{commonbox}

\subsection{Variational interpretation}

In Riemannian geometry, sufficiently short geodesic segments minimize length.

In Lorentzian geometry:

\begin{itemize} \item a timelike geodesic segment in a sufficiently small normal neighborhood locally \emph{maximizes} proper time among nearby causal curves with the same endpoints; \item null geodesics have zero proper time, so proper-time maximization does not characterize them; \item spacelike geodesics are stationary for the appropriate spacelike length or energy functionals but do not obey a universal shortest-path principle analogous to the Riemannian one. \end{itemize}

\begin{warningbox} Geodesic'' does not mean shortest curve.''

The definition is

$$\nabla_{\dot\gamma}\dot\gamma=0.$$

Any minimizing or maximizing property is a further theorem with additional hypotheses. \end{warningbox}

\subsection{Three types of curve}

\begin{figure}[htbp] \centering \begin{tikzpicture}[scale=1] \draw[->] (-3,0)--(3,0) node[right] {$x$}; \draw[->] (0,-2.6)--(0,2.7) node[above] {$t$}; \draw[LorentzPurple,thick] (-1.75,-2.5)--(1.75,2.5); \draw[LorentzPurple,thick] (1.75,-2.5)--(-1.75,2.5);

\draw[CommonBlue,very thick,-{Latex[length=2.5mm]}] (-0.2,-2.0) .. controls (-0.4,-0.8) and (0.1,0.3) .. (0.35,2.0); \node[CommonBlue] at (-1.0,1.3) {timelike};

\draw[LorentzPurple,very thick,-{Latex[length=2.5mm]}] (-1.35,-1.9)--(1.25,1.8); \node[LorentzPurple] at (1.75,1.2) {null};

\draw[IndefOrange,very thick,-{Latex[length=2.5mm]}] (-2.2,-0.6) .. controls (-0.8,-0.2) and (0.9,0.15) .. (2.1,0.55); \node[IndefOrange] at (1.7,-0.6) {spacelike}; \end{tikzpicture} \caption{A schematic timelike, null, and spacelike curve in a spacetime diagram. Their tangents remain respectively inside, on, or outside the local light cones.} \label{fig:three-curves} \end{figure}

% ============================================================ \section{A crucial warning about distance} % ============================================================

\biginsight{Lorentzian geometry does not behave like metric geometry in the Riemannian sense.}

A Riemannian metric produces a positive length for every nonconstant piecewise smooth curve and hence a metric-space distance

$$d(p,q)=\inf_\gamma L(\gamma).$$

A Lorentzian metric does not do this.

\begin{warningbox} The phrase \emph{Lorentzian metric} uses ``metric'' in the differential-geometric sense of a nondegenerate symmetric bilinear form. It is not, by itself, a metric-space distance function. \end{warningbox}

For a future-directed causal curve,

$$L(\gamma) = \int \sqrt{-g(\dot\gamma,\dot\gamma)}\,dt.$$

Null segments contribute zero.

One may define the \emph{Lorentzian time separation}

$$\tau(p,q) = \sup \left\{ L(\gamma): \gamma\text{ future-directed causal from }p\text{ to }q \right\},$$

with the convention

$$\tau(p,q)=0$$

if no such causal curve exists.

Unlike a metric-space distance, $\tau$ may have the following features:

\begin{itemize} \item $\tau(p,q)\neq\tau(q,p)$; \item distinct points may satisfy $\tau(p,q)=0$; \item $\tau(p,q)$ can be infinite on poorly behaved spacetimes; \item along suitable causal chains, the natural inequality is a \emph{reverse} triangle inequality rather than the ordinary one. \end{itemize}

Additional causal hypotheses improve the behavior. For example, global hyperbolicity implies strong finiteness and continuity properties of the time-separation function. Such hypotheses are \emph{extra global assumptions}; they do not follow merely from the existence of a Lorentzian metric.

\begin{changebox} In Riemannian geometry, metric distance organizes much of the global theory.

In Lorentzian geometry, causal relations and light-cone structure take over much of that organizing role. \end{changebox}

% ============================================================ \section{Null geometry} % ============================================================

The existence of nonzero vectors $k$ satisfying

$$g(k,k)=0$$

has no Riemannian analogue.

\subsection{Null curves}

A smooth curve is null if

$$g(\dot\gamma,\dot\gamma)=0$$

and $\dot\gamma\neq0$ everywhere.

Null curves have zero Lorentzian proper time:

$$\int\sqrt{-g(\dot\gamma,\dot\gamma)}\,dt=0.$$

Thus they are geometrically nontrivial despite having zero proper-time length.

\subsection{Null hypersurfaces}

Let $\mathcal H\subset M$ be a hypersurface. Suppose locally

$$\mathcal H=\{F=0\}, \qquad dF\neq0.$$

Define $\grad F$ by

$$g(\grad F,\cdot)=dF.$$

Then

$$T_p\mathcal H = (\grad F_p)^\perp.$$

\begin{definition}[Null hypersurface] A hypersurface $\mathcal H$ is \emph{null} if the induced bilinear form

$$g|_{T\mathcal H}$$

is degenerate. \end{definition}

Equivalently, its normal vector is null.

If $K$ is a nonzero null normal,

$$K\in K^\perp=T_p\mathcal H.$$

Thus $K$ is simultaneously normal and tangent.

\begin{lorentzbox} For a null hypersurface,

$$\boxed{\text{normal direction}=\text{a tangent null direction}.}$$

This phenomenon cannot occur for Riemannian hypersurfaces. \end{lorentzbox}

The induced metric on a null hypersurface has a one-dimensional radical:

$$\rad(T_p\mathcal H)=\Span(K_p).$$

\subsection{Null generators}

Locally choose a nonzero null field $K$ spanning

$$\rad(T\mathcal H).$$

Its integral curves are called \emph{null generators}.

\begin{proposition} For a null hypersurface and its Levi-Civita connection,

$$\nabla_KK = fK$$

for some scalar function $f$. Hence the null generators are pregeodesics and can locally be reparametrized as affine null geodesics. \end{proposition}

\begin{proof} Let $X$ be tangent to $\mathcal H$. Since

$$g(K,X)=0,$$

differentiate along $K$:

$$0 = K[g(K,X)] = g(\nabla_KK,X)+g(K,\nabla_KX).$$

Because the connection is torsion-free,

$$\nabla_KX=\nabla_XK+[K,X].$$

Both $K$ and $X$ are tangent to $\mathcal H$, so $[K,X]$ is tangent and

$$g(K,[K,X])=0.$$

Also

$$g(K,\nabla_XK) = \frac12X[g(K,K)] = 0.$$

Thus

$$g(\nabla_KK,X)=0 \qquad \forall X\in T\mathcal H.$$

Therefore

$$\nabla_KK\in(T\mathcal H)^\perp=\Span(K),$$

as claimed. \end{proof}

% ============================================================ \section{Riemannian and Lorentzian hypersurfaces} % ============================================================

Let $S\subset(M,g)$ be a hypersurface with nonzero normal $N$.

The causal type of $N$ determines the signature of the induced metric on $TS=N^\perp$.

\begin{center} \begin{tabular}{lll} \toprule Normal $N$ & Hypersurface $S$ & Induced metric \ \midrule timelike & spacelike & Riemannian \ spacelike & timelike & Lorentzian \ null & null & degenerate \ \bottomrule \end{tabular} \end{center}

\begin{figure}[htbp] \centering \begin{tikzpicture}[scale=1] \draw[->] (-3.2,0)--(3.2,0) node[right] {$x$}; \draw[->] (0,-2.7)--(0,2.8) node[above] {$t$}; \draw[LorentzPurple!55,thin] (-1.8,-2.5)--(1.8,2.5); \draw[LorentzPurple!55,thin] (1.8,-2.5)--(-1.8,2.5);

\draw[CommonBlue,very thick] (-2.4,1.45)--(2.4,1.45); \node[CommonBlue] at (2.1,1.75) {spacelike hypersurface};

\draw[IndefOrange,very thick] (-1.7,-2.1)--(-1.4,2.2); \node[IndefOrange,align=left] at (-2.0,-2.45) {timelike\hypersurface};

\draw[LorentzPurple,very thick] (-1.6,-2.2)--(1.6,2.2); \node[LorentzPurple] at (1.9,-1.45) {null hypersurface}; \end{tikzpicture} \caption{In $1+1$ dimensions hypersurfaces are curves. Horizontal spacelike, vertical-ish timelike, and $45^\circ$ null examples show the three possibilities.} \label{fig:hypersurfaces} \end{figure}

\begin{changebox} For Riemannian metrics every embedded hypersurface inherits a positive-definite metric.

For Lorentzian metrics an induced metric may be Riemannian, Lorentzian, or degenerate. The null case is genuinely new. \end{changebox}

% ============================================================ \section{Parallel transport in Lorentzian geometry} % ============================================================

Suppose $V,W$ are parallel along $\gamma$:

$$\frac{DV}{dt}=0, \qquad \frac{DW}{dt}=0.$$

\begin{proposition}[Metric preservation under parallel transport] For the Levi-Civita connection,

$$\frac{d}{dt}g(V,W)=0.$$

\end{proposition}

\begin{proof} Metric compatibility gives

$$\frac{d}{dt}g(V,W) = g\!\left(\frac{DV}{dt},W\right) + g\!\left(V,\frac{DW}{dt}\right) = 0.$$

\end{proof}

In particular,

$$g(V,V)$$

is constant.

\begin{corollary}[Causal character is preserved] Lorentzian parallel transport preserves whether a nonzero vector is timelike, null, or spacelike. \end{corollary}

\begin{proof} If $V$ is parallel, then

$$g(V,V)=\text{constant}.$$

Its sign therefore cannot change. \end{proof}

\begin{commonbox} Metric-compatible parallel transport preserves the metric in every semi-Riemannian signature. \end{commonbox}

\begin{lorentzbox} What is specifically Lorentzian is the interpretation of the preserved quantity: preserving $g(V,V)$ also preserves causal character. \end{lorentzbox}

One should therefore not picture Lorentzian parallel transport merely as ``rotating vectors on a Euclidean unit sphere.'' It preserves an indefinite bilinear form and its associated hyperboloids and light cone.

% ============================================================ \section{Pseudo-orthonormal frames and normal coordinates} % ============================================================

At any $p$ in a Lorentzian $n$-manifold, there exists a basis

$$e_0,e_1,\ldots,e_{n-1}$$

of $T_pM$ satisfying

$$g_p(e_0,e_0)=-1,$$ $$g_p(e_i,e_j)=\delta_{ij}, \qquad i,j\ge1,$$

and

$$g_p(e_0,e_i)=0.$$

Such a basis is \emph{pseudo-orthonormal}.

\begin{semibox} Pseudo-orthonormal frames are the indefinite-signature analogue of orthonormal frames. The construction follows from the same nondegenerate bilinear-form theory. \end{semibox}

\subsection{Normal coordinates}

For the Levi-Civita connection of any semi-Riemannian metric, one may choose normal coordinates centered at $p$ such that

$$g_{ij}(p)=\eta_{ij}$$

with

$$\eta=\operatorname{diag}(-1,1,\ldots,1)$$

in the Lorentzian case, and

$$\Gamma^k{}_{ij}(p)=0.$$

Equivalently,

$$\partial_\ell g_{ij}(p)=0.$$

\begin{warningbox}

$$\boxed{\Gamma^k{}_{ij}(p)=0}$$

at one point does not imply that the metric is flat near that point.

Normal coordinates eliminate first-order connection coefficients at the center. Curvature is a second-order obstruction and may remain nonzero. \end{warningbox}

This statement is again common to Riemannian and Lorentzian geometry.

% ============================================================ \section{Curvature: common formalism, Lorentzian interpretation} % ============================================================

For any affine connection,

$$R(X,Y)Z = \nabla_X\nabla_YZ - \nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z.$$

For a Levi-Civita connection, lower the final index:

$$R(X,Y,Z,W) = g(R(X,Y)Z,W).$$

The standard algebraic symmetries remain valid: \begin{align*} R(X,Y,Z,W) &=-R(Y,X,Z,W),\ R(X,Y,Z,W) &=-R(X,Y,W,Z),\ R(X,Y,Z,W) &=R(Z,W,X,Y), \end{align*} together with the first Bianchi identity

$$R(X,Y)Z+R(Y,Z)X+R(Z,X)Y=0.$$

\begin{commonbox} The formal curvature theory of a Levi-Civita connection is essentially the same in every nondegenerate signature. \end{commonbox}

\subsection{Ricci and scalar curvature}

The Ricci tensor is the contraction

$$\operatorname{Ric}_{ij} = R^k{}_{ikj},$$

up to the chosen curvature-index convention.

The scalar curvature is

$$\operatorname{Scal} = g^{ij}\operatorname{Ric}_{ij}.$$

The formulas are formally unchanged. In Lorentzian geometry, contractions involve an indefinite inverse metric, so sign intuition from positive-definite geometry must be used cautiously.

\subsection{Sectional curvature}

For a nondegenerate $2$-plane

$$\Pi=\Span(u,v),$$

define

$$K(\Pi) = \frac{g(R(u,v)v,u)} {g(u,u)g(v,v)-g(u,v)^2}.$$

The denominator is the determinant of the Gram matrix of $u,v$. It is nonzero exactly when $g|_\Pi$ is nondegenerate.

In Lorentzian signature, a nondegenerate $2$-plane is either:

\begin{itemize} \item \emph{spacelike}: $g|_\Pi$ is positive definite; \item \emph{timelike} or \emph{mixed}: $g|_\Pi$ has signature $(-+)$. \end{itemize}

A null $2$-plane is degenerate, and the displayed sectional-curvature formula is not defined for it.

\begin{changebox} Sectional curvature is not attached indiscriminately to every $2$-plane in Lorentzian geometry. The plane must be nondegenerate. Null planes require different invariants and techniques. \end{changebox}

% ============================================================ \section{Worked examples} % ============================================================

\subsection{Example 1: Euclidean space --- the Riemannian baseline}

Let

$$M=\R^n, \qquad g=\sum_{i=1}^n(dx^i)^2.$$

The coefficients are constant:

$$g_{ij}=\delta_{ij}.$$

Hence

$$\partial_k g_{ij}=0$$

and therefore

$$\boxed{\Gamma^k{}_{ij}=0}$$

in Cartesian coordinates.

Thus

$$\nabla_XY = X^i\partial_iY^k\,\partial_k,$$

the ordinary directional derivative.

Geodesics satisfy

$$\ddot x^k=0,$$

so

$$x^k(t)=a^kt+b^k.$$ $$\boxed{\text{Riemannian baseline}}$$

The point of this example is not that Euclidean space has a ``special'' definition of covariant derivative; rather, its standard coordinates happen to make the flat Levi-Civita connection look like ordinary differentiation.

% ------------------------------------------------------------ \subsection{Example 2: Euclidean plane in polar coordinates}

On

$$\R^2\setminus\{0\}$$

use polar coordinates $(r,\theta)$:

$$g=dr^2+r^2d\theta^2.$$

Thus

$$(g_{ij}) = \begin{pmatrix} 1&0\\ 0&r^2 \end{pmatrix}, \qquad (g^{ij}) = \begin{pmatrix} 1&0\\ 0&r^{-2} \end{pmatrix}.$$

The only nonzero derivatives of the metric coefficients are

$$\partial_r g_{\theta\theta}=2r.$$

Hence

$$\Gamma^r{}_{\theta\theta} = -\frac12(2r) = -r,$$

and

$$\Gamma^\theta{}_{r\theta} = \Gamma^\theta{}_{\theta r} = \frac12r^{-2}(2r) = \frac1r.$$

Therefore

$$\boxed{ \nabla_{\partial_r}\partial_\theta = \frac1r\partial_\theta, }$$ $$\boxed{ \nabla_{\partial_\theta}\partial_r = \frac1r\partial_\theta, }$$

and

$$\boxed{ \nabla_{\partial_\theta}\partial_\theta = -r\partial_r. }$$

Yet the Euclidean plane is flat:

$$R=0.$$

\begin{keybox} Nonzero Christoffel symbols do not imply nonzero curvature.

Christoffel symbols can be created by curvilinear coordinates even on flat space. \end{keybox}

% ------------------------------------------------------------ \subsection{Example 3: the round sphere}

On the unit sphere $S^2$, use coordinates

$$(\theta,\phi)$$

with metric

$$g=d\theta^2+\sin^2\theta\,d\phi^2.$$

Then

$$g_{\theta\theta}=1, \qquad g_{\phi\phi}=\sin^2\theta.$$

The representative nonzero Christoffel symbols are

$$\boxed{ \Gamma^\theta{}_{\phi\phi} = -\sin\theta\cos\theta }$$

and

$$\boxed{ \Gamma^\phi{}_{\theta\phi} = \Gamma^\phi{}_{\phi\theta} = \cot\theta. }$$

For the equator

$$\theta=\frac{\pi}{2}, \qquad \phi=t,$$

we have

$$\dot\theta=0, \qquad \dot\phi=1,$$

and

$$\Gamma^\theta{}_{\phi\phi} = -\sin(\pi/2)\cos(\pi/2) = 0.$$

Thus the equator satisfies the geodesic equation. More generally the geodesics of the round sphere are great circles.

Here nonzero Christoffel symbols encode genuine curved geometry, unlike the polar-coordinate example. Curvature, not $\Gamma$ alone, distinguishes the two cases.

% ------------------------------------------------------------ \subsection{Example 4: Minkowski space}

Let

$$\R^{1,n}$$

have coordinates

$$(t,x^1,\ldots,x^n)$$

and metric

$$\eta = -dt^2+\sum_{i=1}^n(dx^i)^2.$$

Its metric coefficients are constant, so

$$\boxed{\Gamma^\mu{}_{\alpha\beta}=0}$$

in Cartesian inertial coordinates.

The geodesics are affine straight lines:

$$x^\mu(\lambda)=A^\mu\lambda+B^\mu.$$

For a vector

$$v=a\partial_t+\sum_i b^i\partial_{x^i},$$ $$\eta(v,v) = -a^2+|\mathbf b|^2.$$

Thus:

$$|a|>|\mathbf b| \quad\Longrightarrow\quad v\text{ timelike},$$ $$|a|=|\mathbf b| \quad\Longrightarrow\quad v\text{ null},$$ $$|a|<|\mathbf b| \quad\Longrightarrow\quad v\text{ spacelike}.$$

The connection theory looks exactly as simple as Euclidean space in Cartesian coordinates, but the causal geometry is completely different.

% ------------------------------------------------------------ \subsection{Example 5: flat Minkowski space in non-Cartesian coordinates}

Consider the right Rindler wedge

$$x>|t|.$$

Introduce coordinates

$$t=\rho\sinh\eta, \qquad x=\rho\cosh\eta, \qquad \rho>0.$$

Then

$$-dt^2+dx^2 = -\rho^2d\eta^2+d\rho^2.$$

Thus

$$g_{\eta\eta}=-\rho^2, \qquad g_{\rho\rho}=1.$$

The nonzero Christoffel symbols are

$$\boxed{ \Gamma^\rho{}_{\eta\eta}=\rho, \qquad \Gamma^\eta{}_{\rho\eta} = \Gamma^\eta{}_{\eta\rho} = \frac1\rho. }$$

They are nonzero even though the spacetime is still Minkowski space.

Indeed, the coordinate transformation explicitly identifies this metric with

$$-dt^2+dx^2,$$

so

$$R=0.$$ $$\boxed{ \Gamma\neq0 \quad\not\Rightarrow\quad R\neq0. }$$

This is the Lorentzian analogue of polar coordinates in the Euclidean plane.

% ------------------------------------------------------------ \subsection{Example 6: the light cone in $1+1$ dimensions}

In Minkowski space,

$$g=-dt^2+dx^2.$$

A curve

$$\gamma(t)=(t,x(t))$$

is null when

$$g(\dot\gamma,\dot\gamma)=0.$$

Since

$$\dot\gamma = \partial_t+\frac{dx}{dt}\partial_x,$$

we obtain

$$-1+\left(\frac{dx}{dt}\right)^2=0.$$

Hence

$$\boxed{ \frac{dx}{dt}=\pm1. }$$

Thus null curves with $t$ as parameter have slopes

$$x(t)=\pm t+C.$$

These are precisely the $45^\circ$ light rays in the standard spacetime diagram.

\begin{figure}[htbp] \centering \begin{tikzpicture}[scale=1] \draw[->] (-3,0)--(3,0) node[right] {$x$}; \draw[->] (0,-2.6)--(0,2.8) node[above] {$t$}; \draw[LorentzPurple,very thick,-{Latex[length=2.5mm]}] (-2.0,-2.0)--(2.0,2.0); \draw[LorentzPurple,very thick,-{Latex[length=2.5mm]}] (2.0,-2.0)--(-2.0,2.0); \node[LorentzPurple] at (2.25,1.6) {$dx/dt=+1$}; \node[LorentzPurple] at (-2.3,1.6) {$dx/dt=-1$}; \fill (0,0) circle (2pt); \end{tikzpicture} \caption{Null lines in $1+1$-dimensional Minkowski spacetime.} \label{fig:light-rays} \end{figure}

% ------------------------------------------------------------ \subsection{Example 7: null orthogonality explicitly}

In $1+2$-dimensional Minkowski space let

$$k=(1,1,0).$$

Then

$$g(k,k) = -1+1 = 0,$$

so $k$ is null.

Let

$$w=(a,b,c).$$

Then

$$g(k,w) = -a+b.$$

Therefore

$$w\in k^\perp \quad\Longleftrightarrow\quad b=a.$$

Hence

$$\boxed{ k^\perp = \{(a,a,c):a,c\in\R\}. }$$

In particular,

$$k=(1,1,0)\in k^\perp.$$

The restriction of the metric to $k^\perp$ is

$$g((a,a,c),(a,a,c)) = c^2.$$

Thus it is degenerate, with kernel

$$\{(a,a,0)\} = \Span(k).$$

This single computation captures the basic linear algebra underlying null hypersurfaces.

% ------------------------------------------------------------ \subsection{Example 8: a curved $1+1$-dimensional Lorentzian metric}

Consider

$$g=-dt^2+a(t)^2dx^2, \qquad a(t)>0.$$

The inverse metric is

$$(g^{ij}) = \begin{pmatrix} -1&0\\ 0&a^{-2} \end{pmatrix}.$$

The nonzero Christoffel symbols are

$$\boxed{ \Gamma^t{}_{xx}=aa', }$$

and

$$\boxed{ \Gamma^x{}_{tx} = \Gamma^x{}_{xt} = \frac{a'}a. }$$

Hence the geodesic equations are

$$\boxed{ \ddot t + aa'\dot x^2 = 0, }$$

and

$$\boxed{ \ddot x + 2\frac{a'}a\dot t\dot x = 0. }$$

The second equation can be rewritten

$$\frac{d}{d\lambda} \left(a(t)^2\dot x\right) = 0.$$

Thus

$$a(t)^2\dot x = C$$

is conserved along geodesics.

A direct curvature calculation gives

$$R(\partial_t,\partial_x)\partial_x = aa''\,\partial_t.$$

The sectional/Gaussian curvature is

$$\boxed{ K=\frac{a''}{a}. }$$

Hence the metric is nonflat whenever

$$a''\neq0.$$

This example separates two layers:

\begin{itemize} \item computing $\Gamma$, geodesics, and $R$ is ordinary semi-Riemannian connection theory; \item interpreting the geodesics as timelike, null, or spacelike is Lorentzian. \end{itemize}

% ============================================================ \section{A same-versus-different synthesis table} % ============================================================

\small \begin{longtable}{@{}p{0.16\textwidth}p{0.22\textwidth}p{0.22\textwidth}p{0.30\textwidth}@{}} \toprule \textbf{Concept} & \textbf{Riemannian} & \textbf{Lorentzian} & \textbf{What changes?} \ \midrule \endfirsthead

\toprule \textbf{Concept} & \textbf{Riemannian} & \textbf{Lorentzian} & \textbf{What changes?} \ \midrule \endhead

Connection & Arbitrary connection on $TM$ & Same definition & Nothing formal; no metric is needed. $$1ex]

Covariant derivative & $\nabla_XY$ & $\nabla_XY$ & Same operation and axioms. \[1ex]

Torsion & $T(X,Y)=\nabla_XY-\nabla_YX-[X,Y]$ & Same & Nothing signature-dependent. \[1ex]

Metric compatibility & $\nabla g=0$ & $\nabla g=0$ & Same equation; $g$ is indefinite in Lorentzian geometry. \[1ex]

Levi-Civita connection & Unique torsion-free metric connection & Unique torsion-free metric connection & Existence and uniqueness require nondegeneracy, not positivity. \[1ex]

Koszul formula & Same formula & Same formula & No formal change. \[1ex]

Christoffel symbols & Local coefficients, not tensors & Same & No transformation-theoretic change. \[1ex]

Tensor covariant derivative & Induced from $\nabla$ & Same & No signature dependence. \[1ex]

Parallel transport & Preserves inner products & Preserves Lorentzian inner products & Lorentzian transport also preserves causal type. \[1ex]

Geodesic equation & $\nabla_{\dot\gamma}\dot\gamma=0$ & Same & Lorentzian geodesics divide into timelike, null, spacelike. \[1ex]

Curvature & Same connection definition & Same & Metric contractions have indefinite signs; null planes require care. \[1ex]

$g(v,v)$ & Positive for $v\neq0$ & Negative, zero, or positive & Fundamental signature change. \[1ex]

Norm & $\sqrt{g(v,v)}$ is a norm & No global norm arises from $g$ & Nonzero null vectors satisfy $g(v,v)=0$. \[1ex]

Orthogonality & $v\notin v^\perp$ for $v\neq0$ & A null $k$ satisfies $k\in k^\perp$ & Null self-orthogonality appears. \[1ex]

Causal character & Absent & Timelike/null/spacelike & Genuinely Lorentzian. \[1ex]

Light cones & Absent & Present in every tangent space & Created by nonzero solutions of $g(v,v)=0$. \[1ex]

Time orientation & No analogue & Choice of future cone & Additional global Lorentzian structure. \[1ex]

Proper time & No causal analogue & $\int\sqrt{-g(\dot\gamma,\dot\gamma)}\,dt$ & Defined along timelike curves. \[1ex]

Null vectors & Only $0$ has $g(v,v)=0$ & Nonzero null vectors exist & Indefinite-signature phenomenon. \[1ex]

Null hypersurfaces & Do not occur & Induced metric is degenerate & Normal null direction is also tangent. \[1ex]

Induced metric on hypersurface & Always Riemannian & May be Riemannian, Lorentzian, or degenerate & Depends on causal type of normal. \[1ex]

Distance / causal structure & Metric-space distance from infimum of lengths & Causal relations and time separation & Lorentzian time separation can be asymmetric, zero, or infinite. \ \bottomrule \end{longtable} \normalsize

[ \boxed{ \text{same formal machinery} \quad+\quad \text{different metric algebra} \quad=\quad \text{Lorentzian geometry}. }

% ============================================================ \section{Common misconceptions and traps} % ============================================================ \begin{enumerate}[label=\textbf{\arabic*.}] \item \textbf{Misconception:} A connection is the same thing as Christoffel symbols. \textbf{Correction:} A connection is an invariant differential operator. Christoffel symbols are its coefficients in a chosen coordinate frame. \item \textbf{Misconception:} The Levi-Civita connection is the only connection. \textbf{Correction:} There are infinitely many affine connections. The Levi-Civita connection is the unique one satisfying

T=0, \qquad \nabla g=0

for a chosen nondegenerate metric. \item \textbf{Misconception:} A Koszul connection is the same as the Levi-Civita connection. \textbf{Correction:} A Koszul connection means a connection on a vector bundle. No metric is required. \item \textbf{Misconception:} The Koszul formula is specific to Riemannian geometry. \textbf{Correction:} It applies to every nondegenerate semi-Riemannian metric, including Lorentzian metrics. \item \textbf{Misconception:} Lorentzian covariant differentiation is a different operation from Riemannian covariant differentiation. \textbf{Correction:} The connection axioms are identical. What changes is the metric algebra and its geometric interpretation. \item \textbf{Misconception:} Christoffel symbols are tensor components. \textbf{Correction:} Their coordinate-change law contains an inhomogeneous second-derivative term. \item \textbf{Misconception:} Nonzero Christoffel symbols mean nonzero curvature. \textbf{Correction:} Polar coordinates in Euclidean space and Rindler coordinates in Minkowski space both give nonzero Christoffel symbols on flat spaces. \item \textbf{Misconception:} A Lorentzian metric defines lengths exactly as a Riemannian metric does. \textbf{Correction:} The sign of $g(v,v)$ depends on causal type, and a nonzero null vector has $g(v,v)=0$. \item \textbf{Misconception:} A null vector has zero length and therefore must be the zero vector. \textbf{Correction:} That implication uses positive definiteness. Lorentzian metrics are indefinite. \item \textbf{Misconception:} Orthogonal complements behave exactly as in Euclidean geometry. \textbf{Correction:} For a nonzero null vector $k$,

k\in k^\perp.

$$\item \textbf{Misconception:} Every Lorentzian manifold is time-orientable. \textbf{Correction:} Time orientation is extra global data. A Lorentzian manifold may fail to admit a globally consistent future cone. \item \textbf{Misconception:} Every nonzero tangent vector is either timelike or spacelike. \textbf{Correction:} There is a third possibility:$$

g(v,v)=0,\qquad v\neq0.

$$\item \textbf{Misconception:} Every geodesic is a shortest curve. \textbf{Correction:} Geodesics satisfy$$

\nabla_{\dot\gamma}\dot\gamma=0.

Timelike geodesics have a local \emph{maximizing} property for proper time under appropriate hypotheses, while null and spacelike cases have different variational interpretations. \item \textbf{Misconception:} Lorentzian distance behaves like an ordinary metric-space distance. \textbf{Correction:} Lorentzian time separation need not be symmetric, strictly positive, or finite. \item \textbf{Misconception:} A null hypersurface inherits a Lorentzian metric. \textbf{Correction:} Its induced metric is degenerate. \item \textbf{Misconception:} Passing from Riemannian to Lorentzian geometry requires a new definition of the Levi-Civita connection. \textbf{Correction:} The same theorem, Koszul formula, and Christoffel formula apply. The difference lies in the indefinite metric. \end{enumerate} % ============================================================ \section{Conceptual hierarchy: which hypothesis is doing the work?} % ============================================================ \begin{tcolorbox}[ breakable, colback=CommonBlue!5, colframe=CommonBlue, title=\textbf{Level I: true for an arbitrary connection} ]

\nabla_XY,\qquad T,\qquad R,\qquad \frac{DV}{dt}, \qquad \text{parallel transport}, \qquad \nabla_{\dot\gamma}\dot\gamma=0.

No metric is required. \end{tcolorbox} \begin{tcolorbox}[ breakable, colback=SemiGreen!5, colframe=SemiGreen, title=\textbf{Level II: requires a nondegenerate metric} ]

\nabla g=0, \qquad \text{Levi-Civita uniqueness/existence}, \qquad \text{Koszul formula},

$$$$

\text{metric-preserving parallel transport}, \qquad \text{raising/lowering indices}.

Positive definiteness is not required. \end{tcolorbox} \begin{tcolorbox}[ breakable, colback=LorentzPurple!6, colframe=LorentzPurple, title=\textbf{Level III: genuinely Lorentzian} ]

\text{timelike/null/spacelike}, \qquad \text{light cones}, \qquad \text{future/past},

$$$$

\text{time orientation}, \qquad \text{proper time}, \qquad \text{null hypersurfaces}, \qquad \text{causal structure}.

These arise from index-one indefinite signature. \end{tcolorbox} % ============================================================ \section{Exercises} % ============================================================ The exercises are arranged by role rather than by section. Concise hints appear afterward; complete solutions are intentionally omitted. % ------------------------------------------------------------ \subsection{Foundational} \begin{exercise} Let $\nabla$ be a connection on a vector bundle $E\to M$. Prove directly from the axioms that

\nabla_0s=0, \qquad \nabla_X0=0.

Then explain why $\nabla_Xs(p)$ depends only on $X_p$. \end{exercise} \begin{exercise} Suppose $\nabla$ and $\widetilde\nabla$ are two connections on $TM$. Define

A(X,Y)

\nabla_XY-\widetilde\nabla_XY.

Show that $A$ is $C^\infty(M)$-linear in both variables and hence is a $(1,2)$-tensor. \end{exercise} \begin{exercise} Show directly that the torsion

T(X,Y)

\nabla_XY-\nabla_YX-[X,Y]

is a tensor. \end{exercise} \begin{exercise} For a connection $\nabla$ and function $f$, determine

\nabla_X(df)

as a bilinear expression involving second derivatives of $f$. What extra symmetry appears when $\nabla$ is torsion-free? \end{exercise} \begin{exercise} Let $g$ be a nondegenerate metric. Show that

\nabla g=0

is equivalent to the assertion that every parallel transport map is a linear isometry between the corresponding tangent spaces. \end{exercise} % ------------------------------------------------------------ \subsection{Computational} \begin{exercise} For the Euclidean plane in polar coordinates

g=dr^2+r^2d\theta^2,

$$rederive all Christoffel symbols and verify explicitly that$$

R(\partial_r,\partial_\theta)\partial_\theta=0.

\end{exercise} \begin{exercise} For the round sphere

g=d\theta^2+\sin^2\theta,d\phi^2,

derive the two independent nonzero Christoffel symbols and write both geodesic equations. \end{exercise} \begin{exercise} Verify directly from the sphere geodesic equations that the equator is a geodesic. What happens for a curve of constant latitude $\theta=\theta_0\neq\pi/2$? \end{exercise} \begin{exercise} For the Rindler metric

g=-\rho^2d\eta^2+d\rho^2,

compute all Christoffel symbols and verify by direct calculation that the curvature vanishes. \end{exercise} \begin{exercise} For

g=-dt^2+a(t)^2dx^2,

$$derive$$

a(t)^2\dot x=\text{constant}

along affine geodesics. \end{exercise} \begin{exercise} Take

a(t)=e^{Ht}, \qquad H\neq0.

$$Compute the curvature of$$

g=-dt^2+e^{2Ht}dx^2.

Is the spacetime flat? \end{exercise} % ------------------------------------------------------------ \subsection{Proof exercises} \begin{exercise} Starting only from metric compatibility and torsion-freeness, derive the Koszul formula carefully. At each step identify where symmetry of $g$ is used. \end{exercise} \begin{exercise} Use the Koszul formula to prove uniqueness of the Levi-Civita connection without using coordinates. \end{exercise} \begin{exercise} Use the Koszul formula as a definition to prove existence of the Levi-Civita connection. In particular, verify explicitly the two connection identities

\nabla_{fX}Y=f\nabla_XY, \qquad \nabla_X(fY)=X(f)Y+f\nabla_XY.

\end{exercise} \begin{exercise} Prove that if $V,W$ are parallel along a curve for a metric-compatible connection, then

g(V,W)

is constant. Deduce that Lorentzian parallel transport preserves the entire causal classification. \end{exercise} \begin{exercise} Let $\gamma$ be a Levi-Civita geodesic. Prove that

g(\dot\gamma,\dot\gamma)

is constant. Explain why this statement simultaneously implies: constant Riemannian speed and constant Lorentzian causal type. \end{exercise} \begin{exercise} Prove that curvature is tensorial in all three input vector fields. Pay special attention to the cancellation of second derivatives of a scalar function. \end{exercise} % ------------------------------------------------------------ \subsection{Lorentzian exercises} \begin{exercise} In $\R^{1,2}$ with

g=-dt^2+dx^2+dy^2,

$$classify each vector:$$

(1,0,0),\qquad (1,1,0),\qquad (1,2,0),\qquad (2,1,1).

\end{exercise} \begin{exercise} Let

k=(1,1,0)

in $\R^{1,2}$. Compute $k^\perp$, determine the radical of the restricted metric, and find a spacelike vector in $k^\perp$ that is not proportional to $k$. \end{exercise} \begin{exercise} Prove that the timelike cone in Minkowski space has exactly two connected components. \end{exercise} \begin{exercise} Let $u,v$ be future-directed timelike vectors in Minkowski space. Prove the reverse Cauchy--Schwarz inequality

-g(u,v) \ge \sqrt{-g(u,u)}\sqrt{-g(v,v)}

by transforming $u$ to a multiple of the time-axis. \end{exercise} \begin{exercise} Show that the set of future-directed timelike vectors in a Lorentzian vector space is convex. \end{exercise} \begin{exercise} For a timelike curve in Minkowski space

\gamma(t)=(t,x(t)), \qquad |x'(t)|<1,

$$show that its proper time is$$

\tau

\int \sqrt{1-(x'(t))^2},dt.

Compare it to the coordinate-time interval. \end{exercise} \begin{exercise} In $1+1$-dimensional Minkowski space, show that every affine null geodesic is a straight line of slope $\pm1$. \end{exercise} \begin{exercise} Let

\mathcal H

{(t,x,y):t=x} \subset\R^{1,2}.

Show that $\mathcal H$ is a null hypersurface and compute the induced degenerate metric explicitly. \end{exercise} \begin{exercise} Suppose $S$ is a hypersurface in a Lorentzian manifold with timelike unit normal $N$. Prove that the induced metric on $S$ is positive definite. \end{exercise} \begin{exercise} Suppose a Lorentzian manifold admits a smooth global timelike vector field. Explain how it determines a time orientation. Then explain why the existence of a Lorentzian metric alone does not specify one of the two cone components as ``future.'' \end{exercise} % ------------------------------------------------------------ \subsection{Challenge exercises} \begin{exercise} Let $g$ be Lorentzian and $S=\{F=0\}$ a regular hypersurface. Show that: \begin{enumerate} \item $S$ is spacelike iff $g(\grad F,\grad F)<0$; \item $S$ is timelike iff $g(\grad F,\grad F)>0$; \item $S$ is null iff $g(\grad F,\grad F)=0$ along $S$. \end{enumerate} \end{exercise} \begin{exercise} Let $K$ span the radical of a null hypersurface. Prove in detail that

\nabla_KK=fK.

Then find a reparametrization of an integral curve of $K$ making it an affinely parametrized null geodesic. \end{exercise} \begin{exercise} Let $g=-dt^2+a(t)^2dx^2$. Show that

K=\frac{a''(t)}{a(t)}

is its Gaussian curvature. Determine all positive functions $a(t)$ for which this metric is flat on an interval. \end{exercise} \begin{exercise} Show that for any two affine connections

R-\widetilde R

$$can be written entirely in terms of the tensor$$

A(X,Y)=\nabla_XY-\widetilde\nabla_XY

and its covariant derivative. This is the first step toward comparing curvatures of different connections. \end{exercise} \begin{exercise} Investigate the quotient of two-dimensional Minkowski space by the isometry

(t,x)\longmapsto(-t,x+1).

Show that the Lorentzian metric descends to the quotient. Analyze why a globally consistent future direction cannot be chosen. \end{exercise} % ============================================================ \section{Hints to the exercises} % ============================================================ \subsection*{Foundational hints} \begin{enumerate}[label=\textbf{F\arabic*.}] \item Use $0=0X$ and the $C^\infty$-linearity in the first slot. \item Expand both connection axioms for $A(fX,Y)$ and $A(X,fY)$; the derivative terms cancel. \item Use

[fX,Y]=f[X,Y]-Y(f)X.

$$\item Start from$$

(\nabla_Xdf)(Y)

X(Yf)-df(\nabla_XY).

Compare the expression with $X$ and $Y$ exchanged. \item Parallel-transport two initial vectors and differentiate their inner product. \end{enumerate} \subsection*{Computational hints} \begin{enumerate}[label=\textbf{C\arabic*.}] \item Only

\Gamma^r{}{\theta\theta}, \quad \Gamma^\theta{}{r\theta}, \quad \Gamma^\theta{}_{\theta r}

are nonzero. \item Differentiate $g_{\phi\phi}=\sin^2\theta$. \item Put

\dot\theta=0

into the $\theta$-equation. \item Use

\Gamma^\rho{}{\eta\eta}=\rho, \qquad \Gamma^\eta{}{\rho\eta}=1/\rho.

\item Multiply the $x$-equation by $a^2$. \item Use

a''/a=H^2.

\end{enumerate} \subsection*{Proof hints} \begin{enumerate}[label=\textbf{P\arabic*.}] \item Write three metric-compatibility identities, add two, subtract one, then replace antisymmetric covariant-derivative combinations with Lie brackets. \item The Koszul right-hand side fixes

g(\nabla_XY,Z)

for every $Z$. \item First prove the Koszul right-hand side is $C^\infty$-linear in $Z$. Nondegeneracy then supplies the vector $\nabla_XY$. \item Differentiate $g(V,W)$ and use $\nabla g=0$. \item Apply the preceding idea with

V=W=\dot\gamma.

$$\item Expand$$

R(X,Y)(fZ)

and watch the $X(Yf)$, $Y(Xf)$, and $[X,Y]f$ terms cancel. \end{enumerate} \subsection*{Lorentzian hints} \begin{enumerate}[label=\textbf{L\arabic*.}] \item Compute

-t^2+x^2+y^2.

$$\item Solve$$

-a+b=0.

\item Write a timelike vector as $(t,\mathbf x)$ and note

|t|>|\mathbf x|.

The sign of $t$ cannot change continuously without leaving the timelike set. \item Normalize $u$ by a Lorentz transformation. \item In a frame where both vectors are future-directed, use the reverse Cauchy--Schwarz inequality or a direct cone estimate. \item Substitute

g(\dot\gamma,\dot\gamma)

-1+(x')^2.

\item With $\Gamma=0$, affine geodesics are straight lines; impose the null condition. \item Tangent vectors satisfy

\delta t=\delta x.

$$Restrict$$

-dt^2+dx^2+dy^2.

\item The orthogonal complement of a timelike vector is positive definite in Lorentzian signature. \item Declare the cone containing the chosen timelike vector field to be future. \end{enumerate} \subsection*{Challenge hints} \begin{enumerate}[label=\textbf{Ch\arabic*.}] \item Use

T_pS=(\grad F)^\perp

$$and classify orthogonal complements by the causal type of the normal. \item If$$

\nabla_KK=fK,

$$rescale the tangent field along each generator to remove the proportionality factor. \item Flatness means$$

a''=0.

Remember also the condition $a>0$. \item Expand

\nabla=\widetilde\nabla+A

inside the definition of curvature. \item Transport a candidate future-directed timelike vector once around the quotient loop and observe that the identification reverses the time component. \end{enumerate} % ============================================================ \section{Final conceptual synthesis} % ============================================================ The subject is best understood as a sequence of added structures. First comes the smooth manifold:

M.

$$A manifold has tangent spaces, vector fields, curves, Lie brackets, and differential forms, but no canonical way to differentiate one vector field in the direction of another. Next comes a vector bundle:$$

E\to M.

$$A Koszul connection gives differentiation of sections:$$

\nabla_Xs.

$$Specializing to$$

E=TM

$$produces an affine connection and therefore the fundamental connection-theoretic constructions:$$

T,\qquad R,\qquad \frac{DV}{dt}, \qquad \text{parallel transport}, \qquad \text{geodesics}.

Only after this do we add a nondegenerate metric $g$. Requiring

\nabla g=0

$$and$$

T=0

$$selects the unique Levi-Civita connection. Thus:$$

\boxed{ \begin{array}{c} \text{smooth manifold }M\[1mm] \Downarrow\ \text{vector bundle }E\to M\[1mm] \Downarrow\ \text{Koszul connection}\[1mm] \Downarrow\ \text{covariant derivative}\[1mm] \Downarrow\ E=TM:\ \text{linear/affine connection}\[1mm] \Downarrow\ \text{add a nondegenerate metric }g\[1mm] \Downarrow\ \nabla g=0,\quad T=0\[1mm] \Downarrow\ \text{Levi-Civita connection}\[1mm] \Downarrow\ \text{parallel transport, geodesics, curvature}. \end{array}}

Up to this stage, positive definiteness has not played the decisive role one might have expected. The branching occurs in the algebra of $g$.

\boxed{ \begin{array}{ccc} \text{\bfseries Riemannian} && \text{\bfseries Lorentzian} \[1mm] g>0 && g\text{ indefinite} \ \downarrow && \downarrow \ \text{ordinary orthogonality} && \text{causal orthogonality} \ \text{positive norms} && \text{timelike/null/spacelike} \ \text{metric distance} && \text{causal structure} \ \text{no nonzero null vectors} && \text{light cones} \ && \text{time orientation} \ && \text{proper time} \ && \text{null hypersurfaces}. \end{array}}

$$The most important intellectual shift is therefore not$$

+\longrightarrow-

$$in some coordinate formula. It is the transition$$

g(v,v)>0\quad(v\neq0)

$$to the three-way Lorentzian possibility$$

g(v,v)<0, \qquad g(v,v)=0, \qquad g(v,v)>0.

$$Once nonzero null vectors exist, a chain of new phenomena follows:$$

\boxed{ \begin{array}{c} \text{indefinite metric}\ \Downarrow\ \text{nonzero null vectors}\ \Downarrow\ \text{light cones}\ \Downarrow\ \text{causal classification}\ \Downarrow\ \text{future/past choice}\ \Downarrow\ \text{causal curves and proper time}\ \Downarrow\ \text{null geodesics and null hypersurfaces}\ \Downarrow\ \text{global causal geometry}. \end{array}}

$$At the same time, the familiar differential-geometric backbone remains intact:$$

\boxed{ \begin{array}{c} \text{Koszul formula}\ \text{Christoffel formula}\ \text{tensor covariant derivative}\ \text{parallel transport equation}\ \text{geodesic equation}\ \text{curvature definition} \end{array} \quad \text{are formally the same in Riemannian and Lorentzian geometry.} }

\begin{keybox} When encountering a new theorem in Lorentzian geometry, ask three questions in order: \begin{enumerate} \item Does this use only a connection? \item Does it use a nondegenerate metric but not its signature? \item Does it use light cones, causal type, or some consequence of indefinite index-one signature? \end{enumerate} These questions usually reveal whether the statement is connection-theoretic, semi-Riemannian, or genuinely Lorentzian. \end{keybox} \biginsight{The Levi-Civita and covariant-derivative machinery is shared; the causal geometry is genuinely Lorentzian.} \end{document}