The effect of time-reversal on worldlines and $n$-point functions

Time‑orientation reversal as a pull‑back

• If $t:M\to\mathbb R$ is a (globally defined) Cauchy temporal function,

  $$ \theta:(t,\mathbf x)\;\longmapsto\;(-t,\mathbf x) $$

  reverses the causal relation: $y\in J^{+}(x)\iff\theta y\in J^{-}(\theta x)$.

• For a classical field $\Phi$ the time‑reversed field is the pull‑back

  $$ \bar\Phi \;=\;\theta^{\!*}\Phi,\qquad (\bar\Phi)(x) := \Phi\!\bigl(\theta x\bigr). $$

Any normally‑hyperbolic operator $P$ (wave, Klein–Gordon, Maxwell, …) commutes with θ because $g$ and the Levi‑Civita connection are θ‑invariant. Consequently the advanced and retarded Green operators $G_{A/R}$ satisfy the fundamental identity

$$ \boxed{ \;G_{R}(x,y)\;=\;G_{A}\bigl(\theta x,\theta y\bigr)\; } \tag{1} $$

Eq. (1) is just the kernel form of $\theta^{\!*}\,G_{R}\,(\theta^{\!*})^{-1}=G_{A}$.

Because $\theta$ exchanges past and future, swapping its two arguments also turns a retarded kernel into an advanced one:

$$ G_{R}(x,y)=G_{A}(y,x).\tag{2} $$

(Many authors quote (2) as definition; (1) shows it is a special case of the pull‑back statement.)

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Why world‑line re‑parameterisation is not enough

Given a world‑line $z:\mathbb R\to M$, the map $\bar z(\tau)=z(-\tau)$ merely reverses the parameter; it is not the same as acting with θ unless

$$ z\bigl(\tau\bigr)=\theta z\!\bigl(2\tau_{0}-\tau\bigr) $$

for some “mid‑time’’ $\tau_{0}$. Only in that (time‑symmetric) case does evaluation of (2) on the curve give

$$ \boxed{ G_{R}\Bigl[z(\tau_{0}{+}\Delta),z(\tau_{0}{-}\Delta)\Bigr] \;=\; G_{A}\Bigl[z(\tau_{0}{-}\Delta),z(\tau_{0}{+}\Delta)\Bigr]. }\tag{3} $$

For a generic curved trajectory there is no such τ₀, hence simply writing $\tau\mapsto -\tau$ does not convert a retarded kernel into an advanced one.

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Retarded vs advanced $n$-point kernels

Let $\mathcal G_{R/A}$ denote the operators (not their integral kernels); for test functions $f_i$,

$$ R^{(n)}(f_1,\dots,f_n) \;=\; (-1)^{n-1}\sum_{\pi\in S_{n-1}} \mathcal G_{R}\,f_{\pi(1)} \; \cdots\; \mathcal G_{R}\,f_{\pi(n-1)}\,f_n, $$

with support restricted to the common causal past of the last argument (Epstein–Glaser retarded products). Advanced products $A^{(n)}$ are defined analogously but with past ↔ future. The pull‑back identity (1) extends multiplicatively because θ leaves the volume form invariant. One obtains the all‑orders rule

$$ \boxed{ \theta^{\!*}\,R^{(n)}(x_1,\dots,x_n) \;=\; A^{(n)}\bigl(\theta x_1,\dots,\theta x_n\bigr). } \tag{4} $$

Hence retarded and advanced $n$-point objects are interchanged by θ; no single permutation of the $x_i$ can do this once $n>2$ because causal support conditions couple all points to each other.

Concrete example:

$$ J_{R}(x,y,z)\;=\;\int\!dV_{x'}\,G_{R}(x,x')G_{R}(x',y)G_{R}(x',z). $$

Apply θ everywhere and use (1):

$$ \theta^{\!*}J_{R}(x,y,z) =\!\int\!dV_{x'}\,G_{A}(\theta x,\theta x')G_{A}(\theta x',\theta y)G_{A}(\theta x',\theta z) = J_{A}(\theta x,\theta y,\theta z).\tag{5} $$

Again the result is advanced and all three external points are simultaneously time‑reversed. There is no simple identity such as $J_{R}(x,y,z)=J_{A}(y,z,x)$.

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Response to comments:

1. When does the global time‑reversal map θ exist?

• Prerequisites. A smooth Cauchy temporal function $T:M\!\to\!\mathbb R$ is a function whose level sets are smooth Cauchy hypersurfaces and whose gradient is everywhere timelike and past‑directed. Bernal & Sánchez proved that every time‑orientable globally hyperbolic spacetime admits such a function $T$ (see arXiv:gr-qc/0512095v1 )

• Definition of θ Once any such $T$ is chosen, one defines

  $$ \theta:M\longrightarrow M,\qquad p\;\mapsto\;T^{-1}\!\!\bigl(-T(p)\bigr), $$

  i.e. in adapted coordinates $(t,\mathbf x)$ it acts as $(t,\mathbf x)\mapsto(-t,\mathbf x)$.

• Killing field not required. A global timelike Killing vector would make $\theta$ an isometry, but it is not needed for $\theta$ to be a well‑defined smooth diffeomorphism. Global hyperbolicity + time‑orientability suffice.

• Non‑globally hyperbolic cases. Without global hyperbolicity one may still construct local time‑reversals in neighbourhoods that admit a temporal coordinate, but a single global diffeomorphism with the required causal property need not exist (e.g. in Gödel space‑time).

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2. Effect of lower‑order terms in the operator

Let

$$ P\phi \;=\;\Box_g\phi\;+\;V(x)\,\phi $$

with real potential $V$. Because the principal symbol is still the metric $g^{\mu\nu}k_\mu k_\nu$, $P$ remains normally hyperbolic and (for real $V$) formally self‑adjoint. Hence

Existence/uniqueness of $G_{R/A}$ is unchanged; Symmetry identity

$$ G_R(x,y)=G_A(y,x)\tag{2} $$

still holds because it follows only from formal self‑adjointness and the causal support properties.

However, the stronger identity

$$ G_R(x,y)=G_A(\theta x,\theta y)\tag{1} $$

requires $P\circ\theta^{\!*}=\theta^{\!*}\!\circ P$. This is true iff all lower‑order coefficients are even under $\theta$ – i.e. $V(\theta x)=V(x)$. If the potential is not time‑reflection symmetric, eq. (1) fails, while eq. (2) survives.

(If $V$ depends on $\phi$ itself the operator is nonlinear; retarded/advanced linear Green functions are then defined for the linearised operator, and the same remarks apply to that linearisation.)

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3. Which identity is “most general’’ – (1) or (2)?

• Equation (2) (argument swap) is the general statement valid on any time‑orientable globally hyperbolic manifold for any formally self‑adjoint normally‑hyperbolic operator.
• Equation (1) (θ‑pull‑back) is an additional refinement that becomes available only when $P$ commutes with θ, i.e. when θ is an isometry of $g$ and leaves the lower‑order terms invariant.

So (2) is logically more fundamental; (1) is a corollary under extra symmetry.

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4. Typo in Eq. (5)

You are right—my integrand inadvertently combined both “retarded → advanced’’ and a second time‑reversal of the arguments. The correct chain is

$$ \begin{aligned} J_R(\theta x,\theta y,\theta z) &= \!\!\int\!dV_{x'}\,G_R(\theta x,x')G_R(x',\theta y)G_R(x',\theta z)\\[4pt] &= \int\!dV_{x'}\,G_A(x,\theta x')G_A(\theta x',y)G_A(\theta x',z)\\[4pt] &=\int\!dV_{x''}\,G_A(x,x'')G_A(x'',y)G_A(x'',z)\qquad (x''=\theta x')\\[4pt] &=J_A(x,y,z). \end{aligned} $$

Hence the clean statement is

$$ \boxed{\;\theta^{\!*}J_R \;=\; J_A,\qquad\text{or equivalently}\quad J_R(x,y,z)=J_A(\theta x,\theta y,\theta z).} $$