1 Introduction

The exact solutions of Einstein equations that can be in some sense interpreted as gravitational waves are interesting in their own right. What is more, the simplest of those, namely plane waves, is in some sense the limit of an arbitrary spacetime in the vicinity of a null geodesic (this is called the Penrose limit) [40].

Into the plane waves, there is an interesting subfamily of spacetimes [20, 35], to wit, that of a special class of exact gravitational waves, namely the plane-fronted parallel waves with parallel rays (PP waves) that belong to the vanishing scalar invariants (VSI) class of spacetimes. That is, all scalars constructed out of the metric, the Riemann tensor and its derivatives vanish identically. This is obviously true also in flat spacetime where Riemann tensor vanishes. The interesting thing is that can be also true in non flat spacetimes, with a non vanishing Riemann tensor.

This fact means that there are no allowed counterterms [27, 28] in the Einstein–Hilbert action when considering gravitons propagating in such a background. Which has led some authors to claim that there are no divergences whatsoever, and quantum gravity must be finite around such spaces. This is not so, and it is enough to remember that Minkowski is just one such space. Quantum gravity in flat space (and its divergences) has been investigated by Capper and coworkers a long time ago [14,15,16].

The S-matrix elements (on shell divergences) however should be finite [36]. There is a general statement which asserts that divergences that vanish on the equations of motion for the quantum fields (as they do in VSI) indicate that modifications of the BRST algebra are necessary at each loop order, but we are not elaborating on that in this paper.

Our aim here instead is to consider some aspects of the physics of such gravitons propagating in a plane gravitational wave. This is interesting for at least two reasons. First of all, we can perform much more detailed computations than in any other background. In this sense, it is an interesting toy model. A more substantial reason is that a gravitational wave is composed of gravitons; understanding the behavior of those gravitons seems a first step in understanding the full quantum origin of the waves. In fact we will find that they behave in a way somewhat similar to partons in a proton.

We review some usuals definitions (see Appendix A for more details). The standard pp-wave metric reads

$$\begin{aligned} \text {d}s^2=\text {d}u\text {d}v-F[u,x^c]\text {d}u^2-\delta _{ab}\text {d}x^a\text {d}x^b \end{aligned}$$
(1)

where \(u\equiv t-z\), \(v\equiv t+z\), and the transverse coordinates are \(x^c=(x,y)\). These spacetimes belong to the Kerr–Schild class [37, 38]. This means that the metric can be written as

$$\begin{aligned} g_{\mu \nu }=\eta _{\mu \nu }+F l_\mu l_\nu \end{aligned}$$
(2)

with

$$\begin{aligned} l_\mu =-\delta _{u\mu } \end{aligned}$$
(3)

and the additional facts that \(l^2=0\), \(R_{\mu \nu \rho \sigma } l^\sigma =0\) and \(\pounds (l) R_{\mu \nu \rho \sigma }=0\). An interesting fact proved in [30, 31] is that Einstein equations can be written in a linear form and the pseudoenergy tensor vanishes.

A plane wave is a classical pp-wave for which the characteristic function F is quadratic in \(x^c\), \(F[u,x^c]=H_{ab}x^ax^b\)

$$\begin{aligned} \text {d}s^2=\text {d}u\text {d}v-H_{ab}[u]x^ax^b\text {d}u^2-\delta _{ab}\text {d}x^a\text {d}x^b \end{aligned}$$
(4)

in harmonic (often called Brinkmann) coordinates.

This metric has an only nonvanishing component of the Riemann tensor

$$\begin{aligned} R_{uaub}=- H_{ab}[u] \end{aligned}$$
(5)

and the Ricci tensor, in turn, has reduced to

$$\begin{aligned} R_{uu}=-\delta ^{ab}H_{ab}[u]\equiv -H^a_a[u] \end{aligned}$$
(6)

Obviously the manifold is Ricci flat whenever the transverse matrix \(H_{ab}\) is traceless, and the matrix is usually written as

$$\begin{aligned} H_{pq}[u] x^p x^q \equiv a_+[u](x^2-y^2)+2 a_\times [u] xy \end{aligned}$$
(7)

here the functions \(a_+[u]\) and \(a_\times [u]\) are the polarization profiles of the wave.

In the non-Ricci-flat case, there is a source

$$\begin{aligned} T_{uu}={1\over 4}\left( T_{tt}+T_{zz}\right) -{1\over 2} T_{tz}=-{1\over \kappa ^2} H^a_a[u] \end{aligned}$$
(8)

Finally, an especially interesting case are the impulsive waves [25, 39], which we will study in the next section

$$\begin{aligned} \text {d}s^2=\text {d}u\text {d}v-\frac{1}{\lambda _a}\delta (u)\,\delta _{ab}x^ax^b\text {d}u^2-\delta _{ab}\text {d}x^a\text {d}x^b \end{aligned}$$
(9)

this metric is Ricci flat if

$$\begin{aligned} \lambda _1+\lambda _2=0 \end{aligned}$$
(10)

It appears a little bit scary to have a distribution in a metric tensor. In fact a quite general study of distributional sources have been made a long time ago by Geroch and Traschen [26]. In this particular case, the solution has been analyzed in detail by Khan and Penrose (cf. the nice review in Griffiths’ book [29]) who showed that it yields an acceptable spacetime.

We shall see in a moment that in some cases, it is convenient to transform to a set of group coordinates or Rosen coordinates [25], where the metric reads

$$\begin{aligned} \text {d}s^2=\text {d}u\text {d}V-h_{ab}[u] \text {d}X^a\text {d}X^b \end{aligned}$$
(11)

The relationship between the two sets of coordinates is given by

$$\begin{aligned}{} & {} v=V+\frac{1}{2}\dot{h}_{ab}(u)X^aX^b\nonumber \\{} & {} x^a=P^a_b(u)X^b \end{aligned}$$
(12)

with

$$\begin{aligned} h_{ab}(u)=\delta _{ij} P_a^i(u)P_b^j(u) \end{aligned}$$
(13)

where the matrix \(P_a^b(u)\) is determined by solving the differential equation

$$\begin{aligned} \delta _{ij}\ddot{P}^i_a(u)=H_{kj}(u)P^k_a(u) \end{aligned}$$
(14)

with initial conditions satisfying the constraint

$$\begin{aligned} \delta _{ij}\left( \dot{P}^i_a(u)P^j_b(u)-\dot{P}^i_b(u)P^j_a(u)\right) =0 \end{aligned}$$
(15)

where the overdot means derivative with respect to u.

These coordinates do not cover the whole manifold because the determinant of the transverse metric vanishes at some point \(u=\overline{u}\). The only nonvanishing component of Ricci tensor in this coordinates reads

$$\begin{aligned} R_{uu}=-{1\over 2} h^{ab}\ddot{h}_{ab}-{1\over 4} \dot{h}^{ab}\dot{h}_{ab} \end{aligned}$$
(16)

There is a theorem, first pointed out in [33, 34], which states that imposing additional Ricci flatness, then all higher order corrections to the equation of motion also vanish. This means the exact equation of motion of the full perturbative effective action.

The most important aspect of VSI spacetimes is that all the DeWitt–Schwinger coefficients of the short time expansion of the heat kernel do necessarily vanish. We will expose this in some detail in the correspondent section.

Is there any physical meaning in the fact that such an action vanishes on shell? In fact there is none. For example, any theory characterized by a bilinear Lagrangian

$$\begin{aligned} \mathcal {L}=\sum \phi _i M_{ij} \phi _j \end{aligned}$$
(17)

however complicated the kernel \(M_{ij}\) also vanishes on shell (like Dirac Lagrangian for example).

In addition, contrary to popular belief, this is not a generic property of covariant actions. A simple example, the Weyl squared action

$$\begin{aligned} S\equiv \int d^4x\sqrt{|g|}W_{\mu \nu \alpha \beta }W^{\mu \nu \alpha \beta } \end{aligned}$$
(18)

which equation of motion is the vanishing of Bach tensor, but they do not imply \(W_{\mu \nu \rho \sigma }^2=0\). That is, the Lagrangian of conformal gravity does not vanish on shell.

The fact that it does for Einstein is just an accident. Another point is that of course this does not mean that the equation of motion are trivial.

The preceding theorem shows that

$$\begin{aligned}{} & {} \left. \Gamma [g]\right| _{on\,shell}=\Gamma _{class}[g]=-{1\over 2 \kappa ^2} \int d^4x\sqrt{|g|}R\nonumber \\{} & {} \left. {\delta \Gamma [g]\over \delta g_{\alpha \beta }(x)}\right| _{on\,shell}=0 \end{aligned}$$
(19)

We have just asserted that this last fact has been shown to be true in VSI backgrounds by Horowitz and Steif [33, 34]; it physically means that the VSI spacetime is the consistent vacuum of quantum gravity. But of course the 1PI two point correlator is different to zero.

$$\begin{aligned} \left. {\delta ^2 \Gamma [g]\over \delta g_{\alpha \beta }(x)\delta g_{\beta \gamma }(x^\prime )}\right| _{on\,shell}\ne 0 \end{aligned}$$
(20)

In the next sections we will show our most relevant results about VSI spacetimes and we will do some illustrative examples.

2 The heat kernel for a plane wave

The most useful tool in order to compute both the propagator as well as the effective action is the heat kernel [4, 5]. In order to do this calculation it pays to work in Rosen coordinates

$$\begin{aligned} ds^2= du dV-h_{ab}[u] dX^a dX^b \end{aligned}$$
(21)

We shall see in a moment that the propagator is the inverse of the plane wave d’Alembertian operator. In fact, the effective action will be determined by the determinant of the same operator. To be specific the d’Alembertian for VSI spacetime reads

$$\begin{aligned}{} & {} \Box =\frac{\dot{h}_{ab}}{h_{ab}}\partial _v+\partial _u\partial _v-h^{ab}\partial _a\partial _b \end{aligned}$$
(22)

There is a theorem by DeWitt [4, 5] on the existence of a small proper time expansion for the heat kernel of any operator which can be obtained by a deformation of the d’Alembertian, given by

$$\begin{aligned} K\left( \tau ;x,y\right) =K_0 \left( \tau ;x,y\right) ~\sum _{p=0}^\infty ~a_p \left( x,y\right) \tau ^p \end{aligned}$$
(23)

with

$$\begin{aligned} a_0(x,y)=1 \end{aligned}$$
(24)

and the \(K_0\left( \tau ;x,y\right) \) is the flat space heat kernel

$$\begin{aligned} K_0 \left( \tau ;x,y\right) ={1\over (4 \pi \tau )^{n\over 2}}\,e^{-{\sigma (x-x^\prime )\over 2\tau }} \end{aligned}$$
(25)

Here \(\sigma (x,x^\prime )\) is the world function corresponding to the points x and \(x^\prime \), [42].

The important thing for our purposes is that the so called DeWitt–Schwinger coefficients, \(a_p\) are given by integrals of pointwise scalars constructed out of Riemann tensor and its covariant derivatives contracted with the metric tensor. But in all VSI metrics, including plane waves, there are none of those. We are forced to conclude that

$$\begin{aligned} a_p=0\hspace{0.5cm}\text {if} \, p\ne 0 \end{aligned}$$
(26)

and the heat kernel reduces to flat space one

$$\begin{aligned} K\left( \tau ;x,y\right) =K_0 \left( \tau ;x,y\right) \end{aligned}$$
(27)

this is a exact expression for the heat kernel.

We want to emphasize that we only need to obtain the world function to know the exact heat kernel.

In order to compute the world function, we consider the Lagrangian

$$\begin{aligned} \mathcal {L}=\frac{du}{d\tau }\frac{dV}{d\tau }-h_{ab}[u]\frac{d X^a}{d\tau }\frac{d X^b}{d\tau } \end{aligned}$$
(28)

where \(\tau \) is the proper time. The canonical momenta read

$$\begin{aligned}{} & {} p_u=\frac{dV}{d\tau }\nonumber \\{} & {} p_V=\frac{du}{d\tau }\nonumber \\{} & {} p_a=-2h_{ab}\frac{d X^b}{d\tau } \end{aligned}$$
(29)

The Euler–Lagrange equation corresponding to the coordinate V implies the conservation of \(p_V\). We can then write

$$\begin{aligned} u-u_0=p_V(\tau -\tau _0) \end{aligned}$$
(30)

The transverse momentum can be written as

$$\begin{aligned} p_a=-2h_{ab}p_V\frac{dX^b}{du} \end{aligned}$$
(31)

therefore

$$\begin{aligned} X^a=X_0^a-\frac{p_b}{2p_V}\int ^u_{u_0} h^{ab}[u']du' \end{aligned}$$
(32)

The first integral of the timelike geodesic equations

$$\begin{aligned} g_{\mu \nu }{d x^\mu \over d\tau }{dx^\nu \over d\tau }=1 \end{aligned}$$
(33)

implies

$$\begin{aligned} \frac{du}{d\tau }\frac{dV}{d\tau }-h_{ab}\frac{dX^a}{d\tau }\frac{dX^b}{d\tau }=\frac{dV}{du}p^2_V-\frac{1}{4}h^{ab}p_ap_b=1 \end{aligned}$$
(34)

so that

$$\begin{aligned}{} & {} V=V_0+\frac{u-u_0}{p_V^2}+\frac{p_ap_b}{4p_V^2}\int ^u_{u_0} h^{ab}[u']du' \end{aligned}$$
(35)

Finally, the world function reads

$$\begin{aligned} \sigma (x,x_0){} & {} =\frac{1}{2}(\tau -\tau _0)^2=\frac{1}{2p_V^2}(u-u_0)^2 \nonumber \\{} & {} =\frac{1}{2}(V-V_0)(u-u_0)\nonumber \\{} & {} \quad -\frac{p_ap_b}{8p_V^2}(u-u_0)\int h^{ab}[u']du'\nonumber \\{} & {} =\frac{1}{2}(V-V_0)(u-u_0)-\frac{1}{2}(X^a-X^a_0)\nonumber \\{} & {} \quad \times C_{ab}[u,u_0](X^b-X^b_0)(u-u_0) \end{aligned}$$
(36)

where, we can define \(C_{ab}\) as

$$\begin{aligned} C_{ab}[u,u_0]\int _{u_0}^u h^{cb}[u']du'=\delta _a^c \end{aligned}$$
(37)

In conclusion, with the characteristic function \(h_{ab}[u]\) of the metric, we can obtain the world function and therefore the heat kernel. We show an explicit example

2.1 Impulsive gravitational plane waves

A particular case worth considering is the impulse gravitational plane waves

$$\begin{aligned} \text {d}s^2=\text {d}u\text {d}v-\frac{1}{\lambda _a}\delta (u)\delta _{ab}x^ax^b\text {d}u^2-\delta _{ab}\text {d}x^a\text {d}x^b \end{aligned}$$
(38)

which in Rosen coordinates translates to

$$\begin{aligned} \text {d}s^2=\text {d}u\text {d}V-\left( 1-\frac{u\Theta [u]}{\lambda _a}\right) ^2\delta _{ab}\text {d}X^a\text {d}X^b \end{aligned}$$
(39)

where \(\Theta [u]\) is the Heaviside step function. This physically represents two flat spaces glued together at \(u=0\), one corresponding to \(u<0\) in Minkowskian coordinates and the other corresponding to \(u>0\) in non-Minkowskian ones.

The world function, using our previous results, reads

$$\begin{aligned} \sigma (x,x_0){} & {} =\frac{1}{2}(V-V_0)(u-u_0)\nonumber \\{} & {} \quad -\frac{1}{2}(X^a-X^a_0)C_{ab}[u,u_0]\nonumber \\{} & {} \quad \times (X^b-X^b_0)(u-u_0) \end{aligned}$$
(40)

where

$$\begin{aligned} C_{a}^{b}[u,u_0]=\frac{1}{\int ^u_{u_0} \left( 1-\frac{u'\Theta [u']}{\lambda _a}\right) ^2du'}\delta _{a}^b \end{aligned}$$
(41)

To be specific,

  • \(u_0,u<0\)

    $$\begin{aligned} C_{a}^{b}[u,u_0]=\frac{1}{(u-u_0)}\delta _{a}^b \end{aligned}$$
    (42)

  • \(u_0<0\) and \(u>0\), in this case

    $$\begin{aligned}{} & {} \int ^u_{u_0} \left( 1-\frac{u'\Theta [u']}{\lambda _a}\right) ^2du'=\int ^0_{u_0} du'\nonumber \\{} & {} \quad +\int ^u_{0} \left( 1-\frac{u}{\lambda _a}\right) ^2du'=u-u_0 - \frac{u^2}{\lambda _a}+\frac{u^2}{3\lambda _a} \end{aligned}$$
    (43)

    then

    $$\begin{aligned} C_{a}^{b}[u,u_0]{} & {} =\frac{1}{u-u_0 - \frac{u^2}{\lambda _a}+\frac{u^2}{3\lambda _a}}\delta _{a}^b \end{aligned}$$
    (44)

  • \(u_0,u>0\), now

    $$\begin{aligned} \int ^u_{u_0} \left( 1-\frac{u'}{\lambda _a}\right) ^2du'{} & {} =-\frac{\lambda _a}{3}\left( 1-\frac{u}{\lambda _a}\right) ^3\nonumber \\{} & {} \quad + \frac{\lambda _a}{3}\left( 1-\frac{u_0}{\lambda _a}\right) ^3\nonumber \\ \end{aligned}$$
    (45)

    then

    $$\begin{aligned} C_{a}^{b}[u,u_0]=\frac{1}{-\frac{\lambda _a}{3}\left( 1-\frac{u}{\lambda _a}\right) ^3+\frac{\lambda _a}{3} \left( 1-\frac{u_0}{\lambda _a}\right) ^3}\delta _{a}^b \end{aligned}$$
    (46)

this world function has a singularity when \(u=\lambda \) (where the determinant \(h=0\)) which marks the boundary of the normal neighborhood. This is the place where the nearest zero of the Jacobi field is located.

3 The propagator of the graviton field

Let us begin with a general remark. In the Ref. [9] a general parametrization of the free graviton propagator in flat space is introduced, namely

$$\begin{aligned}{} & {} {\langle h_{\mu _1\mu _2}(k) h_{\mu _3\mu _4}(-k)\rangle _0} \nonumber \\{} & {} \quad = \,i\dfrac{A_1}{k^2}\left( \eta _{\mu _1\mu _3}\eta _{\mu _2\mu _4}+\eta _{\mu _1\mu _4}\eta _{\mu _2\mu _3} -\eta _{\mu _1\mu _2}\eta _{\mu _3\mu _4}\right) \nonumber \\ {}{} & {} \quad {+}\,i\dfrac{A_2}{k^2}\eta _{\mu _1\mu _2}\eta _{\mu _3\mu _4} {+}\,i\dfrac{A_3}{(k^2)^2}\left( \eta _{\mu _3\mu _4}k_{\mu _1} k_{\mu _2}{+}\eta _{\mu _1\mu _2}k_{\mu _3} k_{\mu _4}\right) \nonumber \\{} & {} \quad {+}\, i\dfrac{A_4}{(k^2)^2}\left( \eta _{\mu _1\mu _3}k_{\mu _2} k_{\mu _4}{+}\eta _{\mu _1\mu _4}k_{\mu _2} k_{\mu _3}\right. \nonumber \\{} & {} \quad \left. +\eta _{\mu _2\mu _3}k_{\mu _1} k_{\mu _4}+\eta _{\mu _2\mu _4}k_{\mu _1} k_{\mu _3}\right) + \,i\dfrac{A_5}{(k^2)^3} k_{\mu _1} k_{\mu _2} k_{\mu _3} k_{\mu _4}. \nonumber \\ \end{aligned}$$
(47)

\(A_i\), \(i=2..5\) are constants. We shall assume that

$$\begin{aligned} A_1=\dfrac{1}{2} \end{aligned}$$
(48)

which is just a normalization. The contribution

$$\begin{aligned} G_{\mu _1\mu _2\mu _3\mu _4}{} & {} \equiv \,\dfrac{1}{p^2}\left( \eta _{\mu _1\mu _3}\eta _{\mu _2\mu _4} +\eta _{\mu _1\mu _4}\eta _{\mu _2\mu _3}\right. \nonumber \\{} & {} \quad \left. -\eta _{\mu _1\mu _2}\eta _{\mu _3\mu _4}\right) \equiv {1\over p^2} P_{\mu _1\mu _2\mu _3\mu _4}\nonumber \\ \end{aligned}$$
(49)

to the propagator can be interpreted as coming from the bit of the graviton field which contains the physical graviton polarizations and the corresponding creation and annihilation operators. This is in fact exactly the propagator used in [14,15,16].

This propagator has several interesting properties. First of all, there is a unique pole at \(p^2=0\), and the residue for \(G_{00,00}\) is positive, although it is neither traceless not transverse.

The need to recover the Newtonian potential in the static case impose some restrictions. In fact, using (47), one readily deduces that

$$\begin{aligned} \left\langle \hat{h}^{00}(k)\hat{h}^{00}(-k)\right\rangle _0=-\dfrac{i}{4\textbf{k}^2}\big [3+\dfrac{1}{4} A_5+ A_4\big ], \end{aligned}$$
(50)

for static sources, namely, \(k^\mu =(0,\textbf{k})^\mu \). Inserting the previous result in the static potential

$$\begin{aligned} V_\textrm{Nw}(\textbf{k})= & {} -i\dfrac{1}{4}m_1 m_2 \left\langle \hat{h}^{00}(k)\hat{h}^{00}(-k)\right\rangle _0\nonumber \\= & {} -\dfrac{m_1 m_2}{16\,\textbf{k}^2}\big [3+\dfrac{1}{4} A_5+ A_4\big ] \end{aligned}$$
(51)

We learn that in order to recover the correct potential,

$$\begin{aligned} V_\textrm{Nw}(\textbf{k})\,=\,-\dfrac{1}{8}\dfrac{m_1 m_2}{\textbf{k}^2}, \end{aligned}$$
(52)

we need that

$$\begin{aligned} A_4+\dfrac{1}{4} A_5=-1, \end{aligned}$$
(53)

We thus conclude that both \(A_4\) and \(A_5\) in the free graviton propagator in (47) cannot vanish at the same time, regardless of the Lorentz covariant gauge-fixing term that one uses.Footnote 1

The definition of the Feyman propagator in curves spacetime is a nontrivial issue [12, 21]. Maybe the cleaner appproach (because it is uniquely defined) is to consider the euclidean propagator

In our case, for a plane wave spacetime, the propagator reads

$$\begin{aligned} G^E_{\mu \nu \rho \sigma }\left( x,y\right){} & {} =\frac{1}{\Box }P_{\mu \nu \rho \sigma }\equiv \Delta ^E(x,y)\, P_{\mu \nu \rho \sigma } \nonumber \\{} & {} =\int _0^{\infty }d\tau K \left( \tau ;x,y\right) P_{\mu \nu \rho \sigma }\nonumber \\{} & {} =\frac{1}{(4\pi )^{\frac{n}{2}}}\sum _{p=0}^\infty \text {tr}~ a_p(x,y)\left( \frac{ \sigma _E}{2 }\right) ^{p-\frac{n}{2}+1}~\nonumber \\{} & {} \quad \times \Gamma \left( {n\over 2}-p-1\right) C_{\mu \nu \rho \sigma } \end{aligned}$$
(55)

where the tensor appropriate for a plane wave background, \(\bar{g}_{\mu \nu }\), reads

$$\begin{aligned} \left. C_{\mu \nu \rho \sigma }\equiv P_{\mu \nu \rho \sigma }\right| _{\eta _{\mu \nu }\rightarrow \bar{g}_{\mu \nu }} \end{aligned}$$
(56)

and the euclidean Synge function is given by

$$\begin{aligned} \sigma _E(x,x_0){} & {} =\frac{1}{2}(V-V_0)(u-u_0)\nonumber \\{} & {} \quad +\frac{1}{2}(X^a-X^a_0)C_{ab}[u,u_0](X^b-X^b_0)(u-u_0)\nonumber \\ \end{aligned}$$
(57)

In conclusion, in the physical dimension \(n=4\) and \(p=0\) (only the \(a_0\) term contributes), the expression of the euclidean graviton propagator in a plane wave background reads

$$\begin{aligned} G_{\mu \nu \rho \sigma }\left( x,y\right) =\frac{1}{(4\pi )^2}\frac{2}{\sigma _E\left( x-y\right) }C_{\mu \nu \rho \sigma } \end{aligned}$$
(58)

What we have computed up to now is the euclidean propagator. In flat space there is a systematic way to get from it Feynman’s propagator by analytic continuation, the simplest form of which is through the heat kernel approach. Assuming the same is true in curved space leads to[12, 21]

$$\begin{aligned} G_F(x,x^\prime )=-i \int _0^\infty \,ds\,e^{-i K s} \end{aligned}$$
(59)

where

$$\begin{aligned} K\equiv |g(x)|^{-1/2}|g(x^\prime |^{-1/2}\left( \Box +m^2-i\epsilon \right) \delta (x-x^\prime ) \end{aligned}$$
(60)

and

$$\begin{aligned}{} & {} e^{-i K s}(x,x^\prime )={i \Delta ^{1/2}(x,x^\prime )\over (4\pi )^2}\,(i s)^{-n/2}\,\nonumber \\{} & {} \quad \times e^{-i\left( m^2+i\epsilon \right) s-{\sigma (x,x^\prime )\over 2 s}}F\left( x,x^\prime |is\right) \end{aligned}$$
(61)

Finally the Schwinger–DeWitt short time expansion reads

$$\begin{aligned} F\left( x,x^\prime |is\right) =a_0+a_1(is)+ a_2(is)^2+\cdots \end{aligned}$$
(62)

The van Vleck–Morette determinant is defined as

$$\begin{aligned} \Delta (x,x^\prime )=|g|^{-1/2}|g^\prime |^{-1/2}\,\text {det}\,\left( -\partial _\mu \partial _{\mu ^\prime }\sigma (x,x^\prime )\right) \end{aligned}$$
(63)

To give an example, when \(h_{ab}\) is constant

$$\begin{aligned} \Delta (x,x_0)=-{1\over 4} \end{aligned}$$
(64)

Feynman’s propagator is then

$$\begin{aligned} G_F(x,x_0)= & {} {2 i\over (4\pi )^2}\Delta (x,x_0)^{1/2}{1\over \sigma (x,x_0)-i\epsilon }\nonumber \\= & {} {2 i\over (4\pi )^2}\Delta (x,x_0)^{1/2}\left( P{1\over \sigma (x,x_0)}+i\pi \delta (\sigma )\right) \nonumber \\ \end{aligned}$$
(65)

It is a well-known fact [23] that when Huygens principle does not hold (like in odd-dimensional spacetimes) there is a tail term in the fundamental solution of the wave equation. This fact has been studied in detail in [1, 2, 43]. It has been claimed [32] that in spite of the fact that in four dimensiona Huygens principle holds, this is not so for the gravitational fluctuations around a plane wave owing to a position-dependent mass-like coupling between Riemann’s tensor and the fluctuations.

The fundamental solution we are talking about here refers to what in quantum field theory is called the retarded propagator appropiate for classical computations. When calculating quantum corrections the appropiate propagator is Feynman’s however.

There is a relationship [12, 27, 28] between the average of the advanced and retarded propagators, the Hadamard function, \(G^{(1)}\) and Feynman’s propagator, namely

$$\begin{aligned} G^F_{\alpha \beta \gamma \delta }(x,x^\prime )= & {} -{1\over 2}\left( G^{adv}_{\alpha \beta \gamma \delta }(x,x^\prime )+\,G^{ret}_{\alpha \beta \gamma \delta }(x,x^\prime )\right) \nonumber \\{} & {} -{i\over 2}\,G^{(1)}_{\alpha \beta \gamma \delta }(x,x^\prime ) \end{aligned}$$
(66)

where Hadamard’s function is defined by

$$\begin{aligned} G^{(1)}_{\alpha \beta \gamma \delta }(x,x^\prime )\equiv \left\langle 0\left| \left\{ g_{\alpha \beta }(x) g_{\gamma \delta }(x^\prime )\right\} \right| 0\right\rangle \end{aligned}$$
(67)

4 The effective action of quantum gravity in a plane wave spacetime

In this section, we want to calculate the effective action of Einstein gravity

$$\begin{aligned} S_{\text {\tiny {EH}}}{} & {} \equiv -\frac{1}{2\kappa ^2}\int d^n x\sqrt{|g|}R \end{aligned}$$
(68)

It should be expanded around a background, \(\bar{g}_{\mu \nu }\), which in our case it will be the plane wave

$$\begin{aligned} g_{\mu \nu }=\bar{g}_{\mu \nu }+\kappa h_{\mu \nu } \end{aligned}$$
(69)

The full quadratic action is obtained after adding the harmonic (de Donder) gauge fixing piece [4, 5, 8]

$$\begin{aligned}{} & {} S_{\text {\tiny {2+gf}}}=-\frac{1}{4}\,\int \,d^nx\,\,h^{\mu \nu }D_{\mu \nu \rho \sigma }h^{\rho \sigma } \end{aligned}$$
(70)

where

$$\begin{aligned} D_{\mu \nu \rho \sigma }= & {} \frac{1}{4}C_{\mu \nu \rho \sigma }\Box +{1\over 2}\left( R_{\mu \rho \nu \sigma }+R_{\nu \rho \mu \sigma }\right) \nonumber \\= & {} \frac{1}{4}(\bar{g}_{\mu \rho }\bar{g}_{\nu \sigma }+\bar{g}_{\mu \sigma }\bar{g}_{\nu \rho }-\bar{g}_{\mu \nu }\bar{g}_{\rho \sigma })\Box \nonumber \\{} & {} +{1\over 2}\left( R_{\mu \rho \nu \sigma }+R_{\nu \rho \mu \sigma }\right) \end{aligned}$$
(71)

This full operator that includes a massive term of sorts for some components of the graviton (the terms involving the Riemann tensor) is too complicated for computing its inverse exactly, which would be the full propagator. Solutions to the linear equation for the fluctuations have been provided in [32] using the retarded Green function.

What we can do is to consider the terms involving the Riemann tensor as a potential and perform a perturbative computation. Let us recall now that the DeWitt–Schwinger coefficients of one-loop counterterm are given by integrals of pointwise scalars constructed out of Riemann tensor and its covariant derivatives contracted with the metric tensor. But in plane wave background, being VSI, there are none of those, so that the piece of the (euclidean) effective action not involving Riemann’s tensor is

$$\begin{aligned} W^0_E= \log \,\det \,\Box _h-2 \log \det \Box _{gh} \end{aligned}$$
(72)

where the determinant of the Laplacian acting on scalars is given by

$$\begin{aligned} \text {log~det}~\Box =-\int _0^{\infty }\frac{d\tau }{\tau }\text {tr} K \left( \tau ;x,y\right) \end{aligned}$$
(73)

again, we directly calculate

$$\begin{aligned} \text {log~det}~\Box= & {} -\frac{1}{(4\pi )^{\frac{n}{2}}}\int \sqrt{|g|}~d^n x\sum _{p=0}^\infty \text {tr}~ a_p(x,y)~\nonumber \\{} & {} \times \lim _{\sigma \rightarrow 0}\left( \frac{ \sigma }{2 }\right) ^{p-\frac{n}{2}}~\Gamma \left( {n\over 2}-p\right) \end{aligned}$$
(74)

as only the \(p=0\) survives

$$\begin{aligned} \text {log~det}~\Box =-\frac{1}{(2\pi )^{n/2}}\int \sqrt{|g|}~d^n x~\lim _{\sigma \rightarrow 0} \frac{1}{ \sigma ^{\frac{n}{2}} }~\Gamma \left( {n\over 2}\right) ~ \end{aligned}$$
(75)

The integrand depends on \({1\over \sigma (0)^2}\) and besides the integral is proportional to the total (divergent) volume of the spacetime manifold. It can be interpreted as a divergent contribution to the cosmological constant, \(\lambda _{\infty }\). In the presence of physical sources,

$$\begin{aligned} e^{-W_E\left[ T_{\mu \nu }\right] }&=e^{-W_E[0]} \,e^{-{1\over 2}\int d(vol) \left( R_{\mu \rho \nu \sigma }+R_{\nu \rho \mu \sigma }\right) {\delta \over \delta T_{\mu \nu }(x)}{\delta \over \delta T_{\rho \sigma }(x)}} \nonumber \\&\quad \times \, e^{,-\int d(vol)_x d(vol)_{x^\prime }\, T^{\mu \nu }(x) G^E_{\mu \nu \rho \sigma }\,(x,x^\prime )\,T^{\rho \sigma }(x^\prime )}\nonumber \\ \end{aligned}$$
(76)

with

$$\begin{aligned} W_E[0]=-{\lambda _\infty \over \kappa ^2}\int d(vol) \end{aligned}$$
(77)

and where \(G^E_{\mu \nu \rho \sigma }\) is the euclidean propagator just discussed in the previous section. The first few terms in the expansion read

$$\begin{aligned}{} & {} e^{-W_E\left[ T_{\mu \nu }\right] }=e^{-W_E[0]} \,\bigg \{1-{1\over 2}\int d(vol) \nonumber \\{} & {} \quad \times \left( R_{\mu \rho \nu \sigma }(x)+R_{\nu \rho \mu \sigma }(x)\right) \nonumber \\{} & {} \quad \times {\delta \over \delta T_{\mu \nu }(x)}{\delta \over \delta T_{\rho \sigma }(x)}+\nonumber \\{} & {} \quad +{1\over 4}\int d(vol)_x \left( R_{\mu \rho \nu \sigma }(x)+R_{\nu \rho \mu \sigma }(x)\right) \nonumber \\{} & {} \quad \times {\delta \over \delta T_{\mu \nu }(x)}{\delta \over \delta T_{\rho \sigma }(x)}\int d(vol)_y \left( R_{\mu \rho \nu \sigma }(y)\right. \nonumber \\{} & {} \quad \left. +R_{\nu \rho \mu \sigma }(y)\right) {\delta \over \delta T_{\mu \nu }(y)}{\delta \over \delta T_{\rho \sigma }(y)}+\cdots \bigg \} \nonumber \\{} & {} \quad \times \, e^{,-\int d(vol)_x d(vol)_{x^\prime }\, T^{\mu \nu }(x) G_{\mu \nu \rho \sigma }\,(x,x^\prime )\,T^{\rho \sigma }(x^\prime )} \end{aligned}$$
(78)

The first terms in the expansion read

$$\begin{aligned}{} & {} e^{-W_E\left[ T_{\mu \nu }\right] }=e^{-W_E[0]} \bigg \{- \left[ R^{\mu \rho \nu \sigma }(x)+R^{\nu \rho \mu \sigma }(x)\right] \nonumber \\{} & {} \qquad \times \left( 2 G^E_{\rho \sigma \mu \nu }+4 G^E_{\rho \sigma \alpha \beta } T^{\alpha \beta } G^E_{\mu \nu \gamma \delta } T^{\gamma \delta }+\cdots \right) \bigg \}\nonumber \\{} & {} \qquad \times \, e^{-\int d(vol)_x d(vol)_{x^\prime }\, T^{\mu \nu }(x) G^E_{\mu \nu \rho \sigma }\,(x,x^\prime )\,T^{\rho \sigma }(x^\prime )}\nonumber \\{} & {} \quad =e^{-W_E[0]} \Bigg \{1-\frac{1}{4\pi ^4}\int ~d^n z \sqrt{|\bar{g}(z)|}\nonumber \\{} & {} \qquad \times \bar{R}_{\alpha \lambda \beta \tau }T^{\alpha \beta }(z)T^{\lambda \tau }(z)+\cdots \Bigg \}\,\nonumber \\{} & {} \qquad \times e^{-\int d^n x\sqrt{|\bar{g}(x)|} ~d^n y \sqrt{|\bar{g}(y)|}\, \frac{1}{\sigma _E\left( x,y\right) }T^{\mu \nu }(x) C_{\mu \nu \rho \sigma }\,(x,y)\,T^{\rho \sigma }(y)}\nonumber \\ \end{aligned}$$
(79)

To sum up, gravitons in this background behave almost as free particles, with a mass term of sorts which is the remembrance of the tail term in Feynman’s propagator. Those are in some sense gravitational partons.Footnote 2

5 The unimodular gauge

In the particular case of plane waves (as in all VSI spacetimes), the equations of motion of unimodular gravity are equivalent to the ones of Einstein General Relativity, because the Ricci tensor is already traceless to begin with. As a matter of fact, in the unimodular gauge the plane wave metric, in Rosen coordinates, reads

$$\begin{aligned} ds^2= {2 d U d V\over \sqrt{ h[u]}}-h_{ab}[u] dX^a dX^b \end{aligned}$$
(82)

where

$$\begin{aligned} h[u]\equiv \det \,h_{ab}[u] \end{aligned}$$
(83)

and U is a function of u given by

$$\begin{aligned} U(u)\equiv \,\frac{1}{2}\int ^u \sqrt{ h(x)}\, dx \end{aligned}$$
(84)

Next, we only need the world function for obtain the propagator and the effective action. The geodesics are determined by the Lagrangian

$$\begin{aligned} \mathcal {L}=\frac{2}{\sqrt{h[u]}}\frac{d U}{d\tau }\frac{dV}{d\tau }-h_{ab}[u]\frac{d X^a}{d\tau }\frac{d X^b}{d\tau } \end{aligned}$$
(85)

where \(\tau \) is the proper time, the momentum about V yields

$$\begin{aligned}{} & {} p_V=\frac{2}{\sqrt{h[u]}}\frac{d U}{d\tau } \end{aligned}$$
(86)

the Euler–Lagrange equation about V implies the conservation of \(p_V\), then we can write

$$\begin{aligned} \frac{dU}{d\tau }=\frac{\sqrt{h[u]}}{2}p_V \end{aligned}$$
(87)

which implies

$$\begin{aligned} \tau -\tau _0=\frac{2}{p_V}\mathcal {F}[U,U_0] \end{aligned}$$
(88)

The function \(\mathcal {F}[U,U_0]\) is then given by

$$\begin{aligned} \mathcal {F}[U,U_0]\equiv \int ^U_{U_0} \frac{dx}{\sqrt{h[u[x]]}} \end{aligned}$$
(89)

this formula yields the function \(\mathcal {F}[U,U_0]\) through the implicit function theorem.

The transverse momentum can be written as

$$\begin{aligned} p_a=-h_{ab}p_V\sqrt{h[u]}\frac{dX^b}{dU} \end{aligned}$$
(90)

then

$$\begin{aligned} X^a(\tau )= & {} X^a_0-{p_b\over p_V}\int _{U_0}^U {h^{ab}(x)\over \sqrt{h(x)}} dx\nonumber \\= & {} X^a_0-{p_b\over p_V}\mathcal {C}^{ab}[U,U_0] \end{aligned}$$
(91)

where we define

$$\begin{aligned} \mathcal {C}^{ab}[U,U_0]\equiv \int _{U_0}^U {h^{ab}(x)\over \sqrt{h(x)}} dx \end{aligned}$$
(92)

with \(\mathcal {C}_{ab}\mathcal {C}^{bc}=\delta _a^c\).

Imposing now the first integral

$$\begin{aligned} \frac{2}{\sqrt{h[u]}}\frac{d U}{d\tau }\frac{dV}{d\tau }-h_{ab}[u]\frac{d X^a}{d\tau }\frac{d X^b}{d\tau }=1 \end{aligned}$$
(93)

it follows that

$$\begin{aligned}{} & {} V=V_0+\frac{2}{p_V^2}\mathcal {F}[U,U_0]+\frac{p_ap_b}{2p_V^2}\mathcal {C}^{ab}[U,U_0] \end{aligned}$$
(94)

and world function in the unimodular gauge reads

$$\begin{aligned} \sigma (x,x_0)= & {} \frac{1}{2}(\tau -\tau _0)^2=\frac{2}{p_V^2}\mathcal {F}^2[U,U_0]\nonumber \\= & {} (V-V_0)\mathcal {F}[U,U_0]-\frac{1}{2}(X^a-X^a_0)\nonumber \\{} & {} \times \mathcal {C}_{ab}[U,U_0](X^b-X^b_0)\mathcal {F}[U,U_0] \end{aligned}$$
(95)

It is to be concluded that with the world function, we can obtain the propagator and the effective action in the unimodular background.

6 Conclusions

In this paper we have determined the exact one loop propagator (albeit in a formal way) as well as the effective action for a graviton propagating in a plane wave. Gravitational plane waves are roughly similar to electromagnetic plane waves, but for the crucial point that a general gravitational wave cannot be written as some superposition of gravitational plane waves, owing to the nonlinear character of Einstein’s equations. There are some exceptions to this whenever the waves are in the Kerr–Schild family in which case Ricci flatness is equivalent to Fierz–Pauli [3].

This work has been possible owing to the fact that plane waves belong to the family of VSI spacetimes (all geometric scalar invariants vanish). Then the Schwinger-DeWitt expansion is trivial, and the heat kernel corresponding to the full laplacian is exact once the world function (one half the squared geodesic distance) is known, which is just an elementary exercise. There is a quadratic coupling to the Riemann tensor, which is like a position dependent mass, and looks like the manifestation of the tail terms is this formalism.

We have also pointed out that for Ricci flat spacetimes (our plane waves are such) are examples of instances where the equations of motion of unimodular gravity are the same as the general relativistic ones in the unimodular gauge.