Causality in Scalar-Einstein Waves

Abstract

A wavelike scalar-Einstein solution is found and indicating vectors constructed from the Bel-Robinson tensor are used to study which objects co-move with the wave and whether gravitational energy transfer is null. It is found that this Bel-Robinson energy criteria gives null energy transfer for both the vacuum plane wave and the scalar plane wave.

1. Introduction

Gravitational waves §5.9 [1] are usually taken to move at the speed of light. Among the many questions that arise are whether the waves gravitational energy travels at the same speed and whether it is possible to have co-moving fields: in particular what happens when the wave moves from a vacuum to a medium such as dust, in this case do both gravity and co-moving fields slow down by the same amount.

The nature of gravitational energy is a problem in its own right. In general one can ‘free-fall’ into a frame where it vanishes. One needs more information than just that given by the gravitational field. It can be given in asymptotically cases, here the extra information is that the spacetime is asymptotically flat. There is also the problem of constructing objects with dimensions of energy, this can be done using the Lanczos potential [2], however the intprepation is not always clear. For dimensions of energy squared the Bel-Robinson [3] has a more straightforward interpretation, for recent work see [4,5,6,7,8]. For present considerations of properties which co-move with a gravitional wave energy squared will do as the co-moving nature is unlikely to change.

The field equations are [1] first form of Equation (3.15)

$$R_{ab}=8\pi \left(T_{ab}-\frac{1}{2}T{g}_{ab}\right)+\mathrm{\Lambda }{g}_{ab},$$

(1)

the equations for a scalar field are given by [1] Equation (3.6)

$$T_{ab}={\psi }_a{\psi }_b-\frac{1}{2}{g}_{ab}\left({\psi }_c^2+\frac{m^2}{{\hslash }^2}{\psi }^2\right),$$

(2)

taking $\mathrm{\Lambda }=0,d=4,m^{\prime }=m/\hslash ,\varphi =2\sqrt{\pi }\psi$ and dropping the prime on m gives

$${\overline{R}}_{ab}\equiv R_{ab}-2{\varphi }_a{\varphi }_b-m^2{\varphi }^2{g}_{ab}=0,$$

(3)

m is nearly always taken to be zero here. A new solution which contains both gravitational wave and has a scalar field obeying (3) is found and investigated by constructing indicator vectors produced by transvecting the Bel-Robinson tensor [3]. Calculations were carried out using Maple.

2. The Plane Wave

2.1. The Plane Wave Line Element

The plane wave has line element §5.9 [1]

$$d{s}^2=W(u,y,z)d{u}^2+2dudv+d{y}^2+d{z}^2,$$

(4)

the determinant of the metric is $g=-1$. This spacetime is very simple as evidenced by the vanishing of the Kretschmann curvature invariant $K=RiemSq=0$. Ihe non-vanishing components of the Riemann tensor are given by

$$R_{uiuj}=-\frac{1}{2}{W}_{,ij}$$

(5)

where $i,j\cdots =1,2,3$. The non-vanishing component of the Ricci tenor is

$$R_{uu}=-\frac{1}{2}{W}_{,yy}-\frac{1}{2}{W}_{,zz}.$$

(6)

2.2. The Vacuum, Bach and Scalar Plane Wave

2.2.1. Vacuum Case

For a vacuum Ricci flat spacetime a choice §5.9 [1] of W is

$$W=(y^2-z^2)f\left(u\right)-2yzg\left(u\right)$$

(7)

where $f,g$ are arbitrary twice differentiable functions of u, this is the same result as §5.9 [1].

2.2.2. Bach Case

The Bach tensor is

$$B_{ab}\equiv 2C_{a..b}^{cd}{R}_{cd}+4C_{a..b;cd}^{cd},$$

(8)

and this tensor can be used in the expression for quadratic field equations

$$R_{ab}+b{B}_{ab}=0.$$

(9)

Quadratic field equations occur in most theories of quantum gravity. For the line element (4) the Bach tensor has non-vanishing component

$$B_{uu}=W_{,yyyy}+2W_{,yyzz}+W_{,zzzz}.$$

(10)

A solution to the field Equation (9) is

$$W=sin\left(\frac{y}{\sqrt{b}}\right)f_1\left(u\right)+cos\left(\frac{y}{\sqrt{b}}\right)f_2\left(u\right)+sin\left(\frac{z}{\sqrt{b}}\right)g_1\left(u\right)+cos\left(\frac{z}{\sqrt{b}}\right)g_2\left(u\right).$$

(11)

2.2.3. Massless Scalar Case

Massless scalar fields typically move at the speed of light and one would anticipate that they would co-move with the gravitational wave. For a scalar-Einstein solution one can choose

$$W=(a{y}^2-b{z}^2)f\left(u\right)-2cyzg\left(u\right),$$

(12)

giving

$$R_{uu}=(b-a)f=2{\varphi }_u^2,$$

(13)

however $\varphi$ can have no $y,z$ dependence as this would entail non-vanishing $R_{yy}$, thus W has a $y,z$ dependence and $\varphi$ does not, and in this sense the scalar field is not co-moving with the gravitational field.

2.2.4. Massive Scalar Case

For $m\ne 0$, $R_{yy}=0$ implies

$$\varphi =A_{+}exp\left(+\frac{imy}{\sqrt{2}}\right)+A_{-}exp\left(-\frac{imy}{\sqrt{2}}\right),$$

(14)

with either $A_{+}\mathrm{or}{A}_{-}$ vanishing, with a similar equation for $R_{zz}$. For simplicity we use

$$\varphi =A\left(u\right)exp\left(+\frac{im}{\sqrt{2}}(y+z)\right),$$

(15)

then the $R_{uu}$ component becomes

$$(b-a)f\left(u\right)=(2A_u^2+m^{2}W{A}^2)exp\left(i\sqrt{2}m(y+z)\right),$$

(16)

which has no y and z dependence on the left hand side but on the right hand side both W and the exponential term have such dependence showing that there is no homogeneous solution, with both W and $\varphi$ having similar $y,z$ dependence.

2.3. Null Tetrads for the Plane Wave

A suitable set of null tetrads is

$$l_a=-\frac{1}{2}W{\delta }_a^u-{\delta }_a^v,n_a={\delta }_a^u,m_a=-\frac{i}{\sqrt{2}}{\delta }_a^y-\frac{1}{\sqrt{2}}{\delta }_a^z,$$

(17)

the Weyl and Ricci scalars are

$${\mathrm{\Phi }}_{22}=-\frac{3}{4}\left(W_{yy}+W_{zz}\right),{\mathrm{\Psi }}_4=\frac{3}{4}\left(W_{yy}+2i{W}_{yz}-W_{zz}\right),$$

(18)

for the particular case (12) with $a,b=1$ (18) reduces to

$${\mathrm{\Phi }}_{22}=0,{\mathrm{\Psi }}_4=3\left(f-ig\right).$$

(19)

3. A Scalar-Einstein Wave

3.1. The Line Element

For a $y,z$ dependent scalar-Einstein wave consider the line element

$$d{s}^2=W(u,x,y)d{u}^2+2A_{3}xydudv+A_{1}d{x}^2+A_{2}d{y}^2,\varphi =\frac{1}{2}ln\left(\frac{kx}{y}\right),$$

(20)

in the case of vanishing gravitational wave it is related to the solution in [9], although the corresponding conformal Killing vector has not been found. After subtracting off the scalar field (3) there remains the Ricci tensor component

$${\overline{R}}_{uu}=-\frac{1}{2A_2{y}^2}\left(W-y{W}_{,y}+y^2{W}_{,yy}\right)-\frac{1}{2A_1{x}^2}\left(W-x{W}_{,x}+x^2{W}_{,xx}\right).$$

(21)

The line element (20) is a scalar-Einstein solution when

$$\begin{array}{ccc}\hfill W& =& \left(B_{1}x\mathrm{J}\left(0,\frac{x}{\sqrt{A_2}}\right)+B_{2}x\mathrm{Y}\left(0,\frac{x}{\sqrt{A_2}}\right)\right)\hfill \\ & \times & \left(C_{1}y\mathrm{J}\left(0,\frac{y}{\sqrt{-A_1}}\right)+C_{2}y\mathrm{Y}\left(0,\frac{y}{\sqrt{-A_1}}\right)\right)f\left(u\right),\hfill \end{array}$$

(22)

where $J,Y$ are Bessel functions, lowest order expansion suggests that the $C_1$ term is real but the $C_2$ term might be complex.

3.2. A Simpler Case

A more simple solution to (21) is

$$H=\left(B_{1}x+B_{2}xln\left(x\right)\right)\left(C_{1}y+C_{2}yln\left(y\right)\right)f\left(u\right).$$

(23)

Curvature is characterized by invariants. $K,WeylSq\equiv C_{abcd}{C}^{abcd},RicciSq\equiv R_{ab}{R}^{ab}$ are the ‘squares’ of the Riemann tensor, Weyl tensor and Ricci tensor, respectively, $R_1\cdots$ are the Carminati McLenaghan invariants [10]. Here the invariants can be expressed in terms of the Ricci scalar

$$R=\frac{A_1{x}^2+A_2{y}^2}{2A_1{A}_2{x}^2{y}^2},$$

(24)

and are

$$\begin{array}{ccc}\hfill & & K=3R^2,WeylSq=\frac{4}{3}{R}^2,RicciSq=R^2,BS=\frac{4}{9}{R}^4\hfill \\ & & R_1=\frac{3}{16}{R}^2,R_2=\frac{3}{64}{R}^3,R_3=\frac{21}{1024}{R}^4,W_{1R}=\frac{1}{6}{R}^2,W_{2R}=\frac{1}{36}{R}^3,\hfill \\ & & M_{2R}=M_3=\frac{1}{96}{R}^4,M_4=\frac{1}{768}{R}^5,M_{5R}=\frac{1}{576}{R}^5.\hfill \end{array}$$

(25)

3.3. Null Tetrads for the Scalar-Einstein Wave

A suitable set of null tetrads is

$$\begin{array}{ccc}& & l_a=-\frac{1}{2}xyH\left(u\right))(B_1+B_{2}ln\left(x\right))(C_1+C_{2}ln\left(y\right)){\delta }_a^u-A_{3}xy{\delta }_a^v,\hfill \\ & & n_a={\delta }_a^u,m_a=\frac{A1}{\sqrt{{}_2}}{\delta }_a^x+\frac{i{A}_2}{\sqrt{2}}{\delta }_a^y,\hfill \end{array}$$

(26)

the Weyl and Ricci scalars are

$$\begin{array}{ccc}& & {\mathrm{\Phi }}_{02}=\frac{{(\sqrt{A_1}x-\sqrt{A_2}y)}^2}{8A_1{A}_2{x}^2{y}^2},{\mathrm{\Phi }}_{11}=\frac{1}{8}R,\hfill \\ & & {\mathrm{\Psi }}_2=\frac{1}{6}R,{\mathrm{\Psi }}_4=i\times \frac{\overline{p}H\left(u\right)}{4\sqrt{A_1{A}_2}{A}_3^2{x}^2{y}^2},\hfill \\ & & \overline{p}\equiv B_1{C}_2+B_2{C}_1+B_2{C}_2\left(2+ln\left(x\right)+ln\left(y\right)\right).\hfill \end{array}$$

(27)

4. The Bel-Robinson Tensor

How good the Bel-Robinson tensor is for describing energy has to be assessed. The problem of whether gravitational waves transfer energy has two parts: finding a good criteria to measure energy and finding suitable wave solutions.

4.1. Definition of the Bel-Robinson Tensor

The dual of a tensor is defined by

$$\ast T_{abm\dots }=\frac{1}{2}{ϵ}_{ab..}^{cd}{T}_{cdm\dots },$$

(28)

where $m\dots$ are a set of indices. The Bel-Robinson tensor is defined by

$$B_{cdef}\equiv C_{acdb}{C}_{.ef.}^{ab}+\ast C_{acdb}\ast C_{.ef.}^{ab},$$

(29)

The four-vector indicator of energy-momentum is

$$P_a=B_{abcd}{V}^b{V}^c{V}^d,$$

(30)

compare [11] Horowitz and Schmidt (1982). Usually V is taken to be a time-like vector field, but other choices are possible: for example when V is replaced by the null tetrad vector l the indicator is referred to a $N_a$.

4.2. The Bel-Robinson Tensor for the Schwarzschild Solution

The Schwarzschild solution has line element

$$d{s}^2=-\left(1-\frac{2m}{r}\right)d{t}^2+\frac{1}{1-\frac{2m}{r}}d{r}^2+r^{2}d{\Sigma }_2^2,d{\Sigma }_2^2\equiv d{\theta }^2+sin{\left(\theta \right)}^{2}d{\varphi }^2,$$

(31)

it has Weyl scalar

$${\mathrm{\Psi }}_2=-\frac{m}{r^3}.$$

(32)

For the time-like vector field

$$T_a\equiv f{\delta }_a^t,T_a{T}^a=-\frac{f^2}{1-\frac{2m}{r}},$$

(33)

and the indicating four-vector (30) is

$$P_a=-\frac{6f^2{\mathrm{\Psi }}_2^2}{1-\frac{2m}{r}}{T}_a,$$

(34)

which is conserved $P_{.;a}^a=0$.

4.3. The Bel-Robinson Tensor for the Plane Wave

The Bel-Robinson tensor for the plane wave has one component

$$B=B_{uuuu}=\frac{1}{4}\left(W_{zz}-W_{yy}\right)+W_{yz}^2,$$

(35)

the indicating four-vector is best (30) with V replaced by l from (17) giving

$$N_a=B{n}_a,$$

(36)

which is conserved. For the particular choice (12) with $a,b=1$

$$B=4\left(f^2+g^2\right).$$

(37)

4.4. The Bel-Robinson Tensor for Imploding Scalar Spacetime

The line element is taken to be [9]

$$d{s}^2=-(1+2\sigma )d{v}^2+2dvdr+r(r-2\sigma v)d{\Sigma }_2^2,\varphi =\frac{1}{2}ln\left(1-\frac{2\sigma v}{r}\right).$$

(38)

with $d{\Sigma }_2^2$ given by (31) and complementary null coordinate $u\equiv (1+2\sigma )v-2r$. In this case study of the Bel-Robinson tensor is not so straightforward and here is approached by four methods.

4.4.1. Time-like Vector Method

A time-like vector is

$$T_a=-\left(\frac{1}{2}(1+T_s)+\sigma \right){\delta }_a^v+{\delta }_a^r,T_a{T}^a=-T_s,$$

(39)

and the indicating four-vector (30) is

$$P_a=-\frac{1}{6}{T}_s{R}^2{T}_a,$$

(40)

where R is the Ricci scalar, (40) is not conserved in general

$$P_{.;a}^a=\frac{2T_s{\sigma }^{4}uv}{3r^5{(-r+2\sigma v)}^5}\left(u{t}_1-T_{s}v{t}_2\right),$$

(41)

where

$$t_1\equiv 3\sigma (1+2\sigma )v^2-(3+4\sigma )rv+2r^2,t_2\equiv t_1+2r(r-2\sigma v),$$

(42)

however it is conserved in the two particular cases $T_s=u{t}_1/v{t}_2$ and $T_s=0$ which is null and we go to next.

4.4.2. Null Tetrad Method

The indicating four-vector (30) with l replacing V just gives $N_a=0$, using mixtures of null tetrad vectors instead of l no simple pattern arises.

4.4.3. Killing Vector Method

The solution (38) has a homeothetic Killing potential $K=cuv$ wich can be partially differentiated to give a homeothetic Killing vector, using this in (30) gives

$$A_a=-\frac{2}{3}cK{K}_a,A_{.;a}^a=\frac{8}{3}{c}^{2}K{R}^2.$$

(43)

4.4.4. Scalar Field Everywhere Method

In (30) one uses the gradient of the scalar field (38) everywhere, then

$$A_a=\frac{1}{12}{R}^3{\varphi }_a,A_{.;a}^a=\frac{(r^2-(1+2\sigma )v(r-\sigma v))R^4}{2\sigma uv}.$$

(44)

4.5. The Bel-Robinson Tensor for the Scalar-Einstein Wave

If one uses a time-like vector in (30) the no simple pattern arises, however for the null tetrads (26) one gets

$$N_a=-4A_3^4{x}^4{y}^4{\mathrm{\Psi }}_4^2{n}_a.N_{.;a}^a=0,$$

(45)

The scalar field everywhere method gives

$$A_a=\frac{1}{12}{R}^3{\varphi }_a,A_{.;a}^a=\frac{1}{2}{R}^4\frac{A_1{x}^2-A_2{y}^2}{A_1{x}^2+A_2{y}^2}.$$

(46)

The stress of the scalar field gives scalar field propagation

$$P_{\varphi }^a=T^{ab}{V}_b=-\frac{1}{2}R{V}^a,$$

(47)

with similar equations for the null tetrad.

5. Conclusions

The indicating four vector (34) is what one would want for the Schwarzschild solution, it is timelike and furthermore co-directional with the chosen vector field (33); however why the proportional function takes the form it does does not seem to be predictable beforehand, one might hope to identify m as the overall energy just from the proportional function. Similarly, the indicating four vector (36) is what one would want for the plane wave, it is null and furthermore co-directional with the chosen null tetrad (17).

The energetics of the imploding scalar solution (38) are important: the solution has no overall energy the negative energy of the gravitational field and the positive energy of the scalar field cancel out, and there is the question of what happens locally where the scalar field energy can be measured but not the gravitational field energy. For the imploding scalar solution (38) the null tetrad method gives vanishing indicator and the other three methods usually have non-vanishing conservation equation, the exception being when $T_s=u{t}_1/v{t}_2$ in (41). The non-vanishing of the conservation equations are of high order in $\sigma$. This leads to the conclusion that there is no local balance in energy exchange: it is only the global energy that cancels, and any detail of how this can happen remains obscure.

For the scalar-Einstein wave the null indicator (45) is conserved, suggesting that the square of gravitational energy is co-moving with the wave. However the solutions scalar field does not seem to co-move (47), suggesting that the scalar field is an ambient medium.

A convenient property of the indicating four vector (30) is that it always turns out to be proportional to a known vector of the spacetime, for the non-null case this is the transvecting vector, for the null tetrad it is the complimentary null vector, for example the indicating four vector (30) is proportional to n after transvecting with l: this appears to be co-incidence. That the conserved indicating vector for the scalar-Einstein wave is null (45) indicates that the transfer of the square of its gravitational energy is null and that the presence of the scalar field does not impede it thus answering the question implied by the title.