The Lanczos Potential for Bianchi Spacetime

Abstract

The form and application of the Lanczos potential for Bianchi spacetime are studied. The Lanczos potential is found in some specific cases, then the general case is studied. It leads to two coupled first order partial differential equations which, although they so far have not been solved in general, can be solved for many configurations. The application is to cosmic energetics: in other words to the study of the energy of the gravitational and other fields in the Universe.

Contents
1. Introduction..........................................................................................................................................	1
2. The Lanczos Potential.........................................................................................................................	2
3. Schwarzschild Spacetime..................................................................................................................	4
4. Vacuum Levi–Cevita Spacetime.......................................................................................................	4
5. Non-Vacuum Levi–Cevita..................................................................................................................	6
6. Exponential Spacetime.......................................................................................................................	8
7. The General Case.................................................................................................................................	9
8. Cosmic Energetics..............................................................................................................................	12
9. Conclusions.........................................................................................................................................	12
References................................................................................................................................................	13

1. Introduction

In cosmology, usually the spherically symmetric spacial part is assumed and called Robertson–Walker spacetime. Spherical symmetry is clearly a big assumption as the Universe is far from a sphere. There are two ways around this problem. The first is two assume a near spherical symmetry and perturb the spacetime, this is called cosmolgical perturbation theory. The second is to try different shapes which have been conveniently categorised by Bianchi. For Biachi spacetimes, the salient feature is that the Weyl tensor no longer vanishes as it does for Robertson–Walker, so it is of interest to study the Weyl tensor of Bianchi spacetimes as much as possible. The Weyl tensor has a potential called the Lanczos potential and this is studied here.

The Lanzcos potential [1] discovered in (1962) is a potential tensor for the Weyl tensor in a similar manner to the vector potential A being a potential for the Maxwell–Faradaty tensor F. Just as for electromagnetism where one can form as stress energy tensor T one can form an energy tensor (13) [2]. After finding exact expressions for the Lanczos potential and its form in general whether associated scalars and energy expressions have application in the early or late universe is discussed. The Lanzcos potential [1] is a first derivative tensor and so might have connection to properties of the covariant derivation of vector fields, such as the shear, and, thus, to chaotic and dynamical systems in cosmology [3,4], this is not looked at here. The only application looked at here is the energetics [2,5] of the various spacetimes.

In Section 2, the Lanczos potential and its associated energy and analogs of the Weyl scalars are defined. In Section 3, for illustration, these objects are derived for Schwarzschild spacetime. The Lanczos potential is known for the Levi–Cevita spacetime and is reproduced in Section 4. In Section 5, this is generalized for the case where the vacuum constraints are not applied, this is done by introducing a ’grand’ shear tensor which is used in all subsequent analysis. In Section 6, the properties of a spacetime with exponentials replacing the powers of Levi–Cevita is discussed. The general case is discussed in Section 7 where the general geometric objects transvected Bel–Robinson tensor, tranvected Lanczos potential energy, and Lanczos scalars are given. Section 8 applies the forgoing to speculation on the nature of gravitational energy in the Universe. Section 9 is the conclusion.

Conventions used include putting a dot where there is a raised indices.

2. The Lanczos Potential

The Weyl tensor can be expressed in terms of the Lanczos potential

$$\begin{array}{ccc}\hfill {\stackrel{}{C}}_{abcd}& =& {\stackrel{1}{C}}_{abcd}+{\stackrel{2}{C}}_{abcd}+{\stackrel{3}{C}}_{abcd}\hfill \\ \hfill {\stackrel{1}{C}}_{abcd}& \equiv & H_{abc;d}-H_{abd;c}+H_{cda;b}-H_{cdb;a},{\stackrel{3}{C}}_{abcd}\equiv \frac{4}{(1-D)(2-D)}{H}_{..e;f}^{ef}(g_{ac}{g}_{bd}-g_{ad}{g}_{bc}),\hfill \\ \hfill {\stackrel{2}{C}}_{abcd}& \equiv & \frac{1}{(2-D)}\left\{g_{ac}(H_{bd}+H_{db})-g_{ad}(H_{bc}+H_{cb})+g_{bd}(H_{ac}+H_{ca})-g_{bc}(H_{ad}+H_{da})\right\},\hfill \end{array}$$

(1)

where the coefficients of $\stackrel{2}{C}$ and $\stackrel{3}{C}$ are fixed by requiring that the Weyl tensor obeys the trace condition $C_{.bad}^a=0$. The higher dimension equations were first given in [6]. $H_{bd}$ is defined by

$$H_{bd}\equiv H_{b.d;e}^e-H_{b.e;d}^e.$$

(2)

The Lanczos potential has the symmetries

$$2H_{\left[ab\right]c}\equiv H_{abc}+H_{bac}=0,6H_{\left[abc\right]}\equiv H_{abc}+H_{bca}+H_{cab}=0.$$

(3)

Equation (1) is invariant under the algebraic gauge transformation

$$H_{abc}\to H_{abc}^{\prime }=H_{abc}+{\chi }_a{g}_{bc}-{\chi }_b{g}_{ac},$$

(4)

where ${\chi }_a$ is an arbitrary four vector, this transformation again fixes the coefficients of $\stackrel{2}{C}$ and $\stackrel{3}{C}$.

In four dimensions, the Lanczos potential with the above symmetries has 20 degrees of freedom, but the Weyl tensor has 10. Lanczos reduced the degrees of freedom to 10 by choosing the algebraic gauge condition

$$3{\chi }_a=H_{a.b}^b=0,$$

(5)

and the differential gauge condition

$$L_{ab}=H_{ab.;c}^c=0.$$

(6)

Introducing a null tetrad $l,n,m,\overline{m}$ for signature $+++-$

$$\begin{array}{ccc}\hfill & & 1=-l·n=-n·l=m·\overline{m}=\overline{m}·m,\hfill \\ & & 0=l^2=n^2=m^2={\overline{m}}^2=l·m=l·\overline{m}=n·m=n·\overline{m},\hfill \\ & & g_{ab}=-l_a{n}_b-n_a{l}_b+m_a{\overline{m}}_b+{\overline{m}}_a{m}_b,\hfill \end{array}$$

(7)

the Weyl scalars are

$$\begin{array}{ccc}\hfill {\Psi }_0& \equiv & C_{abcd}l{}^{a}m{}^{b}l{}^{c}m{}^d,\hfill \\ \hfill {\Psi }_1& \equiv & C_{abcd}l{}^{a}n{}^{b}l{}^{c}m{}^d,\hfill \\ \hfill {\Psi }_2& \equiv & C_{abcd}l{}^{a}m{}^b\overline{m}{}^{c}n{}^d,\hfill \\ \hfill {\Psi }_3& \equiv & C_{abcd}l{}^{a}n{}^b\overline{m}{}^{c}n{}^d,\hfill \\ \hfill {\Psi }_4& \equiv & C_{abcd}n{}^a\overline{m}{}^{b}n{}^c\overline{m}{}^d,\hfill \end{array}$$

(8)

there are similar objects for the Lamczos potential [7,8,9]

$$\begin{array}{ccc}\hfill H_0& \equiv & H_{abc}{l}^a{m}^b{l}^c,\hfill \\ \hfill H_1& \equiv & H_{abc}{l}^a{m}^b{\overline{m}}^c,\hfill \\ \hfill H_2& \equiv & H_{abc}{\overline{m}}^a{n}^b{l}^c,\hfill \\ \hfill H_3& \equiv & H_{abc}{\overline{m}}^a{n}^b{\overline{m}}^c,\hfill \\ \hfill H_4& \equiv & H_{abc}{l}^a{m}^b{m}^c,\hfill \\ \hfill H_5& \equiv & H_{abc}{l}^a{m}^b{n}^c,\hfill \\ \hfill H_6& \equiv & H_{abc}{\overline{m}}^a{n}^b{m}^c,\hfill \\ \hfill H_7& \equiv & H_{abc}{\overline{m}}^a{n}^b{n}^c,\hfill \end{array}$$

(9)

Energy tensors can be constructed from both the Weyl tensor and the Lanczos potential. Define the dual to be on the first two indices

$$\ast H_{abc}\equiv \frac{1}{2}{ϵ}_{abef}{H}_{..c}^{ef},$$

(10)

and similarly dualing over the first two indices for the Weyl tensor. Then, from the Weyl tensor one can construct the Bel–Robinson tensor [5]

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

(11)

which has dimensions energy squared, given a until time-like vector field V the energy squared is

$$B_v\equiv B_{abcd}{V}^a{V}^b{V}^c{V}^d,$$

(12)

which is of a similar form to the energy conditions [10] (p. 95); in most cases looked at here it is proportional to the Weyl tensor squared $\mathrm{WeylSq}=C_{abcd}{C}^{abcd}$. Analogously from the Lanczos potential [2] there is the energy tensor

$$H{E}_{ab}\equiv H_{acd}{H}_{b..}^{cd}+\ast H_{acd}\ast H_{b..}^{cd},$$

(13)

which has dimensions energy, given a until time-like vector field V the transvected energy is

$$H_v\equiv H{E}_{ab}{V}^a{V}^b,$$

(14)

which has the correct dimensions of energy, see also [11]. It has the advantage that it gives a sign for the gravitational energy, but has difficult interpretation as to whether it really is a measure of energy. In many cases, the transvected Bel–Robinson is proportional to the square to the Weyl tensor $C_{abcd}{C}^{abcd}$ and in the Lanczos case proportional to $H_{abc}{H}^{abc}$.

3. Schwarzschild Spacetime

In the most commonly used coordinates the Schwarzschild line element is

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

(15)

The Weyl tensor is given by

$$\begin{array}{ccc}\hfill & C_{r\theta r\theta }=\frac{C_{r\varphi r\varphi }}{sin{(\theta )}^2}=\frac{m}{2m-r},& C_{rtrt}=-\frac{2m}{r},\hfill \\ & C_{\theta t\theta t}=\frac{C_{\varphi t\varphi t}}{sin{(\theta )}^2}=\frac{m(r-2m)}{r^2},& C_{\theta \varphi \theta \varphi \theta }=2mrsin{(\theta )}^2,\hfill \end{array}$$

(16)

and its dual by

$$\ast C_{r\varphi \theta t}=-\ast C_{r\theta \varphi t}=\frac{1}{2}\ast C_{rt\theta \varphi }=\frac{m}{r}sin(\theta ).$$

(17)

The Lanczos potential is given by

$$\begin{array}{ccc}\hfill & H_{\theta r\theta }=\frac{H_{\varphi r\varphi }}{sin{(\theta )}^2}=\frac{mr}{3(r-2m)},& H_{trr}=\frac{1}{r(r-2m)},\hfill \end{array}$$

(18)

$$\begin{array}{ccc}& H_{\theta t\theta }=\frac{H_{\varphi t\varphi }}{sin{(\theta )}^2}=\frac{1}{2r},& H_{trt}=\frac{2m}{3r^2},\hfill \end{array}$$

(19)

and its dual by

$$\begin{array}{ccc}\hfill & & \ast H_{\varphi \theta r}=2\ast H_{\varphi r\theta }=2\ast H_{r\theta \varphi }=\frac{sin(\theta )}{2m-r},\hfill \\ & & \ast H_{t\varphi \theta }=\ast H_{\theta t\varphi }=\frac{1}{2}\ast H_{\theta \varphi t}=\frac{m}{3}sin(\theta ).\hfill \end{array}$$

(20)

The energy squared and energy with respect to the unit time-like vector

$$V_a=\left[0,0,0,\sqrt{1-\frac{2m}{r}}\right],$$

(21)

are (12,14)

$$B_v=\frac{6m^2}{r^6}=\frac{1}{8}\mathrm{WeylSq},H_v=\frac{27-4m^2{r}^2}{18r(r-2m)},$$

(22)

there is the relationship

$$H_1{H}_6=\frac{1}{24}{H}_{abc}{H}^{abc}=-\frac{1}{24}\ast H_{abc}\ast H^{abc}.$$

(23)

Taking the tetrad

$$l_a=\left[1,0,0,1-\frac{2m}{r}\right],n_a=\left[\frac{r}{2(2m-r)},0,0,\frac{1}{2}\right],m_a=\left[0,-\frac{\mathit{i}r}{\sqrt{2}},-\frac{rsin(\theta )}{\sqrt{2}},0\right],$$

(24)

the non-vanishing Weyl and Lanczos scalars are (8) and (9)

$${\Psi }_2=-\frac{m}{r^3},H_1=\frac{3-2mr}{6r^3},H_6=\frac{3+2mr}{12r^2(2m-r)}.$$

(25)

4. Vacuum Levi–Cevita Spacetime

The line element is taken to be

$$d{s}^2=t^{2p_1}d{x}^2+t^{2p_2}d{y}^2+t^{2p_3}d{z}^2-d{t}^2.$$

(26)

The Riemann tensor is

$$R_{xyxy}=p_1{p}_2{t}^{2p_1+2p_2-2},R_{xtxt}=p_1(1-p_1)t^{2p_1-2},$$

(27)

with $R_{xzxz},R_{yzyz},R_{ytyt},R_{ztzt}$ following by symmetry. The Ricci tensor is

$$R_{xx}=p_1(p_1+p_2+p_3-1)t^{2p_1-2},R_{tt}=\frac{1}{t^2}\left(p_1+p_2+p_3-p_1^2-p_2^2-p_3^2\right),$$

(28)

with $R_{yy},R_{zz}$ following by symmetry. The Ricci scalar is

$$R=\frac{2}{t^2}\left(-p_1-p_2-p_3+p_1{p}_2+p_1{p}_3+p_2{p}_3+p_1^2+p_2^2+p_3^2\right).$$

(29)

The Weyl tensor is

$$\begin{array}{ccc}\hfill & & C_{xyxy}=-\frac{1}{6}\left(-p_1-p_2-p_3-2p_1{p}_2+p_1{p}_3+p_2{p}_3+p_1^2+p_2^2-2p_3^2\right)t^{2p_1+2p_2-2},\hfill \\ & & C_{xtxt}=-\frac{1}{6}\left(-2p_1+p_2+p_3-p_1{p}_2-p_1{p}_3+2p_2{p}_3+2p_1^2-p_2^2=p_3^2\right)t^{2p_1-2},\hfill \end{array}$$

(30)

with $C_{xzxz},C_{yzyz},C_{ytyt},C_{ztzt}$ following by symmetry. The dual of the Weyl tensor is

$$\ast C_{xyzt}=-\frac{1}{6}\left(-p_1-p_2+2p_3-2p_1{p}_2+p_1{p}_3+p_2{p}_3+p_1^2+p_2^2-2p_3^2\right)t^{p_1+p_2+p_3-2},$$

(31)

with $\ast C_{xzyt},\ast C_{xtyz}$ following by symmetry.

From the Ricci tensor (28) for a vacuum

$$p_1+p_2+p_3=p_1^2+p_2^2+p_3^2=1,$$

(32)

the first equality in (32) gives the vacuum condition

$$p_3=1-p_1-p_2,$$

(33)

and the second equality in (32) gives the vacuum condition

$$p_2=\frac{(1-p_1)}{2}\pm \frac{1}{2}{\mathrm{rt}}_1,{\mathrm{rt}}_1\equiv \sqrt{(1-p_1)(1+3p_1^2)}.$$

(34)

The transvected Bel–Robinson tensor (12) is

$$\begin{array}{ccc}\hfill B_v& =& \frac{1}{8}\mathrm{WeylSq}\hfill \\ & =& \frac{1}{6t^4}\left[p_1^4+p_2^4+p_3^4-p_1^3(p_2+p_3+2)-p_2^3(p_1+p_3+2)-p_3^3(p_1+p_2+2)\right.\hfill \\ & & +p_1^2(p_2{p}_3+2(p_2+p_3)+1)+p_2^2(p_1{p}_3+2(p_1+p_3)+1)+p_3^2(p_1{p}_2+2(p_1+p_2)+1)\hfill \\ & & \left.-6p_1{p}_2{p}_3-p_1{p}_2-p_1{p}_3-p_2{p}_3\right],\hfill \end{array}$$

(35)

imposing (33)–(35) reduces to

$$B_v=\frac{2(1-p_1)p_1^2}{t^4}.$$

(36)

The unit time-like vector field is

$$V_a=\left[0,0,0,1\right],$$

(37)

which without imposing (33) and (34) it has shear

$${\sigma }_{ab}=\frac{1}{3}\mathrm{diag}\left\{(-2p_1+p_2+p_3)t^{2p_1-1},(p_1-2p_2+p_3)t^{2p_2-1},(p_1+p_2-2p_3)t^{2p_3-1},0\right\}.$$

(38)

The Lanczos potential, compare [12], is

$$H_{abc}=\frac{1}{3}\left({\sigma }_{ac}{V}_b-{\sigma }_{bc}{V}_a\right),$$

(39)

provided the vacuum condition (33) is imposed, it is not necessary to impose (34). The energy (14) is

$$\begin{array}{ccc}\hfill H_v& =& \frac{2}{27t^2}\left(p_1^2+p_2^2+p_3^2-p_1{p}_2-p_1{p}_3-p_2{p}_3\right)\hfill \\ & =& \frac{4}{27}{S}_{.t}^t+\frac{2}{27t^2}(p_1+p_2+p_3),\hfill \end{array}$$

(40)

where $S_{ab}$ is the traceless Ricci tensor, imposing (33) and (34) this becomes

$$H_v=\frac{2}{27t^2},$$

(41)

which is positive and independent of p.

Choosing the tetrad compare [13]

$$l_a=[\sqrt{g_{xx}},0,0,1]/\sqrt{2},n_a=[-\sqrt{g_{xx}},0,0,1]/\sqrt{2},m_a=[0,\sqrt{g_{yy}},\mathit{i}\sqrt{g_{zz}},0]/\sqrt{2},$$

(42)

and imposing (33) and (34), the Weyl scalars (8) are

$${\Psi }_0=4{\Psi }_4=-\frac{p_1}{2t^2}{\mathrm{rt}}_1,{\Psi }_2=\frac{p_1(1-p_1)}{t^2},$$

(43)

the Lanczos scalars (9) are

$$H_1=-H_6=\frac{(3p_1-1)}{18\sqrt{2}t},H_4=-2H_3=\frac{{\mathrm{rt}}_1}{6\sqrt{2}t},$$

(44)

with $r{t}_1$ defined in (34).

5. Non-Vacuum Levi–Cevita

For the line element (26) without imposing the second constraint (34) it is necessary to add a new diagonal tensor, here called the ’grand’ shear, to the shear (38)

$${\Sigma }_{ab}=\mathrm{diag}\left\{f_1(t)g_{xx},f_2(t)g_{yy},f_3(t)g_{zz},0\right\},$$

(45)

where the explicit dependence on t is left out when the ellipsis is clear, i.e., $f_1(t)\to f_1$. the algebraic gauge condition (5) entails that the tensor is traceless here realized by $f_1=-f_2-f_3$. For the line element (26), the f’s are given by

$$\begin{array}{ccc}\hfill & & f_2=\frac{(p_1+p_2+p_3-1)}{6(d-1)t}\left(p_1^2-2p_2^2+p_3^2+2p_1{p}_2-4p_1{p}_3+2p_2{p}_3+p_1-2p_2+p_3\right),\hfill \\ & & f_3=\frac{(p_1+p_2+p_3-1)}{6(d-1)t}\left(p_1^2+p_2^2-2p_3^2-4p_1{p}_2+2p_1{p}_3+2p_2{p}_3+p_1+p_2-2p_3\right),\hfill \\ & & d\equiv (p_1^2+p_2^2+p_3^2-p_1{p}_2-p_1{p}_3-p_2{p}_3),\hfill \end{array}$$

(46)

to which can be added the non-contributing terms

$$\begin{array}{ccc}\hfill & & f_{2C}=C_{1}cos\left(\sqrt{-d-2}ln(t)\right)+C_{2}sin\left(\sqrt{-d-2}ln(t)\right),\hfill \\ & & f_{3C}=C_{3}cos\left(\sqrt{-d-2}ln(t)\right)+C_{4}sin\left(\sqrt{-d-2}ln(t)\right),\hfill \\ & & C_3=\frac{1}{p_1-p_3}\left((p_3-p_2)C_1-\sqrt{d-2}{C}_2\right),C_4=\frac{1}{p_1-p_3}\left((p_3-p_2)C_2+\sqrt{d-2}{C}_1\right),\hfill \end{array}$$

where $C_1,C_2,C_3,C_4$ are constants, these terms do not explicitly contribute to the Weyl tensor via (1) but cancel out. The grand shear’s (45) properties are that it is tracefree, this is a consequence of imposing the algebraic gauge condition (5), and that taking the Lie derivative with respect to the unit time-like vector field (37) gives

$${\mathcal{L}}_v{\Sigma }_{ab}\equiv {\Sigma }_{ab;c}{V}^c=\frac{1}{t}{\Sigma }_{ab}.$$

(47)

Another property is that (45) can be reduced to a covariant derivative of a vector field which, in turn, can be thought of as a potential compare [14], but as this is not useful for present purposes so we stay with (45). The Bel–Robinson tensor is still given by (35). Now the Lanczos potential (39) has an additional term

$$H_{abc}=\frac{1}{3}\left((k{\sigma }_{ac}+{\Sigma }_{ac})V_b-(k{\sigma }_{bc}+{\Sigma }_{bc})V_a\right),$$

(48)

where except for Section 7 $k=1$. The Lanczos potential transvected energy (14) is

$$\begin{array}{ccc}\hfill H_v& =& \frac{1}{6{(d-1)}^2{t}^2}\left[+(p_1^6+p_2^6+p_3^6)-2((p_2+p_3)p_1^5+(p_1+p_3)p_2^5+(p_1+p_2)p_3^5)\right.\hfill \\ & & +(2p_2^2+2p_3^2+2p_2{p}_3+p_2+p_3-2)p_1^4+(2p_1^2+2p_3^2+2p_1{p}_3+p_2+p_3-2)p_2^4\hfill \\ & & +(2p_1^2+2p_2^2+2p_1{p}_3+p_1+p_2-2)p_3^4-2(p_1^3{p}_2^3+p_1^3+p_2^3{p}_3^3)\hfill \\ & & +(-p_2^2-p_3^2-4p_2{p}_3+3p_2+3p_3)p_1^3+(-p_1^2-p_3^2-4p_1{p}_3+3p_1+3p_3)p_2^3\hfill \\ & & +(-p_1^2-p_2^2-4p_1{p}_2+3p_1+3p_2)p_3^3-3p_1^2{p}_2^2{p}_3^2+4(p_2{p}_1^2{p}_3^2+p_3{p}_1^2{p}_2^2+p_1{p}_2^2{p}_3^2)\hfill \\ & & +6p_1{p}_2{p}_3-(p{1}^2{p}_3^3+p_2^2{p}_3^2+p_1^2{p}_3^2++3p_2{p}_3{p}_1^2+3p_1{p}_3{p}_2^2+p_1{p}_2{p}_3^2)\hfill \\ & & +(1-p_2-p_3)p_1^2+(1-p_1-p_3)p_2^2+(1+p_1+p_2)p_3^2\hfill \\ & & \left.-p_1{p}_2-p_1{p}_3-p_2{p}_3\right],\hfill \end{array}$$

(49)

with d defined in (46).

Using the same null tetrad (42) as before the Weyl and Ricci scalars are

$$\begin{array}{ccc}\hfill & & {\Psi }_0=4{\Psi }_4={\Phi }_{02}={\Phi }_{20}=\frac{(p_2-p_3)(p_1-p_2-p_3+1)}{4t^2},\hfill \\ & & {\Psi }_2=\frac{1}{12t^2}\left(-2p_1^2+p_2^2+p_3^2+p_1{p}_3+p_1{p}_3-2p_2{p}_3+2p_1-p_2-p_3\right),\hfill \\ & & {\Phi }_{00}={\Phi }_{22}=\frac{(p_1{p}_2+p_2{p}_3+p_2-p_2^2+p_3-p_3^2)}{4t^2},{\Phi }_{11}=\frac{(-p_1^2+p_1+p_2{p}_3)}{4t^2},\hfill \end{array}$$

(50)

the Lanczos scalars are

$$\begin{array}{ccc}\hfill H_1& =& -H_6=-\frac{1}{2\sqrt{2}(d-1)}\times \hfill \\ & & [-2p_1^3+p_2^3+p_3^2+2(p_2+p_3)p_1^2-(p_1+p_3)p_2^2-(p_1+p_2)p_3^2\hfill \\ & & -p_1{p}_2-p_1{p}_3+2p_2{p}_3-2p_1-p_2-p_3],\hfill \\ \hfill H_4& =& -H_3=\frac{3}{2\sqrt{2}(d-1}\left((p_2-p_3)(p_2^2+p_3^2-(p_2+p_3+1)p_1-1\right),\hfill \end{array}$$

(51)

with d defined in (46).

In order to investigate how the vacuum Levi–Cevita ’sits’ in the non-vacuum transform the p constants to $ϵ$ constants

$$\begin{array}{ccc}\hfill & & {ϵ}_2\equiv p_1+p_2+p_3-1,{ϵ}_3\equiv p_1^2+p_2^2+p_3^2-1,\hfill \\ & & p_2=\frac{1}{2}(1+{ϵ}_2-p_1)+\frac{1}{2}r{t}_2,p_3=\frac{1}{2}(1+{ϵ}_2-p_1)-\frac{1}{2}r{t}_2,\hfill \\ & & r{t}_2^2\equiv (1-p_1)(3p_1+1)+{ϵ}_2(-{ϵ}_2-2(1-p_1))+2{ϵ}_3,\hfill \end{array}$$

(52)

and because there is only one p left drop the index on $p_1\to p$. Despite the $ϵ$ notation all the following are exact expressions, no expansions are used. Firstly, look at the ${ϵ}_2=0$ case, calculations are most easily achieved applying this at the last moment as the are expressions with ${ϵ}_2$ in both numerator and denominator, the Equations (35), (41), (50) and (51) reduce to

$$\begin{array}{ccc}\hfill & & B_v=\frac{1}{8}WeylSq=\frac{1}{6t^4}\left(12p(1-p)+6p{ϵ}_3+{ϵ}_3^2\right),H_v=\frac{2+3{ϵ}_3}{27t^2},\hfill \\ & & {\Psi }_0={\Psi }_4=\frac{6p}{\sqrt{2}t}{H}_3=-\frac{6p}{\sqrt{2}t}{H}_3=\frac{p}{2t^2}\sqrt{1+2p-3p^2+2{ϵ}_3},\hfill \\ & & {\Psi }_2=-\frac{1}{12t^2}\left(6p(p-1)-2{ϵ}_3\right),{\Phi }_{00}={\Phi }_{22}=2{\Phi }_{11}=6{\Phi }_l=-\frac{{ϵ}_3}{4t^2},\hfill \\ & & H_1=-H_6=\frac{\sqrt{2}}{36t}(3p-1),\hfill \end{array}$$

(53)

$H_v$ remains explicitly independent of p as in the vacuum case (41), although they can be thought of as an implicit dependence via ${ϵ}_3$, for ${ϵ}_3<-2/3$ it changes sign. Secondly look at the ${ϵ}_3=0$ case, the Equations (35), (41), (50) and (51) reduce to

$$\begin{array}{ccc}\hfill & & B_v=\frac{2(1-p)p^2}{t^4}+\frac{p(p+2)(p-1)}{t^4}{ϵ}_2+\frac{1-2p^2}{2t^4}{ϵ}_2^2+\frac{p}{2t^4}{ϵ}_2^3-\frac{1}{12t^2}{ϵ}_2^4,\hfill \\ & & H_v=\frac{1}{6{(2+{ϵ}_2)}^2{t}^2}\left[12p^2(1-p)-12p{(p-1)}^2{ϵ}_2+(4-18p+12p^2){ϵ}_2^2+2(2-3p){ϵ}_2^3+{ϵ}_2^4\right],\hfill \\ & & {\Psi }_2=-\frac{1}{12t^2}\left[6p(p-1)+3(1-p){ϵ}_2+{ϵ}_2^2\right],{\Psi }_0={\Psi }_4=\frac{2p-{ϵ}_2}{4t^2}r{t}_3,\hfill \\ & & {\Phi }_{00}={\Phi }_{22}=\frac{1+p}{4t^2}{ϵ}_2,{\Phi }_{11}=\frac{(2(1-p)+{ϵ}_2)}{8t^2}{ϵ}_2,{\Phi }_l=\frac{1}{24t^2}{ϵ}_2^2,\hfill \\ & & H_1=-H_6=-\frac{\sqrt{2}}{12(2+{ϵ}_2)t}\left[3p(p-1)+(2-3p){ϵ}_2+{ϵ}_2^2\right],H_3=-H_4=\frac{\sqrt{2}p}{4(2+{ϵ}_2)t}r{t}_3,\hfill \\ & & r{t}_3^2\equiv 1+2p-3p^2+2(p-1){ϵ}_2-{ϵ}_2^2\hfill \end{array}$$

(54)

and in this case (41) is not explicitly independent of p. The significance of (53) and (54) is discussed in Section 9.

6. Exponential Spacetime

The line element is taken to be

$$d{s}^2=exp(2p_{1}t)d{x}^2+exp(2p_{2}t)d{y}^2+exp(2p_{3}t)d{z}^2-d{t}^2.$$

(55)

The Riemann and Ricci tensors are

$$\begin{array}{ccc}\hfill & & R_{xyxy}=p_1{p}_{2}exp(2(p_1+p_2)t),R_{xtxt}=-p_1^{2}exp(2p_{1}t),\hfill \\ & & R_{xx}=p_1(p_1+p_2+p_3)exp(2p_{1}t),R_{tt}=-p_1^2-p_2^2-p_3^2,\hfill \\ & & R=2(p_1^2+p_2^2+p_3^2+p_1{p}_2+p_1{p}_3+p_2{p}_3),\hfill \end{array}$$

(56)

with $R_{xzxz},R_{yzyz},R_{ytyt},R_{ztzt},R_{yy},R_{zz}$ following by symmetry. the combination of p’s for a vacuum is different than for Levi–Cevita also the tensors can be expressed independently of t. The p’s can be though of as three ’cosmological constants’. The Weyl tensor is

$$\begin{array}{ccc}\hfill & & C_{xyxy}=-\frac{1}{6}exp(2(p_1+p_2)t)\left(p_1^2+p_2^2-2p_3^2-2p_1{p}_2+p_1{p}_3+p_2{p}_3\right),\hfill \\ & & C_{xtxt}=-\frac{1}{6}exp(2p_{1}t)\left(2p_1^2-p_2^2-p_3^2-p_1{p}_2-p_1{p}_3+2p_2{p}_3\right),\hfill \\ & & \ast C_{xyzt}=-\frac{1}{6}exp((p_1+p_2+p_3)t)\left(p_1^2+p_2^2-2p_3^2-2p_1{p}_2+p_1{p}_3+p_2{p}_3\right),\hfill \end{array}$$

(57)

with $C_{xzxz},C_{yzyz},C_{ytyt},C_{ztzt},\ast C_{xzyt},\ast C_{xtyz}$ following by symmetry.

For the line element (55) the f’s are given by

$$\begin{array}{c}\hfill f_2=\frac{p_1+p_2+p_3}{6(d-1)}\left(p_1^2-2p_2^2+p_3^2+2p_1{p}_2-4p_1{p}_3+2p_2{p}_3\right),\\ \hfill f_3=\frac{p_1+p_2+p_3}{6(d-1)}\left(p_1^2+p_2^2-2p_3^2-4p_1{p}_2+2p_1{p}_3+2p_2{p}_3\right),\end{array}$$

(58)

to which can be added the non-contributory terms

$$\begin{array}{ccc}\hfill & & f_{2C}\equiv C_{1}cos\left(\sqrt{1-d}t\right)+C_{2}sin\left(\sqrt{1-d}\right),\hfill \\ & & f_{3C}\equiv C_{3}cos\left(\sqrt{(1-d)}t\right)+C_{4}sin\left(\sqrt{(1-d)}\right),\hfill \\ & & C_1\equiv \frac{1}{p_2-p_1}\left((p_3{p}_2)C_3+\sqrt{1-d}{C}_4\right),C_2\equiv \frac{1}{p_2-p_1}\left((p_3{p}_2)C_4-\sqrt{1-d}{C}_3\right),\hfill \end{array}$$

(59)

The Lie derivative of this with respect to the unit time-like vector field (37) vanishes, compare (47). The expression (48) gives the Lanczos potential. The transvected Bel–Robinson (12) and transvected Lanczos energy (14) are

$$B_v=d_2{H}_v=\frac{1}{6}\left(p_1^4+p_2^4+p_3^4-p_1^3(p_2+p_3)-p_2^3(p_1+p_3)-p_3^3(p_1+p_2)+p_1{p}_2{p}_3(p_1+p_2+p_3)\right)$$

(60)

The Weyl, Ricci, and Lanczos scalars are

$$\begin{array}{ccc}\hfill & & {\Psi }_0={\Psi }_4={\Phi }_{02}={\Phi }_{20}=\frac{1}{4}(p_2-p_3)(p_1-p_2-p_3),\hfill \\ & & {\Psi }_2=\frac{1}{12}\left(-2p_1^2+p_2^2+p_3^2+p_1{p}_2+p_1{p}_3-2p_2{p}_3\right)\hfill \\ & & {\Phi }_{00}={\Phi }_{22}=\frac{1}{4}(p_1{p}_2+p-2p_3-p_2^2-p_3^2),{\Phi }_{11}=\frac{1}{4}(-p_1^2+p_2{p}_3),\hfill \\ & & H_1=-H_6=-\frac{\sqrt{2}}{24(d-1)}\left(-2p_1^3+p_2^3+p_3^3+2p_1(p_2+p_3)-p_2^2(p_1+p_3)-p_3(p_1+p_2)\right),\hfill \\ & & H_3=-H_4=-\frac{\sqrt{2}}{8(d-1)}(p_3-p_2)\left(p_2^2+p_3^2-p_1(p_2+p_3)\right).\hfill \end{array}$$

(61)

7. The General Case

Take line element of the form

$$d{s}^2=A_1{(t)}^{2}d{x}^2+A_2{(t)}^{2}d{y}^2+A_3(t)d{z}^2-d{t}^2,$$

(62)

where the explicit dependence on t is left out when the ellipsis is clear, i.e., $A_1(t)\to A_1$. As before, take unit time-like vector field (37), grand shear (45), and Lanczos potential (48) with k now not necessarily $k=1$. The Lie derivative of the grand shear (45) along (37) is

$${\mathcal{L}}_v{\Sigma }_{ab}=\mathrm{diag}\left\{-\dot{f_1}(t)g_{xx},-\dot{f_2}(t)g_{yy},-\dot{f_3}(t)g_{zz},0\right\},$$

(63)

so it is only in restricted cases that equations, such as (47), happen. Define

$${\beta }_I\equiv \frac{{\dot{A}}_I}{A_I},$$

(64)

where the index I is not summed. The effect of $k\ne 1$ is the same as the transformations

$$f_2\to f_2+\frac{k}{3}\left({\beta }_1-2{\beta }_2+{\beta }_3\right),f_3\to f_3+\frac{k}{3}\left({\beta }_1+{\beta }_2-2{\beta }_3\right).$$

(65)

For (48) to be a Lanczos potential substituting in (1) gives the coupled partial differential equations

$$\begin{array}{ccc}\hfill & & {\gamma }_2+(-{\beta }_2+{\beta }_3)f_2+(-{\beta }_1+{\beta }_3)f_3-\dot{f_2}=0,\hfill \\ & & {\gamma }_3+(-{\beta }_1+{\beta }_2)f_2+({\beta }_2-{\beta }_3)f_3-\dot{f_3}=0,\hfill \end{array}$$

(66)

where

$$\begin{array}{ccc}\hfill 6{\gamma }_2\equiv & & (3-2k)\left(\dot{{\beta }_1}+{\beta }_1^2-2\dot{{\beta }_2}-2{\beta }_2^2+\dot{{\beta }_3}+{\beta }_3^2\right)\hfill \\ & & +(3-4k)\left({\beta }_1{\beta }_2-2{\beta }_1{\beta }_3+{\beta }_2{\beta }_3\right),\hfill \\ \hfill 6{\gamma }_3\equiv & & (3-2k)\left(\dot{{\beta }_1}+{\beta }_1^2+\dot{{\beta }_2}+{\beta }_2^2-2\dot{{\beta }_3}-2{\beta }_3^2\right)\hfill \\ & & +(3-4k)\left(-2{\beta }_1{\beta }_2+{\beta }_1{\beta }_3+{\beta }_2{\beta }_3\right),\hfill \end{array}$$

(67)

as k is free it can be set to remove either the first $k=3/2$ or second $k=3/4$ terms in $\delta$. In the general case the equations remain intractable, exceptions being f’s a constant which is the case of Section 6 and the f’s are proportional to $1/t$ which is the case of Section 5. The equations (66) seem to be the most general because altering them gives functions which are absorable by the f’s..

Seven possible approaches:

• Go for the general case $f(\beta )$ but this has proved intractable so far;
• Consider if the equations are tractable for some examples in the Bianchi classification, which is left for now;
• See if known systems of pdes are compatible with (66);
• Choose A find f, see the next paragraph;
• Choose f as either a function of $\beta$ and/or t then find A, see the paragraph after next;
• Choose new field equations such as:

  $$H{E}_{ab}=\kappa G_{ab},$$

  (68)

  where $H{E}_{ab}$ is given by (13) with interpretation that the matter and gravitational energy are proportional, there are other possibilities such as G proportional to the grand shear $\Sigma$ (48);
• Ignore, rather than exact solutions for f, qualitative properties, such as zeros and sign are what are important.

To illustrate (4) for

$$A_1(t)=a_{1}cos(ct),A_2(t)=a_{2}sin(ct),A_3(t)=a_{3}sin(ct),$$

(69)

solving (66) gives

$$\begin{array}{ccc}\hfill & & f_2=-\frac{c}{4cos(ct)sin(ct)}+C_{3}tan(ct)+\mathit{i}{C}_{1}sinh(\mu )+C_{2}cosh(\mu ),\hfill \\ & & f_3=-\frac{c}{4cos(ct)sin(ct)}+C_{3}tan(ct)-C_{2}sinh(\mu )-\mathit{i}{C}_{1}cosh(\mu ),\hfill \end{array}$$

(70)

where

$$\mu (t)\equiv ln\left(\frac{1-cos(ct)-sin(ct)}{sin(ct)}\right)+ln\left(\frac{1-cos(ct)+sin(ct)}{sin(ct)}\right)-ln\left(\frac{sin(ct)}{1+cos(ct)}\right),$$

(71)

taking $k=C_1=C_2=C_3=0$ leaves just the first terms in (70) which is sufficient to cover the Weyl tensor; $C_1=C_2=C_3\ne 0$ might have application in quantum theory.

To illustrate (5), Levi–Cevita Section 4 can be thought of as $1/t$ Bianchi spacetime and exponential spacetime Section 6 as constant, so what happens for $t^n$, for example take $n=1$

$$f_2(t)=k_{2}t,f_3(t)=k_{3}t,$$

(72)

(66) has solution

$$\begin{array}{ccc}\hfill & & {\beta }_1=-\frac{2}{3}(2k_2+k_3)t,{\beta }_3=-\frac{1}{3t}(3+2(k_3+2k_2)t^2),\hfill \\ & & {\beta }_2=-\frac{2t(k_3+2k_2)(-3+2(k_2+2k_3)t^2}{9+6(k_2+2k_3)t^2}\hfill \end{array}$$

(73)

integrating and exponentiating

$$A_1=exp\left\{-\frac{1}{3}(2k_2+k_3)t^2\right\},A_2={\alpha }_2(t)A1,A_3=\frac{A_1}{t},$$

(74)

where

$$\begin{array}{ccc}\hfill & & {\alpha }_2={\left(3+2(k_2+2k_3)t^2\right)}^{\frac{2k_2+k_2}{k_2+2k_3}},\mathrm{for}k=\frac{3}{2},\hfill \\ & & {\alpha }_2=t^{\frac{k}{2k-3}}exp\left(\frac{(k_2+2k_3)t^2}{2(2k-3)}\right)\times \hfill \\ & & \left\{C_{1}WM(w_1,w_2,w_3)+C_{2}WW(w_1,w_2,w_3)\right\}\mathrm{for}k\ne \frac{3}{2},\hfill \\ & & w_1\equiv \frac{(7k/2-6)k_2+(k-3)k_3}{(2k-3)(k_2+2k_3)},w_2\equiv \frac{\sqrt{45-66k+25k^2}}{2(2k-3)},w_3\equiv \frac{(k_2+2k_3)t^2}{2k-3},\hfill \end{array}$$

(75)

where $WM$ and $WW$ are Whittaker functions. This tells us that for (72) alone, one has Whittaker functions but adding $k=3/2$ times the shear the metric takes a power form. To have Maple compute (75) the Whittaker, expressions had to be done in steps, a metric with Whittaker functions in it does not readily compute, and for the $k=3/2$ case it was necessary to bring a constant factor into the power term. $A_1$ in (74) is in distributional form with standard deviation $\sqrt{3/((2(2k_2+k_3))}$, taking all the A’s in this form but with different standard deviations leads to a simple Ricci tensor which does not seem to be describable in terms of simple matter fields and a Lanczos potential in integral form with integrand quartics times times trignometric functions of quadratics.

For possibility (7), the geometric properties are: transvected Bel–Robinson (12)

$$\begin{array}{ccc}\hfill & & B_v=\frac{1}{6}\left[{(\dot{{\beta }_1}+{\beta }_1^2)}^2+{(\dot{{\beta }_2}+{\beta }_2^2)}^2+{(\dot{{\beta }_3}+{\beta }_3^2)}^2+(\dot{{\beta }_1}+{\beta }_1^2)(-{\beta }_1{\beta }_2-{\beta }_1{\beta }_3+2{\beta }_2{\beta }_3)\right.\hfill \\ & & +(\dot{{\beta }_2}+{\beta }_2^2)(-{\beta }_1{\beta }_2+2{\beta }_1{\beta }_3-{\beta }_2{\beta }_3)+(\dot{{\beta }_3}+{\beta }_3^2)(+2{\beta }_1{\beta }_2-{\beta }_1{\beta }_3-{\beta }_2{\beta }_3)\hfill \\ & & -(\dot{{\beta }_1}+{\beta }_1^2)(\dot{{\beta }_2}+{\beta }_2^2)-(\dot{{\beta }_1}+{\beta }_1^2)(\dot{{\beta }_3}+{\beta }_3^2)-(\dot{{\beta }_2}+{\beta }_3^2)(\dot{{\beta }_3}+{\beta }_3^2)\hfill \\ & & \left.-{\beta }_1^2{\beta }_2{\beta }_3-{\beta }_1{\beta }_2^2{\beta }_3-{\beta }_1{\beta }_2{\beta }_3^2+{\beta }_1^2{\beta }_2^2+{\beta }_1^2{\beta }_3^2+{\beta }_2^2{\beta }_3^2\right],\hfill \end{array}$$

(76)

transvected energy (14)

$$\begin{array}{ccc}\hfill H_v& =& \frac{2}{9}(f_2^2+f_2{f}_3+f_3^2)+\frac{2}{9}{f}_2({\beta }_1-{\beta }_2)+\frac{2}{9}{f}_3({\beta }_1-{\beta }_3)\hfill \\ & & +\frac{2}{27}\left({\beta }_1^2+{\beta }_2^2+{\beta }_3^2-{\beta }_1{\beta }_2-{\beta }_1{\beta }_3-{\beta }_2{\beta }_3\right),\hfill \end{array}$$

(77)

Weyl, Ricci, and Lanczos scalars

$$\begin{array}{ccc}\hfill & & {\Psi }_0={\Psi }_4=-\frac{1}{4}\left({\beta }_1({\beta }_3-{\beta }_2)+\dot{{\beta }_2}+{\beta }_2^2-\dot{{\beta }_3}-{\beta }_3^2\right),\hfill \\ & & {\Psi }_2=\frac{1}{12}\left(-2(\dot{{\beta }_1}+{\beta }_1^2)+\dot{{\beta }_2}+{\beta }_2^2+\dot{{\beta }_3}+{\beta }_3^2+{\beta }_1{\beta }_2+{\beta }_2{\beta }_3-2{\beta }_2{\beta }_3\right),\hfill \\ & & {\Phi }_{00}={\Phi }_{22}=-\frac{1}{4}\left(-{\beta }_1{\beta }_2-{\beta }_1{\beta }_3+\dot{{\beta }_2}+{\beta }_2^2+\dot{{\beta }_3}+{\beta }_3^2\right),{\Phi }_{11}=-\frac{1}{4}\left(\dot{{\beta }_1}+{\beta }_1^2-{\beta }_2{\beta }_3\right)\hfill \\ & & {\Phi }_{02}={\Phi }_{20}=-\frac{1}{4}\left({\beta }_1({\beta }_3-{\beta }_2)-\dot{{\beta }_2}-{\beta }_2^2+\dot{{\beta }_3}+{\beta }_3^2\right),\hfill \\ & & {\Phi }_l=\frac{1}{12}\left(\dot{{\beta }_1}+{\beta }_1^2+\dot{{\beta }_2}+{\beta }_2^2+\dot{{\beta }_3}+{\beta }_3^2+{\beta }_1{\beta }_2+{\beta }_1{\beta }_3+{\beta }_2{\beta }_3\right),\hfill \\ & & H_1=-H_6=-\frac{\sqrt{2}k}{36}\left(-2{\beta }_1+{\beta }_2+{\beta }_3\right)+\frac{\sqrt{2}}{12}(f_2+f_3),\hfill \\ & & H_3=-H_4=\frac{\sqrt{2}k}{12}({\beta }_2-{\beta }_3)+\frac{\sqrt{2}}{12}(f_3-f_2).\hfill \end{array}$$

(78)

For $k=0$ the last two equations can be inverted to give the grand shear (45) with

$$f_1=-6\sqrt{2}{H}_1,f_2=3\sqrt{2}\left(H_1-H_3\right),f_3=3\sqrt{2}\left(H_1+H_3\right).$$

(79)

8. Cosmic Energetics

What has been learnt about early cosmology seems to depend on ones point of view about gravitational energy. The most straightforward approach is to say that gravitational energy is related to the Weyl tensor and that there is Robertson–Walker geometry all the way back to the early eras. There are grounds for think that the geometry is more complicated than that [15,16] and that it becomes Levi–Cevita at times, this would introduce a non-vanishing Weyl tensor the changing nature of which has been conjectured [17,18]. If one assumes positive gravitational energy, then gravitational energy just adds to divergent quantities as one approaches the singularity. If one assumes the negative gravitational energy then there are intriguing possibilities, such as it cancelling out matter field energy and there being no energy at the singularity and perhaps no overall energy at any time. There could be an unbroken supersymmetry regime where the energies cancel themselves. This suggests that the evolution of the universe could be thought of as exchange of energy between gravitation and matter fields. Another way of looking at this is that observations tell us that the Universe is near the critical value between open and closed, the critical value could be thought of as having no overall energy: thus, as the universe evolves, energy transfers from Weyl tensor gravitational energy to matter forms. Yet, another way of looking at this is that if the universe started with a quantum fluctuation [19,20] then this was a departure from zero energy which set off everything. Transfer of energy suggests that entropy is non-vanishing, entropy tensors were first introduced in [6].

9. Conclusions

The equations for the Lanczos potential in Bianchi spacetime (66) have been set up and solved in several cases, although the general case remains intractable. Equation (66) might have solutions with exotic properties. A tensor which generalizes the shear, called the grand shear, is used as an intermediate step in the calculation of the Lanczos potential, alternatively if a Lanczos potential and tetrad are known then the form of this tensor is immediate via (79). The Lie derivative of the grand shear (63) gives $1/t$ of the same object for Levi–Cevita (47) and vanishes for exponential spacetimes. The grand shear might have application to other spacetimes.

Intuitively, what has been learnt about the Lanczos potential is that the non-linearity of the field equations has been ’transferred’. The equations defining the Lanczos potential (1) are linear, but the solutions for the Lanzcos potential are often complicated—the non-linearity has been transferred to the potential. This is hard to make precise because the gauge can be changed (4), however if one takes fixed gauges then (48) is a cubic in p over a quadratic in p which is more complicated than the quadratic in p for the Riemann tensor (27).

Application to cosmic energetics raises more problems than it solves. The energy tensors here do not seem to be directly related to quasi-local energy [21,22]. The good news is that rates of decay of all objects are what would be expected and wanted. The form of (9) is what would be anticipated so there is a case for taking $H_1$ as the overall energy, ${\Psi }_2$ is sometimes though of as a measure of mass/energy but the two are not simply related as connection terms from covariant derivates arise when going from one to the other. The transvected energy tensor (14) has awkward properties. For Levi–Cevita spacetime it is independent of the p’s (41), which is no longer the case (49) if the vacuum constraints (32) are not applied. To see how the non-vacuum case is connected one transfers from p constants to $ϵ$ constants (52): taking the spatial ${ϵ}_2$ to vanish just adds an $ϵ$ term (53) with (14) still independent of p, intuitively one has departed from the vacuum by adding an energy ${ϵ}_3$ and as this is monopolar it does not introduce a p; taking the temporal ${ϵ}_3$ to vanish does introduce a p (54), intuitively this is a spatial departure from the vacuum introducing p. For exponential space time (14) is constant (60). In general, (77) gives no indication of sign. Thus, for the transvected energy tensor (14) the sign and nature of any coupling constant have not been fixed.