# Lanczos potential of Weyl field: interpretations and applications

Ram Gopal Vishwakarma
Unidad Acad$\acute{e}$mica de Matem$\acute{a}$ticas
Universidad Aut$\acute{o}$noma de Zacatecas,
Zacatecas, ZAC, Mexico  E-mail<EMAIL_ADDRESS>
###### Abstract

An attempt is made to uncover the physical meaning and significance of the
obscure Lanczos tensor field which is regarded as a potential of the Weyl
field. Despite being a fundamental building block of any metric theory of
gravity, the Lanczos tensor has not been paid proper attention as it deserves.
By providing an elucidation on this tensor field through its derivation in
some particularly chosen spacetimes, we try to find its adequate
interpretation.

Though the Lanczos field is traditionally introduced as a gravitational
analogue of the electromagnetic 4-potential field, the performed study
unearths its another feature - a relativistic analogue of the Newtonian
gravitational force field. A new domain of applicability of the Lanczos tensor
is introduced which corroborates this new feature of the tensor.

PACS: 04.20.Cv, 04.20.-q, 04.30.-w, 95.30.Sf

Keywords: Lanczos tensor, potential of Weyl tensor, gravitational waves,
gravitational collapse.

##  1 Introduction

General Relativity (GR), which constitutes the current description of
gravitation in modern physics, appears to be afflicted with at least two major
difficulties. Firstly, it shows an intrinsic difficulty in its unification
with the rest of physics, which is perhaps the most profound foundational
problem in physics. Secondly, its cosmological application - the $\Lambda$CDM
model - faces challenges on both theoretical and observational grounds,
despite its overall success and simplicity. It is now claimed that the current
observations show evidence of physics beyond the $\Lambda$CDM model [1]. This
is an alarming signal to scrutinize the very foundations of the theory. It
seems that out of the four fundamental interactions, the most familiar to us
is the most mysterious one. Perhaps the present description of this
interaction is incomplete just because we are not taking into account the
whole system of elements responsible for the structure of spacetime.

Science thrives on crisis. Hard times call for new ideas and insights. In this
view, it would be worthwhile to draw attention towards an unnoticed ingredient
of the spacetime, which is impregnated with extraordinary scientific and
philosophical values and is expected to contribute to our understanding of the
fundamental nature of gravitation itself. This missing link is a rank three
tensor - the Lanczos tensor $L_{\mu\nu\sigma}$ \- appearing as a potential for
the Weyl tensor (which represents the true gravitational degrees of freedom).
As we know, the Riemann curvature tensor plays the nodal point for the
unfolding of gravity in the spacetime manifold. However, strangely enough, the
full curvature tensor has been pushed into background. What comes forth,
through the field equations of gravitation, is only the trace of the Riemann
tensor (the Ricci tensor and Ricci scalar), due to its relation to the energy-
momentum tensor of matter. This has led to a certain eclipse of the full
curvature tensor. It is the Lanczos tensor which fulfills this gap by
providing the trace-free part of the Riemann tensor (the Weyl tensor) -
exactly those components of the tensor that are not embraced by the field
equations. Thus, the understanding of the spacetime structure, and hence that
of gravity, cannot be complete without the Lanczos tensor field.

Interestingly the Lanczos tensor, discovered by Cornelius Lanczos in 1962 [2],
happens to exist, by coincidence or providence, in any Riemannian manifold
with Lorentzian signature in four dimensions. It generates the Weyl tensor
$C_{\mu\nu\sigma\rho}$ of the manifold through the Weyl-Lanczos
equation111Notations adopted: The starred symbol denotes the dual operation
defined by
${*N*}_{\alpha\beta\mu\nu}=\frac{1}{4}e_{\alpha\beta\rho\sigma}e_{\mu\nu\tau\delta}N^{\rho\sigma\tau\delta}$,
with $e_{\mu\nu\sigma\rho}$ representing the Levi-Civita tensor. The semicolon
(comma) followed by an index denotes covariant (ordinary) derivative with
respect to the corresponding variable. (Sometimes we shall also use
$\nabla_{\alpha}$ to represent covariant derivative with respect to
$x^{\alpha}$.) The square brackets [] enclosing indices denote skew-
symmetrization, for instance,
$X_{[\mu\nu]}\equiv\frac{1}{2!}(X_{\mu\nu}-X_{\nu\mu})$. Similarly, the round
brackets () enclosing indices denote symmetrization, i.e.,
$X_{(\mu\nu)}\equiv\frac{1}{2!}(X_{\mu\nu}+X_{\nu\mu})$. The Greek indices
range over the values $0,1,2,3$ where 0 is temporal and 1, 2, 3 are spatial.
For simplicity, we have considered the geometric units with $G=c=1$.

$C_{\mu\nu\sigma\rho}=L_{[\mu\nu][\sigma;\rho]}+L_{[\sigma\rho][\mu;\nu]}-{*L*}_{[\mu\nu][\sigma;\rho]}-{*L*}_{[\sigma\rho][\mu;\nu]}.$
(1)

The Lanczos tensor $L_{\mu\nu\sigma}$ receives a natural and adequate
interpretation in terms of the deep analogy between gravitation and
electrodynamics. In electrodynamics, a crucial ingredient of the
electromagnetic field is its 4-potential $A_{\mu}$, whence emanates the
Faraday tensor $F_{\mu\nu}$:

$F_{\mu\nu}=A_{\mu;\nu}-A_{\nu;\mu}=A_{\mu,\nu}-A_{\nu,\mu},$ (2)

which measures the strength of the electromagnetic field. Let us recall that
the Weyl tensor $C_{\mu\nu\sigma\rho}$ is the gravitational analogue of the
electromagnetic $F_{\mu\nu}$. Hence, the relativistic potential
$L_{\mu\nu\sigma}$ generating the Weyl tensor differentially, should be
regarded as the gravitational analogue of the electromagnetic potential
$A_{\mu}$ generating the field strength tensor $F_{\mu\nu}$. This constitutes
the Lanczos potential as a more fundamental geometrical object than the Weyl
tensor.

The theory of this potential tensor contains many surprises and insights. For
example, it has been shown that the Minkowski spacetime also admits non-
trivial values of the Lanczos potential [3, 4], proclaiming that the Lanczos
potential is a property of spacetime itself rather than an outcome of a
particular theory of gravity. This simple but far-reaching insight of
mathematical and philosophical value, revolutionizes our views on the
foundational nature of spacetime itself indicating towards an important
substantive nature of spacetime and its geometry. This also seems consistent
with quantum theory wherein the vacuum state of a quantum theory is not
nothing. It would be up to the further studies to decide whether this
revolutionary insight is just a mathematical curiosity or leads to a path to
something deeper. Interestingly, the Lanczos potential has already been shown
to be impregnated with various signatures of quantum physics [2, 3, 5].

As mentioned above, the Lanczos tensor emerges as a fundamental geometric
ingredient in any metric theory of gravity (formulated in a 4-dimensional
pseudo Riemannian spacetime), irrespective of the field equations of the
theory. This provides the tensor a status of an inherent structural element in
the geometric embodiment of gravity. Albeit its novelty and importance, this
remarkable discovery is comparatively unfamiliar even now - some sixty years
after Lanczos first introduced it, and his ingenious discovery has remained
more or less a mathematical curiosity. The main reason for this virtual
obscurity is the absence of the physical properties of the tensor. It has not
been possible so far to ascertain what the tensor represents physically,
although some attempts have been made recently in this direction [3]. There is
also another basic question related to the Lanczos potential that needs to be
answered. As the Lanczos tensor appears to be the potential of the Weyl field
and the latter can be written in terms of the connections, then how is the
Lanczos tensor related with the connections?

After exemplifying this tensor field by deriving its values in various
spacetimes in the following sections, we discover some possible applications
thereof in some gravitational phenomena. This may help to assign an adequate
physical interpretation to the tensor.

##  2 Lanczos Potential Tensor and its Non-Uniqueness

As Lanczos derived his tensor from a variational principle and since the
latter need not be physically consistent always, a more robust existence
theorem was needed. This theorem was provided by Bampi and Caviglia, which
guarantees the existence of such a potential in any 4-dimensional222While the
Lanczos potential tensor does not exist in general for dimensions higher than
four [6], a potential tensor of rank 5 indeed exists in all dimensions $\geq
4$. Interestingly, the trace of this potential (which amounts to its double
dual) corresponds, in the case of 4-dimensions, to the usual Lanczos potential
[7]. Riemannian manifold [8]. It should however be noted that this tensor is
not unique in a given spacetime. Let us first emphasize that the tensor
$L_{\mu\nu\sigma}$ satisfies the generating equation (1) only if it possesses
the following two symmetries:

$\displaystyle L_{\mu\nu\sigma}=-L_{\nu\mu\sigma},$ (3a) $\displaystyle
L_{[\mu\nu\sigma]}=0~{}~{}\Leftrightarrow~{}~{}{*L}^{~{}~{}\kappa}_{\mu~{}~{}\kappa}=0.$
(3b)

Let us simplify the Weyl-Lanczos equation (1) by evaluating the duals
appearing in it, giving

$\displaystyle C_{\mu\nu\sigma\rho}$ $\displaystyle=$ $\displaystyle
L_{\mu\nu\sigma;\rho}+L_{\sigma\rho\mu;\nu}-L_{\mu\nu\rho;\sigma}-L_{\sigma\rho\nu;\mu}+g_{\nu\sigma}L_{(\mu\rho)}+g_{\mu\rho}L_{(\nu\sigma)}$
(4) $\displaystyle-$ $\displaystyle
g_{\nu\rho}L_{(\mu\sigma)}-g_{\mu\sigma}L_{(\nu\rho)}+\frac{2}{3}L^{\lambda\kappa}_{~{}~{}~{}\lambda;\kappa}(g_{\mu\sigma}g_{\nu\rho}-g_{\nu\sigma}g_{\mu\rho}),$

where $L_{\mu\nu}\equiv
L^{~{}~{}\kappa}_{\mu~{}~{}\nu;\kappa}-L^{~{}~{}\kappa}_{\mu~{}~{}\kappa;\nu}$.
One may check that this equation indeed admits, by virtue of the two symmetry
conditions of $L_{\mu\nu\sigma}$ given by (3), all the symmetries of the Weyl
tensor: $C_{\mu\nu\sigma\rho}=C_{[\mu\nu][\sigma\rho]}$,
$C_{\mu\nu\sigma\rho}=C_{\sigma\rho\mu\nu}$, $C_{\mu[\nu\sigma\rho]}=0$,
$C^{\lambda}_{~{}~{}\mu\nu\lambda}=0$. While the symmetry condition (3a)
reduces the number of independent components of $L_{\mu\nu\sigma}$ to 24, the
condition (3b) reduces it down to 20. As the Weyl tensor has only 10 degrees
of freedom, Lanczos considered the following additional symmetries

$\displaystyle L^{~{}~{}\kappa}_{\mu~{}~{}\kappa}=0,$ (5a) $\displaystyle
L^{~{}~{}~{}\kappa}_{\mu\nu~{}~{};\kappa}=0,$ (5b)

as two gauge conditions in order to reduce the number of degrees of freedom
present in $L_{\mu\nu\sigma}$. The condition (5a) abolishes the degeneracy in
$L_{\mu\nu\sigma}$ appearing through the gauge transformation

$\bar{L}_{\mu\nu\sigma}=L_{\mu\nu\sigma}+g_{\nu\sigma}X_{\mu}-g_{\mu\sigma}X_{\nu},$
(6)

which leaves equation (4) invariant for an arbitrary vector field
$X_{\alpha}$. This arbitrariness in $L_{\mu\nu\sigma}$ is fixed by the
condition (5a) which gives $X_{\alpha}=0$. Thus the algebraic gauge condition
(5a), taken together with (3a) and (3b), reduces the number of independent
components of $L_{\mu\nu\sigma}$ to 16.

Let us emphasize that the differential gauge condition (5b), which was adopted
by Lanczos just because the divergence
$L^{~{}~{}~{}\kappa}_{\mu\nu~{}~{};\kappa}$ does not participate in equation
(4), does not further reduce the degrees of freedom of $L_{\mu\nu\sigma}$,
contrary to the wide-spread misunderstanding in the literature [3]. Thus ample
degeneracy remains there in the values of the tensor $L_{\mu\nu\sigma}$ even
after the Lanczos gauge conditions (5a) and (5b) are applied, as has been
demonstrated in [3]. In fact, the reason for this degeneracy in
$L_{\mu\nu\sigma}$ is due to the presence of an auxiliary potential to the
Weyl tensor, discovered by Takeno [4]. He noticed that given a Lanczos
potential $L_{\mu\nu\sigma}$ of a spacetime, the quantity

$\bar{L}_{\mu\nu\sigma}=L_{\mu\nu\sigma}+A_{\mu\nu\sigma}$ (7)

is again a Lanczos potential of that spacetime if the tensor
$A_{\mu\nu\sigma}$ (termed as ‘s-tensor’ by Takeno) satisfies

$\displaystyle A_{\mu\nu\sigma}=-A_{\nu\mu\sigma},$ (8a) $\displaystyle
A_{\mu\nu\sigma}+A_{\nu\sigma\mu}+A_{\sigma\mu\nu}=0,$ (8b) $\displaystyle
A_{[\mu\nu][\sigma;\rho]}+A_{[\sigma\rho][\mu;\nu]}-{*A*}_{[\mu\nu][\sigma;\rho]}-{*A*}_{[\sigma\rho][\mu;\nu]}=0.$
(8c)

It has been shown that the gauge condition (5b) of divergence-freeness of the
Lanczos tensor, which implies that
$A^{~{}~{}~{}\kappa}_{\mu\nu~{}~{};\kappa}=0$, does not fix $A_{\mu\nu\sigma}$
uniquely and hence causes degeneracy in $L_{\mu\nu\sigma}$ [3]. Thus, the
cause of the degeneracy in $L_{\mu\nu\sigma}$ is the redundant degrees of
freedom of the tensor, and it is not an issue with the gauge conditions. That
is, the conditions (5a) and (5b) alone, taken together with (3a) and (3b),
cannot supply a unique value of $L_{\mu\nu\sigma}$ in a given spacetime. In
order to fix this arbitrariness, one needs additional conditions/assumptions
of plausibility to be imposed on $L_{\mu\nu\sigma}$, which is still missing.
In the absence of this, any arbitrary value of $L_{\mu\nu\sigma}$, just
satisfying the existing conditions/symmetries, may not constitute a physically
viable quantity.

Although there exists no algorithm for finding the Lanczos potential tensor in
general, the tensor has indeed been found explicitly in certain special
situations, for instance the spacetimes of physical significance like
Schwarzschild [9] and Kerr [10], besides many others (see [11] and the
references therein). In order to elucidate the obscure theory of Lanczos
potential tensor, we derive its values in some particularly chosen spacetimes
in the following. These values will also be used in the later sections while
studying some possible applications of the tensor.

###  2.1 Lanczos tensor for Schwarzschild Spacetime

The simplest example of the Lanczos tensor, which we take from the literature,
corresponds to the Schwarzschild line element

$ds^{2}=\left(1-\frac{2m}{r}\right)dt^{2}-\frac{dr^{2}}{(1-2m/r)}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta~{}d\phi^{2},$
(9)

which represents a static spherically symmetric spacetime outside an isotropic
mass $m$ placed in a Ricci-flat ($R_{\mu\nu}=0$) manifold. Novello and Velloso
[9] have shown that if a unit time-like vector field $V^{\alpha}\equiv
dx^{\alpha}/ds$ tangential to the trajectory of an observer in a given
spacetime is irrotational and shear-free, the Lanczos potential of the
sapcetime is given by

$L_{\mu\nu\sigma}=V_{\mu;\lambda}V^{\lambda}V_{\nu}V_{\sigma}-V_{\nu;\lambda}V^{\lambda}V_{\mu}V_{\sigma}.$
(10)

By considering $V^{\alpha}=\left(\frac{1}{\sqrt{1-2m/r}},0,0,0\right)$, which
comes out as irrotational and shear-free in the spacetime (9) hence satisfying
all the requirements of the Novello-Velloso formula (10):
$V^{\alpha}V_{\alpha}=1$, shear $=$ rotation $=0$, acceleration $\neq 0$. The
formula (10) then provides the Lanczos tensor with only one non-vanishing
independent component

$L_{100}=-\frac{m}{r^{2}}.$ (11)

By chance this satisfies Lanczos’s gauge condition (5b) of divergence-
freeness, though it does not satisfy the condition (5a) of trace-freeness. A
trace-free potential can be obtained by using relation (6), which allows to
cancel the trace of the tensor by choosing
$X_{\mu}=-L_{\mu~{}~{}\lambda}^{~{}~{}\lambda}/3$ giving

$\left.\begin{aligned} &\bar{L}_{100}=-\frac{2m}{3r^{2}}\\\
&\bar{L}_{122}=-\frac{m}{3(1-2m/r)}\\\
&\bar{L}_{133}=-\frac{m\sin^{2}\theta}{3(1-2m/r)}\end{aligned}\right\\},$ (12)

which satisfy both Lanczos gauge conditions given in (5) fortunately. We find
another solution of equation (4), not realized so far in the literature, for
the Schwarzschild line element (9):

$\left.\begin{aligned} &\tilde{L}_{100}=-\frac{1}{2}\frac{m}{r^{2}}\\\
&\tilde{L}_{133}=\frac{1}{2}r\sin^{2}\theta\\\
&\tilde{L}_{233}=\frac{1}{4}r^{2}\sin 2\theta\end{aligned}\right\\},$ (13)

satisfying the gauge condition (5b) of the divergence freeness but not
satisfying the gauge condition (5a) of trace-freeness. The trace-free form can
similarly be obtained by using (6), though at the cost of letting more non-
vanishing components enter it.

###  2.2 Lanczos Tensor for the Schwarzschild-de Sitter Spacetime

The Lanczos potential of the Schwarzschild spacetime can easily be generalized
by adding the cosmological constant $\Lambda$ in (9) thereby resulting in the
Schwarzschild-de Sitter solution of $G_{\mu\nu}+\Lambda g_{\mu\nu}=0$:

$ds^{2}=\left(1-\frac{2m}{r}-\frac{\Lambda
r^{2}}{3}\right)dt^{2}-\frac{dr^{2}}{(1-2m/r-\Lambda
r^{2}/3)}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta~{}d\phi^{2}.$ (14)

The choice $V^{\alpha}=\left(\left(1-\frac{2m}{r}-\frac{\Lambda
r^{2}}{3}\right)^{-1/2},0,0,0\right)$ satisfies all the requirements of the
Novello-Velloso formula (10) for the line element (14). The formula then gives
only one non-vanishing independent component

$L_{100}=\frac{\Lambda r}{3}-\frac{m}{r^{2}}$ (15)

satisfying the gauge condition (5b) but not (5a). A trace-free potential is
obtained by the use of (6) giving

$\left.\begin{aligned} &\bar{L}_{100}=\frac{2}{3}\left(\frac{\Lambda
r}{3}-\frac{m}{r^{2}}\right)\\\ &\bar{L}_{122}=\frac{\frac{\Lambda
r^{3}}{3}-m}{3(1-2m/r-\Lambda r^{2}/3)}\\\
&\bar{L}_{133}=\frac{\left(\frac{\Lambda
r^{3}}{3}-m\right)\sin^{2}\theta}{3(1-2m/r-\Lambda
r^{2}/3)}\end{aligned}\right\\},$ (16)

which satisfies both gauge conditions (5).

###  2.3 Lanczos Tensor for the Schwarzschild Interior Spacetime

It may appear surprising that the Lanczos tensor, that appears as the
potential of the Weyl tensor, may be non-vanishing even when the Weyl tensor
vanishes. In order to exemplify this, let us first consider the Schwarzschild
interior solution

$ds^{2}=\frac{1}{4}\left(3\sqrt{1-\frac{r_{0}^{2}}{R^{2}}}-\sqrt{1-\frac{r^{2}}{R^{2}}}\right)^{2}dt^{2}-\frac{dr^{2}}{1-r^{2}/R^{2}}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta~{}d\phi^{2},$
(17)

which, being a static and isotropic solution of Einstein equation, represents
the spacetime inside a non-rotating, incompressible perfect fluid sphere of
constant density $\rho_{0}$ and radius $r_{0}$. Here
$R\equiv\sqrt{3/8\pi\rho_{0}}$. As the solution is conformally flat, its Weyl
tensor vanishes identically. It would not be difficult to check that a
$V^{\alpha}=\left(\left(\frac{3}{2}\sqrt{1-\frac{r_{0}^{2}}{R^{2}}}-\frac{1}{2}\sqrt{1-\frac{r^{2}}{R^{2}}}\right)^{-1},0,0,0\right)$
in the spacetime (17), satisfies all the requirements of the Novello-Velloso
formula (10): $V^{\alpha}V_{\alpha}=1$, shear $=$ rotation $=0$, acceleration
$\neq 0$. The formula then gives

$L_{100}=\frac{5r\left[1-\frac{r^{2}+9r_{0}^{2}}{10R^{2}}-\frac{3}{5}\sqrt{\left(1-\frac{r^{2}}{R^{2}}\right)\left(1-\frac{r_{0}^{2}}{R^{2}}\right)}\right]}{2R^{2}\left[\left(1-\frac{r^{2}}{R^{2}}\right)-3\sqrt{\left(1-\frac{r^{2}}{R^{2}}\right)\left(1-\frac{r_{0}^{2}}{R^{2}}\right)}\right]}$
(18)

as the only non-vanishing independent component of $L_{\mu\nu\sigma}$, which
is divergence-free but not trace-free. A trace-free form can similarly be
obtained by the use of (6), giving

$\left.\begin{aligned}
&\bar{L}_{100}=\frac{5r\left[1-\frac{r^{2}+9r_{0}^{2}}{10R^{2}}-\frac{3}{5}\sqrt{\left(1-\frac{r^{2}}{R^{2}}\right)\left(1-\frac{r_{0}^{2}}{R^{2}}\right)}\right]}{3R^{2}\left[\left(1-\frac{r^{2}}{R^{2}}\right)-3\sqrt{\left(1-\frac{r^{2}}{R^{2}}\right)\left(1-\frac{r_{0}^{2}}{R^{2}}\right)}\right]}\\\
&\bar{L}_{122}=\frac{10r^{3}\left[1-\frac{r^{2}+9r_{0}^{2}}{10R^{2}}-\frac{3}{5}\sqrt{\left(1-\frac{r^{2}}{R^{2}}\right)\left(1-\frac{r_{0}^{2}}{R^{2}}\right)}\right]}{3R^{2}\left[\left(1-\frac{r^{2}}{R^{2}}\right)^{2/3}-3\left(1-\frac{r^{2}}{R^{2}}\right)^{1/6}\left(1-\frac{r_{0}^{2}}{R^{2}}\right)^{1/2}\right]^{3}}\\\
&\bar{L}_{133}=\frac{10r^{3}\left[1-\frac{r^{2}+9r_{0}^{2}}{10R^{2}}-\frac{3}{5}\sqrt{\left(1-\frac{r^{2}}{R^{2}}\right)\left(1-\frac{r_{0}^{2}}{R^{2}}\right)}\right]}{3R^{2}\left[\left(1-\frac{r^{2}}{R^{2}}\right)^{2/3}-3\left(1-\frac{r^{2}}{R^{2}}\right)^{1/6}\left(1-\frac{r_{0}^{2}}{R^{2}}\right)^{1/2}\right]^{3}}\sin^{2}\theta\end{aligned}\right\\},$
(19)

which satisfies both gauge conditions documented in (5).

Let us note that if $L_{\mu\nu\sigma}$ is a Lanczos potential of a conformally
flat (or flat) spacetime, then for an arbitrary constant $\kappa$, the tensor
$\kappa L_{\mu\nu\sigma}$ is also its Lanczos potential by virtue of the
vanishing Weyl tensor and the linearity of equation (4) in $L_{\mu\nu\sigma}$.

###  2.4 Lanczos Tensor for the Robertson-Walker Spacetimes

As another example of a spacetime with vanishing Weyl tensor, let us consider
the Robertson-Walker (R-W) line element (which too is conformally flat) in the
form

$ds^{2}=dt^{2}-S^{2}(t)\left[\frac{dr^{2}}{1-\alpha
r^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta~{}d\phi^{2})\right],$ (20)

where $\alpha$ has been kept as an arbitrary constant (not normalized, as in
the usual standard form) to be used later in section 6.2. The Lanczos tensor
for some particular cases of this spacetime has been discovered in [3]. Here
we discover its value in a more general case. For this purpose, we follow a
heuristic approach and take guidelines from the Schwarzschild case (9). Since
both the line elements (9) and (20) are spherically symmetric, we expect that
it should be possible to have $L_{\mu\nu\sigma}$ for the R-W line element (20)
with a single (independent) non-vanishing component, like the one given by
(11) for the Schwarzschild line element (9). But apart from the radial
coordinate $r$, now in the case of the line element (20) we should also expect
contributions from $S$ and possibly from $(1-\alpha r^{2})$ also. Hence, we
expect

$L_{100}=\kappa r^{\ell}S^{n}(1-\alpha r^{2})^{p},$ (21)

to form the only independent non-vanishing component of the Lanczos potential
for the spacetime (20) for suitable values of the parameters $\ell,n$ and $p$.
These parameters are to be determined by the condition that (21) should
satisfy the Weyl-Lanczos generating equation (4) for the line element (20)
with $C_{\mu\nu\sigma\rho}=0$. Let us recall that the spacetime (20) is
conformally flat and hence its Weyl tensor vanishes identically. The
multiplicative constant $\kappa$ has been introduced in (21) for the same
reason, as explained earlier, that if $L_{\mu\nu\sigma}$ is a solution of (4)
for the line element (20), then so is $\kappa L_{\mu\nu\sigma}$.

If we denote the right hand side of equation (4) by $W_{\mu\nu\sigma\rho}$,
then its non-vanishing components generated by the definition (21) for the
spacetime (20) yield

$\left.\begin{aligned} &W_{1010}=\frac{2\kappa}{3}r^{\ell-1}(1-\alpha
r^{2})^{p-1}S^{n}[1-\ell+\alpha r^{2}(\ell+2p)],\\\
&W_{1212}=\frac{\kappa}{3}r^{\ell+1}(1-\alpha
r^{2})^{p-1}S^{n+2}[1-\ell+\alpha r^{2}(\ell+2p)],\\\
&W_{1313}=\frac{\kappa}{3}r^{\ell+1}(1-\alpha
r^{2})^{p-1}S^{n+2}[1-\ell+\alpha r^{2}(\ell+2p)]\sin^{2}\theta,\\\
&W_{2020}=-\frac{\kappa}{3}r^{\ell+1}(1-\alpha
r^{2})^{p}~{}S^{n}[1-\ell+\alpha r^{2}(\ell+2p)],\\\
&W_{3030}=-\frac{\kappa}{3}r^{\ell+1}(1-\alpha
r^{2})^{p}~{}S^{n}[1-\ell+\alpha r^{2}(\ell+2p)]\sin^{2}\theta,\\\
&W_{2323}=-\frac{2\kappa}{3}r^{\ell+3}(1-\alpha
r^{2})^{p}~{}S^{n+2}[1-\ell+\alpha
r^{2}(\ell+2p)]\sin^{2}\theta.\end{aligned}\right\\}$ (22)

Obviously, the vanishing of all these components simultaneously (for all $r$,
$S$ and $\kappa\neq 0$), requires

$1-\ell+\alpha r^{2}(\ell+2p)=0,$

which gives a unique solution

$\ell=1,~{}~{}~{}~{}p=-\frac{1}{2},$

to be compatible with an arbitrary value of $\alpha$. Thus

$L_{100}=\kappa\frac{rS^{n}}{\sqrt{1-\alpha r^{2}}},$ (23)

constitutes the Lanczos potential for the spacetime (20) with $\kappa$, $n$
and $\alpha$ as arbitrary constants. The trace
$L^{~{}~{}\lambda}_{\mu~{}~{}\lambda}$ and the divergence
$L^{~{}~{}~{}\lambda}_{\mu\nu~{}~{};\lambda}$ of the tensor can be calculated,
giving the non-vanishing components as

$L^{~{}~{}\lambda}_{1~{}~{}\lambda}=\kappa\frac{rS^{n}}{\sqrt{1-\alpha
r^{2}}},$ (24)
$L^{~{}~{}~{}\lambda}_{10~{}~{};\lambda}=-L^{~{}~{}~{}\lambda}_{01~{}~{};\lambda}=\kappa(n+2)\frac{rS^{n-1}\dot{S}}{\sqrt{1-\alpha
r^{2}}}.$ (25)

Clearly the solution (23) does not satisfy the condition (5a) [nor does it
satisfy the condition (5b)]. As we have mentioned earlier, a trace-free
potential can be obtained by using relation (6), which allows to cancel the
trace of the tensor by choosing
$X_{\mu}=-L_{\mu~{}~{}\lambda}^{~{}~{}\lambda}/3$ giving now the following
three independent non-vanishing components:

$\left.\begin{aligned} &L_{100}=\frac{2\kappa}{3}\frac{rS^{n}}{\sqrt{1-\alpha
r^{2}}},\\\ &L_{122}=\frac{\kappa}{3}\frac{r^{3}S^{n+2}}{\sqrt{1-\alpha
r^{2}}},\\\ &L_{133}=\frac{\kappa}{3}\frac{r^{3}S^{n+2}}{\sqrt{1-\alpha
r^{2}}}\sin^{2}\theta.\end{aligned}\right\\}$ (26)

This can be generalized into yet another solution

$\left.\begin{aligned} &L_{100}=\kappa_{1}\frac{rS^{n}}{\sqrt{1-\alpha
r^{2}}},\\\ &L_{122}=\kappa_{2}\frac{r^{3}S^{n+2}}{\sqrt{1-\alpha r^{2}}},\\\
&L_{133}=\kappa_{2}\frac{r^{3}S^{n+2}}{\sqrt{1-\alpha
r^{2}}}\sin^{2}\theta,\end{aligned}\right\\}$ (27)

though at the cost of loosing the gauge symmetry (5a). Here
$\kappa_{1},\kappa_{2}$ are arbitrary constants.

Our elucidation on the Lanczos tensor is now complete to some extent, and it
is ready to be involved in applications. Although much attention has been
devoted to the Weyl tensor in the research on a gravitational theory, the same
did not occur with the Lanczos potential. Since the Lanczos tensor appears as
a fundamental property of spacetime itself irrespective of any particular
theory of gravitation, it is expected to be imbued with more interesting
physical properties than the Weyl tensor. With the values of this tensor
derived in these sections, we explore what physical information this potential
tensor may convey in a physical situation.

##  3 Conformal Properties of Lanczos Tensor

Conformal symmetry plays an important role in field theories including gravity
and leads to interesting insights about the physical phenomena related to the
dynamics of spacetime. Under this symmetry for instance, all the features of
gravitational wave propagation and the global causal structure of spacetime
remain conserved, as the conformal transformations leave the null geodesics
and hence the light cones invariant.

A conformal transformation is a rescaling of the space and time intervals
performed without changing the coordinates used to describe events in a
spacetime manifold. This rescaling is brought about through the metric
transformation

$g_{\mu\nu}\rightarrow\tilde{g}_{\mu\nu}=\Omega^{2}g_{\mu\nu},$ (28)

where the conformal factor $\Omega(x^{\alpha})$ is a dimensionless, positive
and smooth but otherwise arbitrary function of the spacetime coordinates. This
gives rise to a new spacetime and its geometry characterized by the metric
$\tilde{g}_{\mu\nu}$, wherein the inverse metric and the Christoffel symbols
lead to

$\displaystyle\tilde{g}^{\mu\nu}=\Omega^{-2}g^{\mu\nu},$ (29a)
$\displaystyle\tilde{\Gamma}^{\alpha}_{\beta\gamma}=\Gamma^{\alpha}_{\beta\gamma}+\Omega^{-1}\left(\delta^{\alpha}_{\beta}~{}\Omega_{,\gamma}+\delta^{\alpha}_{\gamma}~{}\Omega_{,\beta}-g_{\beta\gamma}~{}g^{\alpha\lambda}~{}\Omega_{,\lambda}\right).$
(29b)

The tilde over a quantity is going to denote its value in the conformally
rescaled spacetime with metric $\tilde{g}_{\mu\nu}$. As the Weyl tensor is
fully invariant under the transformation (28), it is natural to ask how this
conformal invariance is manifested in the Lanczos tensor which plays the role
of the potential for the Weyl tensor. But, unlike the Weyl tensor, it has not
so far been possible to obtain an expression for the Lanczos tensor in terms
of the metric tensor so that its conformal invariance can be studied directly.
Nevertheless one can still analyze the conformal invariance of the Weyl-
Lanczos equation (4) by considering a conformal weight for $L_{\mu\nu\sigma}$:

$\tilde{L}_{\mu\nu\sigma}=\Omega^{s}L_{\mu\nu\sigma},$ (30)

where $s$ is a real number (see the Appendix D of Reference [12]). If we
denote the right hand side of equation (4) with $W_{\mu\nu\sigma\rho}$ as we
have done earlier, then the application of the rescaling (30) must render
$W^{\mu}_{~{}~{}\nu\sigma\rho}$ unchanged under the transformation (28), i.e.
$\tilde{W}^{\mu}_{~{}~{}\nu\sigma\rho}=W^{\mu}_{~{}~{}\nu\sigma\rho}$, since
so is the left hand side of equation (4):
$\tilde{C}^{\mu}_{~{}~{}\nu\sigma\rho}=C^{\mu}_{~{}~{}\nu\sigma\rho}$.

In order to look into this matter directly, let us compute the right hand side
of equation (4) in the conformally transformed spacetime. Use of the conformal
rescalings (29a, 29b, 30) and a long but straightforward calculation produce

$\left.\begin{aligned}
\tilde{W}^{\mu}_{~{}~{}\nu\sigma\rho}=\Omega&{}^{(s-2)}W^{\mu}_{~{}~{}\nu\sigma\rho}\\\
&+\frac{(s-3)}{2}\Omega^{(s-3)}g^{\mu\alpha}\big{[}2\big{\\{}L_{\alpha\nu\sigma}~{}\Omega_{,\rho}+L_{\nu\alpha\rho}~{}\Omega_{,\sigma}+L_{\sigma\rho\alpha}~{}\Omega_{,\nu}+L_{\rho\sigma\nu}~{}\Omega_{,\alpha}\big{\\}}\\\
&+\big{\\{}g_{\alpha\rho}(L^{~{}\kappa}_{\nu~{}\sigma}+L^{~{}\kappa}_{\sigma~{}\nu})+g_{\nu\sigma}(L^{~{}\kappa}_{\alpha~{}\rho}+L^{~{}\kappa}_{\rho~{}\alpha})-g_{\alpha\sigma}(L^{~{}\kappa}_{\nu~{}\rho}+L^{~{}\kappa}_{\rho~{}\nu})-g_{\nu\rho}(L^{~{}\kappa}_{\alpha~{}\sigma}+L^{~{}\kappa}_{\sigma~{}\alpha})\big{\\}}\Omega_{,\kappa}\\\
&+\big{\\{}\big{(}g_{\nu\rho}L^{~{}\kappa}_{\sigma~{}\kappa}-g_{\nu\sigma}L^{~{}\kappa}_{\rho~{}\kappa}\big{)}\Omega_{,\alpha}+\big{(}g_{\alpha\sigma}L^{~{}\kappa}_{\rho~{}\kappa}-g_{\alpha\rho}L^{~{}\kappa}_{\sigma~{}\kappa}\big{)}\Omega_{,\nu}+\big{(}g_{\rho\nu}L^{~{}\kappa}_{\alpha~{}\kappa}-g_{\rho\alpha}L^{~{}\kappa}_{\nu~{}\kappa}\big{)}\Omega_{,\sigma}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\big{(}g_{\sigma\alpha}L^{~{}\kappa}_{\nu~{}\kappa}-g_{\sigma\nu}L^{~{}\kappa}_{\alpha~{}\kappa}\big{)}\Omega_{,\rho}\big{\\}}\\\
&-\frac{4}{3}L^{\lambda\kappa}_{~{}~{}~{}\kappa}~{}\Omega_{,\lambda}\big{(}g_{\alpha\sigma}~{}g_{\nu\rho}-g_{\nu\sigma}~{}g_{\alpha\rho}\big{)}\big{]}.\end{aligned}\right.$
(31)

Thus in general the potential tensor $L_{\mu\nu\sigma}$, defined through (4),
fails to produce
$\tilde{W}^{\mu}_{~{}~{}\nu\sigma\rho}=W^{\mu}_{~{}~{}\nu\sigma\rho}$ if the
tensor is limited to admit the symmetries (3) only. This is due to the
presence of the second term on the right hand side of equation (31). Although
this undesired term vanishes for $s=3$, this choice does not fulfill the
requirement
$\tilde{W}^{\mu}_{~{}~{}\nu\sigma\rho}=W^{\mu}_{~{}~{}\nu\sigma\rho}$. The
only way to achieve this goal is to have $s=2$ together with imposing a
condition on $L_{\mu\nu\sigma}$ given by

$\left.\begin{aligned}
2\big{\\{}&L_{\alpha\nu\sigma}~{}\Omega_{,\rho}+L_{\nu\alpha\rho}~{}\Omega_{,\sigma}+L_{\sigma\rho\alpha}~{}\Omega_{,\nu}+L_{\rho\sigma\nu}~{}\Omega_{,\alpha}\big{\\}}+\big{\\{}g_{\alpha\rho}(L^{~{}\kappa}_{\nu~{}\sigma}+L^{~{}\kappa}_{\sigma~{}\nu})+g_{\nu\sigma}(L^{~{}\kappa}_{\alpha~{}\rho}+L^{~{}\kappa}_{\rho~{}\alpha})\\\
&-g_{\alpha\sigma}(L^{~{}\kappa}_{\nu~{}\rho}+L^{~{}\kappa}_{\rho~{}\nu})-g_{\nu\rho}(L^{~{}\kappa}_{\alpha~{}\sigma}+L^{~{}\kappa}_{\sigma~{}\alpha})\big{\\}}\Omega_{,\kappa}+\big{\\{}\big{(}g_{\nu\rho}L^{~{}\kappa}_{\sigma~{}\kappa}-g_{\nu\sigma}L^{~{}\kappa}_{\rho~{}\kappa}\big{)}\Omega_{,\alpha}\\\
&+\big{(}g_{\alpha\sigma}L^{~{}\kappa}_{\rho~{}\kappa}-g_{\alpha\rho}L^{~{}\kappa}_{\sigma~{}\kappa}\big{)}\Omega_{,\nu}+\big{(}g_{\rho\nu}L^{~{}\kappa}_{\alpha~{}\kappa}-g_{\rho\alpha}L^{~{}\kappa}_{\nu~{}\kappa}\big{)}\Omega_{,\sigma}+\big{(}g_{\sigma\alpha}L^{~{}\kappa}_{\nu~{}\kappa}-g_{\sigma\nu}L^{~{}\kappa}_{\alpha~{}\kappa}\big{)}\Omega_{,\rho}\big{\\}}\\\
&-\frac{4}{3}L^{\lambda\kappa}_{~{}~{}~{}\kappa}~{}\Omega_{,\lambda}\big{(}g_{\alpha\sigma}~{}g_{\nu\rho}-g_{\nu\sigma}~{}g_{\alpha\rho}\big{)}=0.\end{aligned}\right.$
(32)

It is not clear at this point what kind of symmetry this equation imposes on
$L_{\mu\nu\sigma}$ in addition to the two obligatory symmetries (3a) and (3b).
In the following we discuss a simple possibility which is suggested by
breaking the long equation (32) in four parts according to the symmetries of
the terms:

$\displaystyle
L_{\alpha\nu\sigma}~{}\Omega_{,\rho}+L_{\nu\alpha\rho}~{}\Omega_{,\sigma}+L_{\sigma\rho\alpha}~{}\Omega_{,\nu}+L_{\rho\sigma\nu}~{}\Omega_{,\alpha}=0,$
(33a)
$\displaystyle\big{\\{}g_{\alpha\rho}(L^{~{}\kappa}_{\nu~{}\sigma}+L^{~{}\kappa}_{\sigma~{}\nu})+g_{\nu\sigma}(L^{~{}\kappa}_{\alpha~{}\rho}+L^{~{}\kappa}_{\rho~{}\alpha})-g_{\alpha\sigma}(L^{~{}\kappa}_{\nu~{}\rho}+L^{~{}\kappa}_{\rho~{}\nu})-g_{\nu\rho}(L^{~{}\kappa}_{\alpha~{}\sigma}+L^{~{}\kappa}_{\sigma~{}\alpha})\big{\\}}\Omega_{,\kappa}=0,$
(33b)
$\displaystyle\big{(}g_{\nu\rho}L^{~{}\kappa}_{\sigma~{}\kappa}-g_{\nu\sigma}L^{~{}\kappa}_{\rho~{}\kappa}\big{)}\Omega_{,\alpha}+\big{(}g_{\alpha\sigma}L^{~{}\kappa}_{\rho~{}\kappa}-g_{\alpha\rho}L^{~{}\kappa}_{\sigma~{}\kappa}\big{)}\Omega_{,\nu}+\big{(}g_{\rho\nu}L^{~{}\kappa}_{\alpha~{}\kappa}-g_{\rho\alpha}L^{~{}\kappa}_{\nu~{}\kappa}\big{)}\Omega_{,\sigma}$
$\displaystyle+\big{(}g_{\sigma\alpha}L^{~{}\kappa}_{\nu~{}\kappa}-g_{\sigma\nu}L^{~{}\kappa}_{\alpha~{}\kappa}\big{)}\Omega_{,\rho}=0,$
(33c) $\displaystyle
L^{\lambda\kappa}_{~{}~{}~{}\kappa}~{}\Omega_{,\lambda}=0.$ (33d)

The last equation, taken together with the arbitrariness in $\Omega$, implies
that the Lanczos potential tensor $L_{\mu\nu\sigma}$ should be trace-free:

$L^{~{}\kappa}_{\mu~{}\kappa}=0.$ (34)

This not only satisfies identically the condition (33c) but also appears
consistent with the general tenet that the conformally invariant tensors must
be trace-free, thus contributing further to the aesthetic appeal of (34). Let
us note that the scaling law (30), for $s=2$, implies that the tensor
$L_{\mu\nu\sigma}$ (with one index raised) is conformally invariant:
$\tilde{L}^{\mu}_{~{}~{}\nu\sigma}=L^{\mu}_{~{}~{}\nu\sigma}$, akin to
$\tilde{C}^{\mu}_{~{}~{}\nu\sigma\rho}=C^{\mu}_{~{}~{}\nu\sigma\rho}$.
(However, this is not so with the tensor with all covariant indices:
$\tilde{L}_{\mu\nu\sigma}=\Omega^{2}L_{\mu\nu\sigma}$, akin to
$\tilde{C}_{\mu\nu\sigma\rho}=\Omega^{2}C_{\mu\nu\sigma\rho}$.)

Thus under this simple case, the constraints (33a), (33b) and (34) are imposed
on the Lanczos tensor by the conformal invariance of the Weyl tensor (which
must be met) brought about through the Weyl-Lanczos equation. This may supply
the missing plausibility conditions to be imposed on the Lanczos tensor as has
been mentioned in section 2. Further study is required to decipher the meaning
of the more general case (32).

##  4 Possible Interpretations of Lanczos Tensor

We have derived the Lanczos potential tensor in various cases of spacetime in
the above examples. However, unlike the Weyl tensor, it has not so far been
possible to obtain a general expression for this tensor in terms of the metric
tensor. Though this expression does exist in the case of a weak gravitational
field, wherein the spacetime metric differs minutely from the Minkowskian
metric $\eta_{\mu\nu}$:

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},~{}~{}~{}~{}~{}~{}{\rm
where}~{}~{}~{}~{}|h_{\mu\nu}|<<1.$ (35)

In this case, the Lanczos tensor can be written as [2]

$L_{\mu\nu\sigma}=\frac{1}{4}\left(h_{\mu\sigma,\nu}-h_{\nu\sigma,\mu}+\frac{1}{6}h_{,\mu}\eta_{\nu\sigma}-\frac{1}{6}h_{,\nu}\eta_{\mu\sigma}\right),$
(36)

to linear order in the metric perturbations $h_{\mu\nu}$. Here $h\equiv
h_{\mu\nu}\eta^{\mu\nu}$.

###  4.1 Minkowskian Lanczos Tensor as a Ground State ‘Potential’

Let us note that the covariant and ordinary differentiations coincide in the
first order of approximation in the case of a weak gravitational field and
hence the differential and algebraic tensor operations are performed by the
use of the Minkowskian metric $\eta_{\mu\nu}$, as is the case with
$L_{\mu\nu\sigma}$ defined by (36). This is exactly the case with the Lanczos
potential for the Minkowski spacetime, which has been elaborated on in [3,
4]333The Lanczos tensor of any conformally flat spacetime including the flat
spacetime itself can be defined by equations (8a-8c) (with the understanding
of $A_{\mu\nu\sigma}$ denoting the Lanczos tensor of the corresponding
spacetime).. This implies that if $L_{\mu\nu\sigma}$ is the Lanczos potential
of a given spacetime in a weak field, then so is
$L_{\mu\nu\sigma}+\overset{\rm{\scriptscriptstyle GS}}{L}_{\mu\nu\sigma}$,
where $\overset{\rm{\scriptscriptstyle GS}}{L}_{\mu\nu\sigma}$ is the Lanczos
potential of the Minkowskian spacetime. Thus $\overset{\rm{\scriptscriptstyle
GS}}{L}_{\mu\nu\sigma}$ appears as a ‘ground state’ potential field which
makes an essential contribution to the Lanczos potential fields of all the
curved spacetimes in a weak gravitational field. This provides the Minkowskian
Lanczos potential a natural and adequate interpretation portraying it as the
‘weight’ or the ‘metrical elasticity’ of spacetime, which being an integral
part of all the spacetime geometries, opposes the curving of spacetime.

Assigning a ‘ground state’ potential field to the Minkowskian spacetime in the
absence of any curvature, may appear puzzling and surprising at the first
glance. However, there does exist a parallel to this situation in the quantum
realm wherein one can have a vanishing electromagnetic field in a region but a
non-vanishing potential, explaining the conspicuous physical effects which do
exist in that region (Aharonov-Bohm effect) [3].

A word of caution is in order here. It should be noted that the Lanczos
‘potential’ does not have the dimensions of the potential energy, in the same
way as the Weyl tensor does not have the dimensions of force. Thus the
denomination of the Lanczos tensor as the ‘Lanczos potential’, refers to just
the electromagnetic analogy. Let us recall that the Lanczos tensor is regarded
as the gravitational analogue of the electromagnetic potential, since it
generates the Weyl tensor differentially, in the same way as the
electromagnetic potential generates the Faraday tensor. Hence, the Lanczos
tensor is generally termed as the potential to the Weyl tensor.

###  4.2 Lanczos Tensor as a Relativistic Analogue of the Newtonian Force

Since the Lanczos tensor generates the Weyl tensor and since the latter is
linked with the (free) gravitational field, one may naturally expect that the
Lanczos tensor too has something to do with the gravitational field. In this
view, the quantity $L_{\mu\nu\sigma}$ given by expression (36) in terms of the
derivatives of the metric tensor, gives a clue that the Lanczos tensor may
represent a relativistic analogue of the Newtonian force. However if this is
so, the tensor should reduce to the Newtonian gravitational force in a
physical situation in the weak field and low velocity limit. As the Newtonian
theory of gravitation provides excellent approximations under a wide range of
astrophysical cases, the first crucial test of any theory of gravitation is
that it reduces to the Newtonian gravitation.

In order to check this, let us consider a static spherical mass $m$ placed at
the origin of a centrally symmetric coordinate system $r,\theta,\phi$. In the
Newtonian theory of gravity, the gravitational field produced by the mass at a
point $r$ is represented in terms of the gravitational potential
$\Phi(r)=-m/r$ at that point. In a relativistic theory of gravitation, for
example GR, the gravitational field of the mass is well-described by the
Schwarzschild line element (9). In a weak field, the line element (9) reduces
to

$ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}-\frac{2m}{r}(dt^{2}+dr^{2}),$ (37)

at large $r$ so that $2m/r<<1$. Its comparison with the condition (35) gives
the only non-vanishing components of $h_{\mu\nu}$ as $h_{00}=h_{11}=-2m/r$.
For this, the definition (36) provides

$L_{100}=-\frac{m}{2r^{2}},$ (38)

as the only non-vanishing independent component of $L_{\mu\nu\sigma}$, which
amounts to

$L_{100}=-\frac{1}{2}\frac{\partial\Phi}{\partial r},$ (39)

by virtue of the well-known relation $g_{00}=(1+2\Phi)$ valid in the weak
field and low velocity limit. This can be written in terms of $F_{\rm N}$, the
corresponding Newtonian force experienced by a unit test mass placed at a
distance $r$ from the source mass:

$L_{100}=\frac{1}{2}F_{\rm N}.$ (40)

Thus the Lanczos tensor indeed represents the relativistic analogue of the
Newtonian gravitational force in a curved spacetime in the weak field and low
velocity limit (the factor $\frac{1}{2}$ can be considered as a relativistic
effect). It may be mentioned that conventionally it is the Christoffel symbol
which is considered as the analogue of the Newtonian force. However, as the
Christoffel symbols are not tensors, the Lanczos tensor suits better to a
covariant theory in representing the analogue of the Newtonian force when the
analogue of the Newtonian potential is ascribed to a tensor (the metric
tensor).

This new understanding of the Lanczos tensor also makes a prediction: the
tensor should match at the boundary of the two spacetime regions on the
surface of a gravitating body. In order to exemplify this, let us apply the
force analogy on a perfect fluid ball of constant density wherein the
spacetime inside the ball is given by the Schwarzschild interior line element
(17), and that outside the ball by the Schwarzschild (exterior) line element
(9). According to the force analogy, the Lanczos tensor would approximate the
force experienced by a unit test mass placed on the surface of the ball. But
the Lanczos tensor can be calculated from either of the two line elements,
which of course match at the boundary. It is then expected that the Lanczos
tensor too should match smoothly at the boundary, if the force analogy is
correct. This is indeed the case, as we shall witness in section 6.1. This
scenario is also corroborated by the case of the collapsing ball discussed in
section 6.2.

This analogy however also poses a question: How is the gravitational
information encoded in the Lanczos tensor? It is clear that in the above-
described example, the Newtonian force on the test mass is estimated in terms
of the mass of the gravitating ball, which enters the scene through the
Einstein field equation. In other words, the information on this force (and
hence that on matter and gravity) gets encoded in Lanczos tensor through
Einstein’s tensor, i.e. through Ricci’s tensor and scalar. Let us also recall
that the Riemann tensor is decomposed in terms of its two mutually independent
parts - its trace-free part (the Weyl tensor) and its trace (the Ricci tensor
and Ricci scalar) as

$R_{\mu\nu\sigma\rho}=C_{\mu\nu\sigma\rho}-g_{\mu[\rho}R_{\sigma]\nu}-g_{\nu[\sigma}R_{\rho]\mu}-\frac{1}{3}Rg_{\mu[\sigma}~{}g_{\rho]\nu}.$
(41)

As the energy-momentum tensor of matter is correlated with the trace of
Riemann tensor (through the field equation), equation (41) indicates that the
Weyl tensor is algebraically independent of the matter tensor. Hence so is the
Lanczos tensor (which is the potential of Weyl tensor), since the Ricci’s
tensor and scalar cannot admit a Lanczos tensor by virtue of the absence of a
Lanczos potential tensor for the Riemann tensor [13]. Then how is the above-
mentioned encoding of the gravitational information into the Lanczos tensor
realized? Perhaps some another unknown feature of the tensor is at work,
further studies may unearth it.

It may however be noted that the force analogy of the Lanczos tensor has
limitations. The analogy does not imply that a geometric theory of
gravitation, for instance GR, would mimic the Newtonian action-at-a-distance
concept (which requires an infinite speed of propagation of gravity) for
explaining observations. The domain of applicability of the force analogy of
the Lanczos tensor is the same as that of the potential analogy of the metric
tensor, i.e. the slow motion and the weak field limit of gravity.

##  5 Gravitational Wave and its Energy-Momentum

Existence of gravitation waves is one of the most important features of GR,
which was first predicted by Einstein. In the light of the direct detection of
the gravitational waves from binary mergers, it is now possible to extract
information about matter and spacetime under extreme conditions.

On the theoretical side however, there is no general agreement on the
description of the energy-momentum carried by the gravitational waves. The
reason is that there is no existence of a localizable (i.e., tensor-
represented) point-wise gravitational energy-momentum density in GR. The use
of the non-covariant energy-momentum pseudo-tensors totally obscures the
analysis of the subject. Besides their non-uniqueness, they can be annihilated
at will at any given spacetime point by the right choice of coordinates. Thus,
in the framework of pseudo tensor-formulation of the energy-momentum, one
concludes that the energy carried by a gravitational wave is coordinate
dependent. Thus, the pseudo tensor-formulation of the energy-momentum density
of the gravitational waves is though practically useful, it is not
satisfactory from the theoretical point of view. We need a fully covariant
tensorial formulation of this quantity with the requirement that it reduces to
the pseudo tensor-formulation in a weak field.

In our search for a covariant tensorial formulation of the energy-momentum of
the gravitational waves, we get a clue from the results obtained in the
preceding section that the Lanczos tensor is a relativistic analogue of the
Newtonian force. Hence its square has the same dimensions as that of the
energy density (in geometric units). This means that, following the
gravitation-electrodynamics correspondence, an energy-momentum tensor can be
formulated in terms of the Lanczos tensor, along the lines of the energy-
momentum tensor of the electromagnetic field, in which the Lanczos tensor
appears quadratically. It is shown in the following that the so framed energy-
momentum tensor gives the same results in the linearized theory as do the
conventional pseudo tensor-based formulations for the energy-momentum of the
gravitational waves.

###  5.1 Linearized Gravity and Isaacson’s Energy-Momentum Pseudo Tensor

A simple wave equation in GR, is a linearized approximation of Einstein’s
equation where the velocities are small and the gravitational fields are weak,
so that the spacetime metric is given by (35). Einstein’s equation, in a
Ricci-flat spacetime, then yields

$\partial^{\kappa}\partial_{\kappa}\bar{h}_{\mu\nu}=0~{}~{}~{}~{}~{}{\rm
with}~{}~{}~{}~{}~{}\bar{h}_{\mu\nu}\equiv h_{\mu\nu}-\eta_{\mu\nu}h/2,$ (42)

which is a spin-2 field equation. The linearized Einstein equation takes the
simple form (42) only if the Lorenz gauge conditions
$\partial_{\mu}\bar{h}^{\mu\nu}=0$ are satisfied. This requirement makes the
metric perturbation in equation (42) look like a transverse wave. By imposing
the Lorenz gauge conditions, we have reduced the 10 independent components of
the symmetric tensor $\bar{h}_{\mu\nu}$ to six. Since there are really only
two independent components of the Riemann tensor in the present case, the
remaining freedom is used to choose a gauge in which the perturbation becomes
traceless ($\bar{h}\equiv\bar{h}_{\mu\nu}\eta^{\mu\nu}=0=h\equiv
h_{\mu\nu}\eta^{\mu\nu}$). This defines the transverse and traceless (TT)
gauge leaving the only independent components of $\bar{h}_{\alpha\beta}$
($=h_{\alpha\beta}$) as

$\displaystyle h_{11}$ $\displaystyle=-h_{22}\equiv h_{+},$ (43a)
$\displaystyle h_{12}$ $\displaystyle=h_{21}\equiv h_{\times}.$ (43b)

In this case, the plane wave solution of equation (42) traveling along the
$z=x^{3}$-direction can be written as

$h_{\mu\nu}=A_{\mu\nu}\cos\big{(}\omega(t-z)\big{)},$ (44)

where $A_{\mu\nu}$ is the constant amplitude of the wave and $\omega$ its
angular frequency.

As has been mentioned earlier, the description of the energy-momentum carried
by the gravitational waves is still a disputed topic. Generally, the energy-
momentum tensor for gravitational waves is obtained from Isaacson’s stress-
energy pseudo-tensor $\tau_{\mu\nu}$ [14] by averaging the squared gradient of
the wave field over a length much larger than the typical gravitational
wavelength:

$\tau_{\mu\nu}=\frac{1}{32\pi}\langle\nabla_{\mu}h_{\alpha\beta}\nabla_{\nu}h^{\alpha\beta}\rangle.$
(45)

In the above-defined TT gauge, the non-vanishing components of $\tau_{\mu\nu}$
are

$\tau_{00}=-\tau_{03}=-\tau_{30}=\tau_{33}=\frac{1}{16\pi}\langle(\dot{h_{+}})^{2}+(\dot{h_{\times}})^{2}\rangle,$
(46)

where an overhead dot denotes derivative with respect to $t$.

###  5.2 Lanczos Wave and its Energy-Momentum

As has been mentioned earlier, there exists a perfect correspondence between
electrodynamics and gravitation and various analogues of electromagnetic
phenomena have been discovered in gravitation [15]. Electrodynamics admits a
wave equation for the electromagnetic 4-potential. Note that this is so not
only in flat spacetime, but also in the curved one. It is already known (see,
for example, [12, 16]) that a Killing vector field $A^{\mu}$ in a Ricci-flat
spacetime plays the role of the electromagnetic 4-potential and the source-
free Maxwell equations, in Lorenz gauge ($A^{\kappa}_{~{}~{};\kappa}=0$),
reduce to the homogeneous wave equation

$\nabla^{\kappa}\nabla_{\kappa}A_{\mu}=0.$ (47)

As the Lanczos potential constitute a gravitational analogue of
electromagnetic 4-potential, we expect a similar equation for
$L_{\mu\nu\sigma}$ if the correspondence between electrodynamics and gravity
has any real meaning. Remarkably, the Lanczos potential indeed satisfies a
homogeneous wave equation

$\nabla^{\kappa}\nabla_{\kappa}L_{\mu\nu\sigma}=0,$ (48)

in Lanczos gauge, in any Ricci-flat spacetime [11, 17] (not necessarily in a
weak field), strengthening our belief in the correspondence. Akin to the wave
equation (42), this simple and beautiful exact analytical solution is admitted
in a Ricci-flat spacetime in any metric theory of gravity formulated in a
4-dimensional pseudo-Riemannian manifold.

This cannot be just a formal accident and must have some deeper meaning. We
know that a plane wave describes, from the quantum point of view, a collection
of massless quanta - photons - with helicity $\pm 1$ for electrodynamics.
Hence, the electrodynamics-gravity correspondence, in view of equations (47)
and (48), insinuates that the plane wave (48) transports the excitations of a
massless field. That is, it describes at the quantum level, a collection of
massless quanta - gravitons - with helicity $\pm 2$ mediating gravity. Hence,
its energy-momentum must necessarily be represented by a trace-free symmetric
tensor, which is expected to reduce to the pseudo tensor-formulation (45) in a
weak field. This might be helpful to develop an effective description of the
physics of graviton which captures some quantum effects but is otherwise based
on classical concepts. This would also fit nicely with the earlier findings
that the Lanczos potential is impregnated with various signatures of quantum
physics [2, 3, 5].

Supplemented with the novel insight that the energy-momentum tensor for the
wave (48) must be symmetric and trace-free, and the earlier-noted cue that
this tensor should have $L_{\mu\nu\sigma}$ appearing quadratically, we define
the following tensor along the lines of the energy-momentum tensor of the
electromagnetic field $(\overset{\rm
EM}{T}_{\alpha\beta}=F^{~{}\sigma}_{\alpha}F_{\beta\sigma}+*F^{~{}\sigma}_{\alpha}~{}{*F}_{\beta\sigma})$:

$T_{\alpha\beta\gamma\delta}=\kappa(L^{~{}~{}\sigma}_{\alpha~{}~{}~{}\gamma}~{}L_{\beta\sigma\delta}+*L^{~{}~{}\sigma}_{\alpha~{}~{}~{}\gamma}~{}{*L}_{\beta\sigma\delta}),$
(49)

which is though not symmetric in all pair of indices. [A dimensional constant
$\kappa$ has been inserted in equation (49), which is given in terms of $G$
and $c$ in ordinary units.] A symmetric tensor can be obtained from (49) by
contracting over the last pair of indices of $T_{\alpha\beta\gamma\delta}$,
giving

$T_{\alpha\beta}=\kappa\left(L^{~{}~{}\sigma\rho}_{\alpha}~{}L_{\beta\sigma\rho}+*L^{~{}~{}\sigma\rho}_{\alpha}~{}{*L}_{\beta\sigma\rho}\right),$
(50)

which is indeed symmetric and trace-free (even if $L_{\mu\nu\sigma}$ does not
satisfy the trace-free condition) like the energy-momentum tensor of the
electromagnetic field. By evaluating the duals, equation (50) can be recast in
the form

$T_{\alpha\beta}=\kappa\left(2L^{~{}~{}\sigma\rho}_{\alpha}~{}L_{\beta\sigma\rho}-\frac{1}{2}g_{\alpha\beta}L^{\kappa\sigma\rho}L_{\kappa\sigma\rho}\right).$
(51)

In order to compare this tensor-based definition of the energy-momentum of the
gravitational waves with the traditional pseudo tensor one, let us calculate
it in the weak gravity case (35). But before that, let us first check if the
wave equation (48) is admitted in a weak field. That is, if the Lanczos gauge
conditions are admitted in a weak gravity field. For this purpose, we
calculate the trace and the divergence of the Lanczos tensor field given by
(36) in a weak field. They lead to

$\displaystyle
L^{~{}~{}\lambda}_{\mu~{}~{}\lambda}=\frac{1}{4}\bar{h}^{~{}\lambda}_{\mu~{},\lambda},$
(52a) $\displaystyle
L^{~{}~{}~{}\lambda}_{\mu\nu~{}~{},\lambda}=\frac{1}{4}\left(\bar{h}^{~{}\lambda}_{\mu~{},\lambda,\nu}-\bar{h}^{~{}\lambda}_{\nu~{},\lambda,\mu}\right).$
(52b)

Thus the tensor $L_{\mu\nu\sigma}$ does admit the Lanczos gauge conditions and
hence the wave equation (48) in the TT gauge in a weak field. We can now come
back to equation (51). In order to calculate it in a weak field, we use the
definition of $L_{\mu\nu\sigma}$ given by (36). In the TT gauge, this
evaluates the following two terms appearing in (51) as

$\displaystyle L^{~{}\sigma\rho}_{\alpha}L_{\beta\sigma\rho}$
$\displaystyle=\frac{1}{16}\big{(}2h_{11,\alpha}~{}h_{11,\beta}+2h_{12,\alpha}~{}h_{12,\beta}+h_{\alpha
1,3}~{}h_{\beta 1,3}+h_{\alpha 2,3}~{}h_{\beta 2,3}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-h_{\alpha 1,0}~{}h_{\beta
1,0}-h_{\alpha 2,0}~{}h_{\beta 2,0}\big{)},$ (53a) $\displaystyle
L^{\kappa\sigma\rho}L_{\kappa\sigma\rho}$
$\displaystyle=\frac{1}{4}\big{(}(h_{11,0})^{2}+(h_{12,0})^{2}-(h_{11,3})^{2}-(h_{12,3})^{2}\big{)}.$
(53b)

By virtue of $\partial h_{\mu\nu}/\partial z=-\partial h_{\mu\nu}/\partial t$,
which readily follows from solutions (44),
$L^{\kappa\sigma\rho}L_{\kappa\sigma\rho}$ vanishes identically and hence the
second term on the r.h.s. of (51) does not contribute anything to
$T_{\alpha\beta}$. Similarly, by the use of this identity, equation (53a)
reduces to

$L^{~{}\sigma\rho}_{\alpha}L_{\beta\sigma\rho}=\frac{1}{8}\big{(}\partial_{\alpha}h_{+}~{}\partial_{\beta}h_{+}+\partial_{\alpha}h_{\times}~{}\partial_{\beta}h_{\times}\big{)},$

giving

$T_{\alpha\beta}=\frac{\kappa}{4}\big{(}\partial_{\alpha}h_{+}~{}\partial_{\beta}h_{+}+\partial_{\alpha}h_{\times}~{}\partial_{\beta}h_{\times}\big{)}.$
(54)

Thus the only non-vanishing components of $T_{\alpha\beta}$ are obtained as

$T_{00}=-T_{03}=-T_{30}=T_{33}=\frac{\kappa}{4}\left((\dot{h}_{+})^{2}+(\dot{h}_{\times})^{2}\right).$
(55)

Remarkably, these values do match with their pseudo-tensor counterparts given
by (46) for $\kappa=1/4\pi$. But, here they emerge from a fully covariant
tensorial definition, which is intuitively appealing.

One can guess, how this energy-momentum tensor interacts with that of the
dynamical matter fields when the gravitational wave travels through matter.
There will necessarily be an exchange of energy and momentum between matter
and the Lanczos field, and what should be conserved is not the energy and
momentum of the matter alone, but that of the sum of all the fields
contributing to the structure of spacetime.

##  6 Matching of the Lanczos Tensor at the Surface of a Gravitating Body

In the general-relativistic stellar modelling, the exterior and the interior
spacetime geometries are glued together at the hypersurface by using the
Israel-Darmois junction conditions [18]. This requires that the extrinsic
curvature and the induced metric on the hypersurface must be the same on both
sides of the hypersurface. Well-known examples are:

  1. 1.

The Schwarzschild interior metric (17), representing the spacetime inside an
idealized star of constant density, is joined with the Schwarzschild exterior
metric (9).

  2. 2.

The dynamical cosmological interior (20), modeling an idealized collapsing
star of spatially constant density, is joined with the Schwarzschild static
exterior metric (9).

Why are the junction conditions limited to matching only the extrinsic
curvature and the induced metric at the boundary? Because the union of the two
metrics form a valid solution to Einstein field equation in this case. However
in a geometric theory of gravitation wherein gravity appears through the
curvature, we should be able to calculate the gravitational effect on a test
mass placed at the boundary, from either of the two metrics. Should we then
expect the curvature of the spacetimes too to match on the both sides of the
boundary? This expectation is however not fulfilled in the above examples. In
both these examples, the interiors are given by the conformally flat
spacetimes giving vanishing Weyl tensor. Whereas the exteriors are Ricci-flat
spacetimes. Thus, the Riemann tensor cannot be matched at the boundary, as is
ascertained by equation (41).

How else can we measure the gravitational effect experienced by the test mass?
In terms of a quantity which is an analogue of Newtonian gravitational force -
the Lanczos tensor! Interestingly, the Lanczos tensor indeed matches at the
boundary, as we shall see in the following.

###  6.1 Matching the Lanczos Tensor on an Idealized Static Star

As has been mentioned earlier, the line element given by (17), i.e.

$ds^{2}=\frac{1}{4}\left(3\sqrt{1-\frac{r_{0}^{2}}{R^{2}}}-\sqrt{1-\frac{r^{2}}{R^{2}}}\right)^{2}dt^{2}-\frac{dr^{2}}{1-r^{2}/R^{2}}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta~{}d\phi^{2},$

represents the static and isotropic spacetime inside a non-rotating star of
radius $r_{0}$, composed of an incompressible perfect fluid of constant
density $\rho_{0}$ and variable pressure $p$. This spacetime is glued with the
Schwarzschild static exterior spacetime given by the line element (9), i.e.

$ds^{2}=\left(1-\frac{2m}{r}\right)dt^{2}-\frac{dr^{2}}{(1-2m/r)}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta~{}d\phi^{2}.$

It is well-known that these interior and exterior spacetimes have a perfect
matching at the boundary $r=r_{0}$ (see for example, [19]).

We have already calculated the Lanczos tensor for these interior and exterior
spacetimes in sections 2.3 and 2.1 respectively. Let us first consider the
values of the tensor in these spacetimes satisfying the gauge condition of
divergence-freeness (5b). These are given by equations (18) and (11)
respectively for the interior and the exterior spacetimes:

$\overset{\rm
int}{L}_{100}=\frac{5r\left[1-\frac{r^{2}+9r_{0}^{2}}{10R^{2}}-\frac{3}{5}\sqrt{\left(1-\frac{r^{2}}{R^{2}}\right)\left(1-\frac{r_{0}^{2}}{R^{2}}\right)}\right]}{2R^{2}\left[\left(1-\frac{r^{2}}{R^{2}}\right)-3\sqrt{\left(1-\frac{r^{2}}{R^{2}}\right)\left(1-\frac{r_{0}^{2}}{R^{2}}\right)}\right]}~{}~{}~{}~{}~{}~{}~{}~{}\leftrightarrow~{}~{}~{}~{}~{}~{}~{}~{}\overset{\rm
ext}{L}_{100}=-\frac{m}{r^{2}},$

which become equal at the boundary $r=r_{0}$ by virtue of
$R^{2}=3/8\pi\rho_{0}$ and realizing that the total mass of the interior
$=4\pi r_{0}^{3}\rho_{0}/3$ constitutes the source mass $m$ for the exterior
spacetime. The same is true for the Lanczos tensor satisfying both Lanczos
gauge conditions (5a) and (5b). In this case, the respective values of the
tensor are given by equations (19) and (12) respectively for the interior and
the exterior spacetimes. At the boundary $r=r_{0}$, these values reduce to

$\left.\begin{aligned} &\overset{\rm
int}{L}_{100}=-\frac{r_{0}}{3R^{2}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leftrightarrow~{}~{}~{}~{}~{}~{}~{}~{}~{}\overset{\rm
ext}{L}_{100}=-\frac{2m}{3r_{0}^{2}},\\\ &\overset{\rm
int}{L}_{122}=-\frac{r_{0}^{3}}{6R^{2}\left(1-\frac{r_{0}^{2}}{R^{2}}\right)}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leftrightarrow~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\overset{\rm
ext}{L}_{122}=-\frac{m}{3(1-2m/r_{0})},\\\ &\overset{\rm
int}{L}_{133}=-\frac{r_{0}^{3}\sin^{2}\theta}{6R^{2}\left(1-\frac{r_{0}^{2}}{R^{2}}\right)}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leftrightarrow~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\overset{\rm
ext}{L}_{133}=-\frac{m\sin^{2}\theta}{3(1-2m/r_{0})},\end{aligned}\right.$

which coincide by virtue of $1/R^{2}=2m/r_{0}^{3}$ as before. Thus the Lanczos
tensor does match smoothly at the common boundary despite the Weyl tensor
differing in the two regions. This strengthens our observation that the
Lanczos tensor is a relativistic analogue of the Newtonian gravitational
force.

It however appears that the interior (17) considered in these examples, which
was derived by Schwarzschild to have a mathematically simple solution, is not
physically viable. As this solution assumes a static sphere of matter
consisting of a perfect fluid of constant density $\rho$ but a variable
pressure $p(r)$ that vanishes at the boundary, the speed of sound
$=\sqrt{dp/d\rho}$ becomes infinite in the fluid. In order to avoid this
situation, let us consider, in the following section, an interior composed of
a dust cloud with vanishing pressure, which gives rise to a collapsing
interior.

###  6.2 Matching the Lanczos Tensor on a Collapsing Star

The solution of Einstein field equation for a homogeneous dust cloud of
spherical symmetry, leads to the well-known problem of spherical gravitational
collapse. The spacetime inside the collapsing sphere is represented by the
dynamical R-W metric (20) with the spatially closed case, which is matched
with the Schwarzschild static exterior (9). This problem of the imploding dust
ball was considered by Datt in 1938 [20] and by Oppenheimer & Snyder in 1939
[21]. For the sake of completeness, we have summarized the formal aspects of
the theory in the Appendix.

Let us note that unlike the case studied in the preceding section, here the
matching of the two spacetime regions take place at a moving boundary between
the dynamical cosmological interior (20) and the static Schwarzschild exterior
(A-3). In order to study the matching of the accompanied Lanczos potentials in
the two regions, we first consider the potentials with single non-vanishing
components. These are given by equations (23) and (11) for the interior and
the exterior spacetimes respectively:

$\overset{\rm int}{L}_{100}=\frac{\kappa rS^{n}}{\sqrt{1-\alpha
r^{2}}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leftrightarrow~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\overset{\rm
ext}{L}_{100}=-\frac{m}{{\bar{r}}^{2}}.$ (56)

As the coordinates used in the two spacetime regions are different, the
accompanied potentials shown in (56) also have different coordinates. In order
to make the comparison possible, we use the transformation (A-5) to bring the
potentials in a single (comoving) coordinate system. This gives

$\overset{\rm int}{L}_{100}=\frac{\kappa rS^{n}}{\sqrt{1-\alpha
r^{2}}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leftrightarrow~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\overset{\rm
ext}{L}_{100}=-\frac{m}{r^{2}S^{2}}.$ (57)

Let us recall that the coordinates $r,\theta,\phi$ are the comoving
coordinates of the spacetime (20) and hence are independent of time. Hence, a
continuous contraction in the surface (boundary) of the ball can be brought
about only through the function $S=S(t)$, which is a decreasing function of
the proper time $t$, as is indicated by (A-1). This implies that both the
potentials in (57) must have the same $S$-dependence to ensure a proper
matching and continuity across the boundary throughout the collapse. This
uniquely determines the parameter $n$ appearing in (23) as $n=-2$.
Interestingly, this value of $n$ brings the potential (23) in the same
(divergence-free) gauge, as is (11), which is clear from equation (25). Thus
the final form of the potentials to be matched is

$\overset{\rm int}{L}_{100}=\frac{\kappa r}{\sqrt{1-\alpha
r^{2}}~{}S^{2}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leftrightarrow~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\overset{\rm
ext}{L}_{100}=-\frac{m}{r^{2}S^{2}},$ (58)

which indeed match at the boundary $r=r_{\rm b}$ throughout the collapse as
the matching becomes independent of $S(t)$, and hence independent of time.
This determines the constant $\kappa$ uniquely as
$\kappa=-\alpha\sqrt{1-\alpha r_{\rm b}^{2}}/2$ by virtue of
$\alpha=8\pi\rho_{0}/3=2m/r_{\rm b}^{3}$, thus resulting in a smooth fit which
appropriately accounts for continuity in the Lanczos potentials across the
sphere boundary.

Let us now consider the matching of the Lanczos tensor having more than one
non-vanishing components in the two regions of spacetime, for instance (27)
versus (12):

$\left.\begin{aligned} &\overset{\rm
int}{L}_{100}=\frac{\kappa_{1}r}{\sqrt{1-\alpha
r^{2}}S^{2}}~{}~{}~{}~{}~{}~{}~{}~{}\leftrightarrow~{}~{}~{}~{}~{}~{}~{}~{}~{}\overset{\rm
ext}{L}_{100}=-\frac{2m}{3\bar{r}^{2}}=-\frac{2m}{3r^{2}S^{2}},\\\
&\overset{\rm int}{L}_{122}=\frac{\kappa_{2}r^{3}}{\sqrt{1-\alpha
r^{2}}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leftrightarrow~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\overset{\rm
ext}{L}_{122}=-\frac{m}{3(1-2m/\bar{r})}=-\frac{m}{3\left(1-\frac{2m}{rS}\right)},\\\
&\overset{\rm
int}{L}_{133}=\frac{\kappa_{2}r^{3}\sin^{2}\theta}{\sqrt{1-\alpha
r^{2}}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leftrightarrow~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\overset{\rm
ext}{L}_{133}=-\frac{m\sin^{2}\theta}{3(1-2m/\bar{r})}=-\frac{m\sin^{2}\theta}{3\left(1-\frac{2m}{rS}\right)}.\end{aligned}\right.$
(59)

An inspection reveals that only the first component (i.e., $L_{100}$) of the
interior and exterior potentials diverge at $S=0$, but the rest two components
do not. While the last two interior ones remain fixed on the boundary, the
corresponding exterior ones vanish as $S\rightarrow 0$. Hence they do not seem
to match in this form. Let us recall that in the absence of sufficient
conditions of plausibility that should be imposed to give a unique value of
the tensor $L_{\mu\nu\sigma}$, there remain redundant degrees of freedom in it
and hence some of its values (in some arbitrary gauge) may not be physically
meaningful. However, it would be interesting to see under what conditions the
values given in (59) can match. In this context, one can check that if the
collapsing $S$ can halt at a non-vanishing minimum of $S$, say $S_{\rm min}$,
then all the three components of the tensor match at the boundary after the
ball settles to the minimum radius. This determines
$\kappa_{1}=-\alpha\sqrt{1-\alpha r_{\rm b}^{2}}/3$ and
$\kappa_{2}=-\frac{\alpha}{6}\frac{\sqrt{1-\alpha r_{\rm b}^{2}}}{(1-\alpha
r_{\rm b}^{2}/S_{\rm min})}$.

The above result does not mean that the Lanczos tensor acts as a repulsive
agent that can halt the collapse. Rather it provides an indication that there
must exist an era when the quantum gravity-effects take over at some $S_{\rm
min}$, arresting the collapse and avoiding the singularity. There have been
various claims that the singularities of GR can be avoided by using quantum
effects. Although a self-consistent theory of quantum gravity remains elusive,
there is a general agreement that removal of classical gravitational
singularities is not only a crucial conceptual test of any theory of quantum
gravity but also a prerequisite for any reasonable theory of quantum gravity.
Let us note that the existence of an $S_{\rm min}$ is also not inconsistent
with the matching discussed in equation (58).

##  7 Concluding Remarks

Einstein’s revolutionary insight - the local equivalence of gravitation and
inertia - paved the way to the lofty goal of the geometrization of gravitation
by introducing (pseudo) Riemannian geometry wherein the Riemann-Christoffel
curvature tensor plays the central role for the unfolding of gravity in a
four-dimensional spacetime manifold. However, it is not the full curvature
tensor which takes part in the field equations of gravitation. Rather, only
its trace - the Ricci tensor and Ricci scalar - appear in Einstein’s equation.
It is the Lanczos tensor which fulfills this gap by generating the trace-free
part of the Riemann tensor - exactly those components of the tensor which are
missing in Einstein’s equation.

Despite its importance and novelty, the Lanczos tensor has not been paid
proper attention as it deserves. One of the reasons of its obscurity is that
it is not clear what it represents physically. By deriving expressions for the
tensor in some particularly chosen spacetimes, we have attempted to find its
physical meaning and an adequate interpretation. It has been shown that the
especial property of the Weyl tensor - its conformal invariance - imposes
additional constraints on the Lanczos tensor, not realized before.

The traditional viewpoint on the Lanczos tensor comes from the well-studied
correspondence between electrodynamics and gravitation. Thence appears the
perspective that the Lanczos tensor is a gravitational analogue of the
electromagnetic 4-potential since it generates Weyl tensor differentially. A
complementary perspective on the tensor has been discovered which emerges from
the weak field and low velocity limit of GR: One may interpret the Lanczos
tensor in a curved spacetime as a relativistic, tensorial analogue of the
Newtonian gravitational force. Additionally, the Minkowskian Lanczos tensor
appears as part and parcel of the Lanczos tensor field of any curved spacetime
in a weak field and hence can be interpreted as the ‘weight’ of the spacetime,
providing a substantive status to it. This emphasizes the importance of the
Lanczos tensor in terms of a multi-featured quantity, which is also
corroborated by its applications devised in the preceding sections.

The relation of this tensor to the energy-momentum of the gravitational waves,
and its matching on the surface of a gravitating body give added proof of its
fundamental significance. Interestingly, the matching of the potential at the
boundary of a collapsing mass provides an indication of the existence of
quantum gravity-effects avoiding the singularity. This finding attributes a
quantum signature to the Lanczos potential, which appears consistent with some
other studies revealing a quantum character of the potential tensor.

It appears that the theory of the Lanczos tensor field contains surprises and
insights which have only just started to reveal. The performed study has
introduced a new domain of applicability of this tensor field. It reveals that
the tensor field, which requires for its construction merely the basic
concepts of spacetime, is impregnated with interesting physical properties and
opens up a new research avenue with rich prospects.

## Appendix: Dutt-Oppenheimer-Snyder model of collapsing dust cloud $-$ A
Brief Review

This is a simple model of spherical gravitational collapse of a dust ball,
which has considerable methodological importance. Let us consider a non-
rotating ball of electrically neutral matter having a spherical symmetry in
its physical parameters such as density and pressure. Let us ignore the effect
of pressure as an opposing agency to gravity. This is a reasonable assumption
for the stars in which the fusion process has nearly ceased and hence little
radiation pressure remains. Moreover, the pressure generated during collapse
is not adequate to halt the collapse.

By solving Einstein’s field equation for a dynamic, spherically symmetric line
element with the energy-momentum tensor for a pressure-less perfect fluid, it
has been shown that such a ball of pressure-less dust of uniform but time-
dependent density $\rho$, must contract without limit under self gravitation
[22]. The spacetime interior to the dust ball turns out to be given by the
spatially closed R-W line element (20):

$ds^{2}=dt^{2}-S^{2}(t)\left[\frac{dr^{2}}{1-\alpha
r^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta~{}d\phi^{2})\right],$

where $\alpha\equiv 8\pi\rho_{0}/3$ with $\rho_{0}$ being the initial density
of the dust particles measured by the observer riding the surface of the
collapsing ball. Clearly such an observer uses comoving coordinates
$r,\theta,\phi$ and thus measures proper time $t$. He thus measures the total
mass of the ball as $m=4\pi\rho_{0}r_{\rm b}^{3}/3$, with $r=r_{\rm b}$ being
the radial coordinate of the boundary of the dust ball. The dynamics of the
collapse is regulated by the dynamics of the function $S(t)$ through

$\dot{S}(t)=-\sqrt{\alpha}\left[\frac{1}{S(t)}-1\right]^{1/2},$ (A-1)

obtained by normalizing the radial coordinate so that $S(0)=1$, and by
assuming that the dust particles are at rest initially in the chosen
coordinates, so that $\dot{S}(0)=0$. Equation (A-1) can easily be integrated
in the following parametric form:

$S=\frac{1}{2}(1+\cos\psi),~{}~{}~{}~{}~{}~{}~{}t=\frac{\psi+\sin\psi}{2\sqrt{\alpha}}.$
(A-2)

This indicates that $S=0$ when $\psi=\pi$. Thus the observer riding the
surface of the ball, registers it collapsing into the centre with $S=0$ in
proper time $\frac{\pi}{2\sqrt{\alpha}}.$ However, a distant external observer
measures a different time interval for this process. Let us note that the
spacetime exterior to the boundary is described by the Schwarzschild line
element (9)

$ds^{2}=\left(1-\frac{2m}{\bar{r}}\right)d\bar{t}^{2}-\frac{d\bar{r}^{2}}{(1-2m/\bar{r})}-\bar{r}^{2}(d\bar{\theta}^{2}+\sin^{2}\bar{\theta}~{}d\bar{\phi}^{2}),$
(A-3)

as may be inferred from the Birkhoff theorem. As this line element is not in
the Gaussian normal form as in (20), we have departed from the earlier
notation used for coordinates in (9). It can now be shown [22] that a light
signal emitted in the radial direction at time ${\bar{t}_{1}}$ from a point
$\bar{r}={\bar{r}_{1}}$ on the surface of the collapsing ball, will arrive at
an external observer located at $\bar{r}={\bar{r}_{2}}$ at time
${\bar{t}_{2}}$ given by

${\bar{t}_{2}}={\bar{t}_{1}}+\int^{\bar{r}_{2}}_{\bar{r}_{1}}\left(1-\frac{2m}{x}\right)^{-1}dx,$
(A-4)

indicating that ${\bar{t}_{2}}\rightarrow\infty$ as ${\bar{r}_{1}}\rightarrow
2m$ (Schwarzschild radius). Thus the collapse of the ball to the Schwarzschild
radius, appears to an outside observer to take an infinite time. Hence, the
collapse to $S=0$ is utterly unobservable from outside. This classical,
unavoidable collapse of all the particles of the ball into the centre, is a
true singularity of spacetime, as $S=0$ signifies a state wherein the entire
space shrinks to zero volume with the density going to infinity.

As the coordinates used in the interior and the exterior line elements (20)
and (A-3) are different, we have to convert one into the other in order to
facilitate the matching of the two line elements at the boundary of the dust
ball. Comparing the angular parts of the line elements (20) and (A-3), it is
evident that the comoving R-W coordinates in (20) and those in (A-3) are
related by

$\bar{r}=rS(t),~{}~{}~{}~{}\bar{\theta}=\theta,~{}~{}~{}~{}\bar{\phi}=\phi.$
(A-5)

By the use of these transformations, the interior line element (20) can be
expressed in Schwarzschild coordinates $\bar{t}$, $\bar{r}$, showing that the
interior and the exterior line elements fit continuously at the boundary
$\bar{r}=r_{\rm b}S(t)$ by virtue of $\alpha\equiv 8\pi\rho_{0}/3=2m/r_{\rm
b}^{3}$ [22].

Acknowledgements: The author would like to thank Sanjeev V. Dhurandhar, Jayant
V. Narlikar and José M. M. Senovilla for many insightful discussions. This
paper has benefited from suggestions made by an anonymous reviewer.

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