# Holomorphic gravity and its regularization of Locally Signed Coordinate
Invariance 111Essay written for the Gravity Research Foundation 2024 Awards
for Essays on Gravitation

Eduardo Guendelman<EMAIL_ADDRESS>Department of Physics, Ben-Gurion
University of the Negev, Beer-Sheva, Israel.
Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Strasse 1, 60438
Frankfurt am Main, Germany.
Bahamas Advanced Study Institute and Conferences, 4A Ocean Heights, Hill View
Circle, Stella Maris, Long Island, The Bahamas.

(29.01.2024)

###### Abstract

We expect the final theory of gravity to have more symmetries than we suspect
and our research points in this direction, to start with, standard general
coordinate invariance can be extended to complex holomorphic general
coordinate transformations This is possible by introducing a non Riemannian
Measure of integration (NRMI) and where we avoid the non holomorphic standard
$\sqrt{-g}$. measure of integration. Second, although globally signed
transformations produce a change of boundary conditions and like global parity
and global time reversal are not symmetries of nature, locally signed
coordinate transformations where the jacobian changes sign locally but the
jacobian approaches one asymptotically should be symmetries of nature, The
holomorphic extension can regularize the regions of space time where the
jacobian changes sign. Consequences for Quantum Gravity are discussed

##  I Introduction

The development of quantum theories of gravity has lead to the notion that the
quantum gravity theory produces a phenomenology that involves the breakdown of
symmetries, like Lorentz symmetry, etc. For a review see
Quantum:gravityphenomenology . We would like to explore the opposite
possibility, that a more complete theory of gravity should have more space
time symmetries than the ones considered so far.

As we will see. this can be done in the framework of the metric independent
non Riemannian measures has been used for the construction of modified gravity
theories Refs.TMT-orig-1 -TMT-orig-3  (see also Refs.TMT-recent-1-a -TMT-
recent-2 ), in some instances we have included the standard measure as well,
where the standard Riemannian integration measure might also contain a Weyl-
scale symmetry preserving $R^{2}$-term TMT-orig-3 . Some applications have
been, (i) $D=4$-dimensional models of gravity and matter fields containing the
new measure of integration appear to be promising candidates for resolution of
the dark energy and dark matter problems, the fifth force problem, and a
natural mechanism for spontaneous breakdown of global Weyl-scale symmetry TMT-
orig-1 -TMT-recent-2 , (ii) To study in Ref.susy-break  of modified
supergravity models with an alternative non-Riemannian volume form on the
space-time manifold, non singular emergent models leading to inflation and
then decaying into a dark enery and dark matter phase ourquintessence  \-
ourquintessencewithEDE ,(iv) Gravity-Assisted Emergent Higgs Mechanism in the
Post-Inflationary Epoch, Gravityassisted , (v) To study of reparametrization
invariant theories of extended objects (strings and branes) based on employing
of a modified non-Riemannian world-sheet/world-volume integration measure
mstring , nishino-rajpoot , leads to dynamically induced variable string/brane
tension and to string models of non-abelian confinement, interesting
consequences from the modified measures spectrum mstringspectrum , and
construction of new braneworld scenarios mstringbranes , . Modifed Measures
Theories have been discussed as effective theories for causal fermion theories
MMT .

We will see here how in this framework one can construct a General coordinate
invariant theory which has extended general coordinate transformations that
includes also transformations with positive and negative jacobian, including
locally alternating the sign in space time . This is possible by introducing a
non Riemannian Measure of integration, which transforms according to the
jacobian of the coordinate transformation, not the absolute value of the
jacobian of the coordinate transformation as it is the case with $\sqrt{-g}$.

It is very important to notice that the signed general coordinate invariance
is a particular case of the holomorphic general coordinate invariance in
complex space, which can be used to regularize locally signed general
coordinate invariance so the jacobian can go from positive to negative values
without going through zero, using the complex plane to achieve this.

##  II  General Relativity and other theories use a Riemannian volume element
that is not invariant under signed general coordinate transformations

The action of GR, and other theories that use the standard Riemannian volume
element $d^{4}x{\sqrt{-g}}$ is of the form,

$S=\int d^{4}x{\sqrt{-g}}L$ (1)

where $L$ is a generally coordinate invariant lagrangian. Now notice that
under a general coordinate transformation,

$d^{4}x\rightarrow Jd^{4}x$

, while

$\sqrt{-g}\rightarrow\mid J\mid^{-1}\sqrt{-g}$

where $J$ is the jacobian of the transformation and $\mid J\mid$ is the
absolute value of the transformation. Therefore
$d^{4}x{\sqrt{-g}}\rightarrow\frac{J}{\mid J\mid}d^{4}x{\sqrt{-g}}$, so
invariance is achieved only for $J=\mid J\mid$, that is if $J>0$, that is
signed general coordinate transformations are excluded.

One could argue that when taking the square root of the determinant of the
metric one may choose the negative solution when it suit us, but this would be
an arbitrary procedure if no specific rule is given to choose the positive or
the negative root. We choose instead to declare that $\sqrt{-g}$ is always
positive and replace it in the measure by something else whose sign is well
defined.

###  II.1 Invariance of the action with non invariant lagrangian density
(integrand) and compensating non invariant manifold of integration?

If conditions are optimal, the non invariance of the lagrangian density
(integrand) , which includes the measure, in a signed coordinate
transformation, could be compensated by the non invariance of the manifold of
integration. For example, in a time reversal transformation, the integrand
will change sign, but , if there no obstructions, and we can then change the
limits of integrations, the exchange of the limits of integrations will
involve an additional exchange of signs that can compensate for the sign
change in the integrand. Such transformations where the jacobian of the
transformation is negative all over space are for example a global time
reversal transformation or a global party transformation, which by the way are
not symmetries of nature, since they are broken by the weak interactions, see
for example a modern textbook like WEAKINTERACTION  for a review.

Also, if the manifold has boundaries in coordinate space time, introducing
coordinate transformations that change the boundaries involve compensating
terms at the boundaries, even when the transformations are infinitesimal, so
at this point the invariance of the action becomes complicated and
problematic, in particular in cosmology where the universe may have a
beginning in time.

A very important issue is to separate signed general coordinate transformation
which change boundary conditions that do not, since those transformations that
change boundary conditions, like time reversal or parity, everywhere in space
time do not appear to be symmetries of nature, they are broken by the weak
interactions , so they seems not so relevant. More interesting would be signed
general coordinate transformations in some regions of space time, which the
asymptotically at large times and space the jacobian becomes one, This
possibility we will discuss as the most relevant,

Such transformations that are locally signed are best formulated when non
Riemannian measures are introduced and for a resulting theory that allows a
holomorphic extension.

##  III Metric Independent Non-Riemannian Volume-Forms and Volume elements
invariant under locally signed general coordinate transformations

One can define a metric independent measure from a totally anti symmetric
tensor gauge field, for example

$\Phi(A)=\frac{1}{3!}\varepsilon^{\mu\nu\kappa\lambda}\partial_{\mu}A_{\nu\kappa\lambda}\quad,$
(2)

Then, under a general coordinate transformation

$\Phi(A)\rightarrow J^{-1}\Phi(A)$

. . Therefore $d^{4}x\Phi(A)\rightarrow d^{4}x\Phi(A)$, so invariance is
achieved regardless of the sign of $J$.

##  IV Theory using Metric Independent Non-Riemannian Volume-Forms

First we review our previous papers where we have considered the action of the
general form involving two independent non-metric integration measure
densities generalizing the model analyzed in quintess  is given by

$S=\int d^{4}x\,\Phi_{1}(A)\Bigl{[}R+L^{(1)}\Bigr{]}+\int
d^{4}x\,\Phi_{2}(B)\Bigl{[}L^{(2)}+\epsilon
R^{2}+\frac{\Phi(H)}{\sqrt{-g}}\Bigr{]}\;.$ (3)

Here the following definitions are used:

  * •

The quantities $\Phi_{1}(A)$ and $\Phi_{2}(B)$ are two densities and these are
independent non-metric volume-forms defined in terms of field-strengths of two
auxiliary 3-index antisymmetric tensor gauge fields

$\Phi_{1}(A)=\frac{1}{3!}\varepsilon^{\mu\nu\kappa\lambda}\partial_{\mu}A_{\nu\kappa\lambda}\quad,\quad\Phi_{2}(B)=\frac{1}{3!}\varepsilon^{\mu\nu\kappa\lambda}\partial_{\mu}B_{\nu\kappa\lambda}\;.$
(4)

The density $\Phi(H)$ denotes the dual field strength of a third auxiliary
3-index antisymmetric tensor

$\Phi(H)=\frac{1}{3!}\varepsilon^{\mu\nu\kappa\lambda}\partial_{\mu}H_{\nu\kappa\lambda}\;.$
(5)

  * •

The scalar curvature $R=g^{\mu\nu}R_{\mu\nu}(\Gamma)$ and the Ricci tensor
$R_{\mu\nu}(\Gamma)$ are defined in the first-order (Palatini) formalism, in
which the affine connection $\Gamma^{\mu}_{\nu\lambda}$ is a priori
independent of the metric $g_{\mu\nu}$.

  * •

The two different Lagrangians $L^{(1,2)}$ correspond to two matter field
Lagrangians

On the other hand, the variation of (3) w.r.t. auxiliary tensors
$A_{\mu\nu\lambda}$, $B_{\mu\nu\lambda}$ and $H_{\mu\nu\lambda}$ becomes

$\partial_{\mu}\Bigl{[}R+L^{(1)}\Bigr{]}=0\quad,\quad\partial_{\mu}\Bigl{[}L^{(2)}+\epsilon
R^{2}+\frac{\Phi(H)}{\sqrt{-g}}\Bigr{]}=0\quad,\quad\partial_{\mu}\Bigl{(}\frac{\Phi_{2}(B)}{\sqrt{-g}}\Bigr{)}=0\;,$
(6)

whose solutions are

$\frac{\Phi_{2}(B)}{\sqrt{-g}}\equiv\chi_{2}={\rm
const}\;\;,\;\;R+L^{(1)}=-M_{1}={\rm const}\;\;,\;\;L^{(2)}+\epsilon
R^{2}+\frac{\Phi(H)}{\sqrt{-g}}=-M_{2}={\rm const}\;.$ (7)

Here the parameters $M_{1}$ and $M_{2}$ are arbitrary dimensionful and the
quantity $\chi_{2}$ corresponds to an arbitrary dimensionless integration
constant.

The resulting theory is called a Two Measure Theory, due to the presence of
the Two measures $\Phi_{1}(A)$ and $\Phi_{2}(A)$. But for the purpose of this
paper this is two general, since we want to restrict to a theory that will
give us ordinary General Relativity, and we want to keep the general
coordinate invariance under signed general coordinate invariance.

For obtaining GR dynamics, we can restrict to one measure, so let us take

$\Phi_{1}(A)=\Phi_{2}(B)=\Omega$

we can also express $\Phi$ in terms of four scalar fields

$\Omega=\frac{1}{3!}\varepsilon^{\mu\nu\kappa\lambda}\varepsilon^{abcd}\partial_{\mu}\varphi_{a}\partial_{\nu}\varphi_{b}\partial_{\kappa}\varphi_{c}\partial_{\lambda}\varphi_{d}\quad$
(8)

(one has to point out that in the earlier formulations of modified measures
theories we used the 4 scalar field representation for the measure, see TMT-
orig-1  , ) The mapping of the four scalars to the coordinates $x^{\mu}$ may
be topologically non trivial. Finally, we have to correct the equation

$\frac{\Phi_{2}(B)}{\sqrt{-g}}\equiv\chi_{2}={\rm const}.$ (9)

for another equation the will be invariant under signed general coordinate
invariant transformations, which will be

$\frac{\Omega^{2}}{(-g)}\equiv\chi=K^{2}={\rm const}>0.$ (10)

without loss of generality we define $K$ to be positive. The resulting action
that replaces (4) is,

$S=\int d^{4}x\,\Omega\Bigl{[}R+L\Bigr{]}+\int
d^{4}x\,\Omega^{2}\Bigl{[}\frac{\Phi(H)}{{(-g)}}\Bigr{]}\;.$ (11)

the density $\Phi(H)$ remains defined eq. (5) so the integration obtained from
the variation of the $H$ gauge field is eq. (10) now. The solution of eq. (10)
are

$\frac{\Omega}{\sqrt{(-g)}}=\pm{K}.$ (12)

where the sign in (12) will be dynamically determined

Another possibility for a measure that would transform like the the jacobian
of the coordinate transformation, not the absolute value of the jacobian,
would be the determinant of the vierbein. This will destroy however (up to a
sign) the invariance of the theory under signed local Lorentz transformation
of the vierbeins. that is Lorentz transformations with negative determinants,
so, it is not a solution, rather we trade one asymmetry for another.

##  V Holomorphic general coordinate invariance invariance of the modified
measure theory and regularization of signed coordinate transformations

It is very useful to note that the action (11), when extended to complex space
time, a much larger coordinate invariance appears, larger than signed general
coordinate transformations, which is holomorphic general coordinate invariance
Holomorphic , the reason being that in the action (11) the expression
$\sqrt{-g}$, which is non holomorphic, does not appear anymore now, and this
is enough to extend the general coordinate invariance in the complexified
extension of the theory to holomorphic general coordinate invariance.

This allows us to regularize signed general coordinate transformations at some
points in space time where the jacobian would be zero if we have stayed in the
real domain, while returning to the identity at very large times or very large
values of the spatial coordinates.

Indeed, the problem with considering signed general coordinate
transformations, when restricting ourselves to a strictly real spacetime, is
that change from positive jacobian to negative jacobian and then say
considering the jacobian approach one at large values of the coordinates, is
that when restricting to strictly real values, the jacobian goes through zero
, or it can go from positive to negative values through infinity (approaching
plus infinity then going to minus infinity). In either case, zero jacobian or
infinite jacobian represent a singular transformations.

Using a holomorphic complex extension, we can regularize such sign changing
jacobian transformations, for example when just transforming one dimension,
which we call $x$, like in,

$x\rightarrow\bar{x}=\bar{x}(x)$

where,

$J=\frac{\partial\bar{x}}{\partial
x}=\tanh^{2}(Ax)+B(x+i\epsilon)exp(-\frac{x^{2}}{\Delta^{2}})$ (13)

Where $A$, $B$ , $\Delta$ and $\epsilon$ are real parameters and we take
$\epsilon$ very small, we can see that,

1\. going through the real $x$ axis. we do not encounter any singularity,

2\. As $x\rightarrow\pm{\infty}$, $J\rightarrow 1$. So asymptotically we go to
a trivial mapping, not affecting boundary conditions at infinity.

3\. In the region close to $x=0$ the real part of J changes sign and no
singularity in $J$ appears at $x=0$ due to the extension to complex space that
allows going from positive to negative values of the jacobian without
singularities, that is due to the parameter $\epsilon$. Since the
transformation (13) becomes trivial at asymptotic values, the boundary
conditions are not affected, and the manifold is invariant.

Boundary conditions may be real, at least in the classical theory. All kind of
features, like regularization of of symmetries, as we have seen above become
more transparent by extending to the complexified theory and in the quantum
theory, when tunneling and other effects are considered, the complex space
time becomes inevitable.

##  VI A note on proper and improper signed coordinate transformations

In gauge theories, gauge transformations that do not vanish fast enough at
infinity, are classified as improper gauge transformations, the
transformations described in the previous sections, where we demand the
jacobian goes to one at infinity is a proper one, like in (13).

In contrast a signed transformation all over space is necessarily an improper
one, since by definition the jacobian does not return to one at infinity.

The analysis for the invariance of the action for the case of a signed
transformation all over space involves changes in the limits of integration
when the coordinates $x$ are considered. There may not be a need to change the
limits of integration when the scalars used in the integration measure are
considered as we will see in the next section.

##  VII Only proper signed coordinate transformations have a chance to be
symmetries of nature

An important point to be made is that only proper signed coordinate
transformations like in LABEL:signed_holomorphic_transformation have a chance
to be symmetries of nature, That is, only transformations with Jacobian
approaching one at infinity.

This of course excludes global Time Reversal and global Parity
transformations, which are of course known to be violated by the weak
interactions.

But then of course the proper signed coordinate transformations need that the
jacobian change sign in order to be negative in some regions of space and to
approach one at infinity. But then the change of sign cannot go strictly in
the real space, because then the jacobian either goes through zero or through
infinity when the sign changes, so we invoke the holomorphic regularization,
like in (13) requiring part of the transition to take place in the complex
domain so as to avoid those singularities.

##  VIII The invariant scalar integration manifold and invariant lagrangian
density, another way to extend the space time manifold

Notice that using the volume element converts the the integration over
coordinates in the action into integration over scalar fields, since

$\Phi d^{4}x=d\varphi_{1}d\varphi_{2}d\varphi_{3}d\varphi_{4}$

The integration manifold existing in the four scalar field manifold is in fact
completely unaffected by any coordinate transformation taking place in the $x$
space. The lagrangian density is also a scalar not affected by any coordinate
transformation, the theory formulated in this way does not require any
boundary terms if the boundaries are for example formulated in the scalar
field space.

In the case of (11) for example,

$S=\int d\varphi_{1}d\varphi_{2}d\varphi_{3}d\varphi_{4}L$

, where

$L=\Bigl{[}R+L\Bigr{]}+\,\Omega\Bigl{[}\frac{\Phi(H)}{{(-g)}}\Bigr{]}\;$

We have seen that complexifying the space can be a useful way to extend the
space time manifold. Another way is to consider the measure scalars instead of
the original coordinates, since the mapping between these two spaces may not
be one to one.

In this respect, one issue that should be addressed is that of the gauge
fixing in the $\varphi_{a}$ space. Indeed, we notice that the only thing where
these fields appear in the equations of motion is $\Omega$, but this quantity
is invariant under volume preserving diffeomorphisms of the fields
$\varphi_{a}$, $\varphi^{\prime}_{a}=\varphi^{\prime}_{a}(\varphi_{a})$ which
satisfy

$\epsilon_{a_{1}a_{2}a_{3}a_{4}}\frac{\partial{\varphi^{\prime}}_{b_{1}}}{\partial\varphi_{a_{1}}}\frac{\partial{\varphi^{\prime}}_{b_{2}}}{\partial\varphi_{a_{2}}}\frac{\partial{\varphi^{\prime}}_{b_{3}}}{\partial\varphi_{a_{3}}}\frac{\partial{\varphi^{\prime}}_{b_{4}}}{\partial\varphi_{a_{4}}}=\epsilon_{b_{1}b_{2}b_{3}b_{4}}$
(14)

so the study of the best gauge for the $\varphi_{a}$ fields for further
comparison with the $x^{\mu}$ space could be a very important subject. Of
course when we say that the mapping between the $\varphi_{a}$ and the
$x^{\mu}$ spaces, we want to exclude multi valuedness due to volume preserving
diffeomorphisms of the fields $\varphi_{a}$, if for example different signs
for $\Omega$ are associated to the same point in $x^{\mu}$ space, it is clear
that there are at least two points in $\varphi_{a}$ space associated to one
point in $x^{\mu}$ space, and these two points in the $\varphi_{a}$ are not
related through a volume preserving diff.

##  IX Gravitational Equations of motion

We start by considering the equation that results from the variation of the
degrees of freedom that define the measure $\Omega$, that is the scalar fields
$\varphi_{a}$, these are,

$A^{\mu}_{a}\partial_{\mu}(R+L+2\Omega\frac{\Phi(H)}{{(-g)}})=0$ (15)

where

$A^{\mu
a}=\frac{1}{3!}\varepsilon^{\mu\nu\kappa\lambda}\varepsilon^{abcd}\partial_{\nu}\varphi_{b}\partial_{\kappa}\varphi_{c}\partial_{\lambda}\varphi_{d}\quad$
(16)

Notice that the determinant of $A^{\mu a}$ is proportional to $\Omega^{3}$, so
if the measure is not vanishing, the matrix $A^{\mu a}$ is non singular and
therefore $\partial_{\mu}(R+L+2\Omega\frac{\Phi(H)}{{(-g)}})=0$, so that,

$R+L+2\Omega\frac{\Phi(H)}{{(-g)}}=M=constant$ (17)

The variation with respect to the metric $g^{\mu\nu}$, we obtain.

$\Omega(R_{\mu\nu}+\frac{\partial L}{\partial
g^{\mu\nu}})+g_{\mu\nu}\Omega^{2}\frac{\Phi(H)}{{(-g)}}=0$ (18)

solving $\Omega\frac{\Phi(H)}{{(-g)}}$ from (17) and inserting into (18), we
obtain,

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\frac{1}{2}Mg_{\mu\nu}+\frac{\partial
L}{\partial g^{\mu\nu}}-\frac{1}{2}g_{\mu\nu}L=0$ (19)

which gives exactly the form of Einstein equation with the canonical energy
momentum defined from $L$

$T_{\mu\nu}=g_{\mu\nu}L-2\frac{{\partial{}}}{{\partial{g^{\mu\nu}}}}L\;.$ (20)

The equations of motion of the connection (in the first order formalism)
implies that the connection is the Levi Civita connection. L can describe a
scalar field with the potential and the term $\frac{1}{2}M$ can be interpreted
as a shift of the scalar field potential by a constant or a floating
contribution to the cosmological constant. In the modified measure approach to
this problem is best to consider the embedding space as the one defined by the
four scalar fields $\varphi_{a}$ that define the measure $\Omega$ (8).
Therefore the most fundamental space is the $\varphi_{a}$ , since only in this
space we can formulate the full description and solution of the problem.

##  X Discussion and Conclusions

In this essay we have seen that general coordinate invariance can be extended
to locally signed transformations, where the jacobian locally changes sign .
This is possible in models with modified measures of integration which are
independent of the metric and avoid the square root ofthe determinant of the
metric which is not holomorphic. The holomorphic nature of the theory allows
to change the jacobian of the transformation without any singularity, which is
avoided through a consistent holomorphic regularization, where a small
imaginary component is added to the jacobian when we go from positive to
negative values.

This small imaginary component in the jacobian is reminiscent of the small
imaginary component necessary to define the Feynman propagator so that
positive and negative waves propagate forward or backwards in time. Recall
also that a locally signed general coordinate transformation can indeed
involve a local change in the direction of time, that could affect our notions
of virtual pair creation which may become a pair annihilation process and vice
versa.

Other interesting aspect is the necessary co existence of positive and
negative measures in the theory, as expressed in eq. (12), so that according
to this we could expect positive and negative measures of integration in the
quantum gravity theory, which is a surprise, but recall that the appearance of
positive and negative energies in Relativistic Quantum Mechanics was also a
surprised, but Feynman provided us with a way to deal with this. In some sense
this issue has been considered to some extent, For example, Farhi et al
FARHIGUVEN , when considering the quantum nucleation of a baby universe, the
authors are forced to consider a larger space than the original coordinate
space, which would be the analog of our scalars which can serve as integration
variables as discussed in section VIII, as Farhi et al FARHIGUVEN  had also
have to consider that the additional spave covers multiple times the
coordinate space, and with each covering, a sign for the measure could be
positive or negative, so the calculations of Farhi et al FARHIGUVEN  can be
interpreted in the context of the formalism of section VIII.

Negative measures have been considered also by Linde in a non local model
Lindemultiplication

$S=\int
d^{4}xd^{4}y{\sqrt{-g(x)}}{\sqrt{-\bar{g}(y)}}(\frac{M_{P}^{2}}{16\pi}R(x)+L(\phi(x))-\frac{M_{P}^{2}}{16\pi}R(y)-L((\bar{\phi}(y))$
(21)

$R(x)$ is defined in terms of a $g_{\mu\nu}$ metric, while $R(y)$ is defined
in terms of a $\bar{g}_{\mu\nu}$ metric and where $L(x)$ and $L(y)$ have
exactly the same functional form with respect to their corresponding mirror
fields, like for a scalar field $\phi(x)$, there will be a potential
$V(\phi(x))$, same with kinetic terms, etc. that are appearing in $L(x)$ ,
while in $L(y)$ there will be corresponding field $\bar{\phi}(y)$ with a
potential $V(\bar{\phi}(y))$, then in $L(y)$ the metric $\bar{g}_{\mu\nu}$
appears instead of the metric $g_{\mu\nu}$, then the theory is obviously
invariant under $V\rightarrow V+constant$ Lindemultiplication . This non local
coexistence of a spacetime and an antispacetime was shown by Linde to have
remarkable properties concerning its behavior with respect to the cosmological
constant problem. We can think of this in the context of our framework as
having regions where the measure of integration can change sign, an effect
that must take place at the same time as we double the space time, to realize
Linde´s ideas. Spaces with negative measures in gravity or in theories of
extended objects we have called anti spaces, anti strings, anti branes, etc.
Pair production of strings anti strings (formulated with a non Riemannian
measure) can be considered stringsandantistrings .

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# Comparison of Shock-Boundary Layer Interactions in
Adiabatic and Isothermal Supersonic Turbine Cascades

Hugo F. S. Lui111Graduate Student, School of Mechanical Engineering, E-mail:
<EMAIL_ADDRESS>University of Campinas, UNICAMP, Campinas, SP, 13083-860,
Brazil  Tulio R. Ricciardi 222Postdoctoral Research Associate, National Center
for Supercomputing Applications, University of Illinois Urbana-Champaign,
E-mail<EMAIL_ADDRESS>and William R. Wolf333Associate Professor, School
of Mechanical Engineering, Department of Energy, E-mail<EMAIL_ADDRESS>University of Campinas, UNICAMP, Campinas, SP, 13083-860, Brazil  Carlos A. J.
B. Junior444Research Engineer, Laboratoire de Dynamique de Fluides (Dynfluid),
E-mail<EMAIL_ADDRESS>École Nationale Supérieure d’Arts et
Métiers, ENSAM, Paris, 75013, France

###### Abstract

Wall-resolved large eddy simulations are employed to investigate the shock-
boundary layer interactions (SBLIs) in a supersonic turbine cascade. An
analysis of the suction side separation bubbles forming due to the SBLIs is
presented for adiabatic and isothermal (cooled) walls. Flow snapshots indicate
that the separation bubble contracts and expands in a similar fashion for both
thermal boundary conditions. However, the skin-friction coefficient
distributions reveal a downstream displacement of the separation region when
cooling is applied. The separation bubble is also smaller for this setup
compared to the adiabatic one. A steeper pressure rise is observed for the
isothermal wall downstream of the incident oblique shock, and this occurs
because the incident shock wave gets closer to the blade surface when cooling
is applied. The Reynolds stresses are computed to investigate the effects of
wall temperature on the turbulence activity. While the levels of the
tangential stresses are similar for the cases analyzed, those for the wall-
normal component are higher for the cooled wall.

##  1 Introduction

Supersonic fluid machinery are applied in high-speed propulsion and power
generation systems due to their high power density [1]. In supersonic
turbines, inlet shock waves are formed and interact with the boundary layers
of neighboring blades. The shock-boundary layer interactions (SBLIs) can
increase the aerodynamic drag due to flow separation and induce higher heat
transfer rates to the blade surface. They can also be a source of flow
unsteadiness, where multiple frequencies are excited due to motion of the
incident and reflected shock waves, breathing of the separation bubble,
besides the incoming turbulent boundary layer. Typically, the shock wave
motion leads to strong pressure fluctuations that can compromise the system’s
structural integrity [2, 3, 4, 5].

Most studies of SBLIs have considered adiabatic wall conditions and, thus, the
effects of surface heat transfer are not fully explored. Schülein [6] used
non-intrusive techniques to perform heat transfer and skin-friction
measurements in the impingement of an incident oblique shock wave on a flat
plate with isothermal wall conditions. Their results show that within the
separation region, the heat flux increases in the streamwise direction, while
the skin-friction decreases. Jaunet et al. [7] investigated experimentally the
impact of the wall temperature on a shock-induced boundary layer separation.
They observed that the interaction length considerably increases when the wall
temperature is raised. Bernardini et al. [8] and Volpiani et al. [9] carried
out direct numerical simulations (DNS) to investigate the wall temperature
effects on the physics of SBLIs. Results revealed that wall cooling
significantly reduces the size of the separation bubble and interaction
scales, while the opposite behavior is noticed in the case of wall heating.

In the present work, a high-order overset compressible large eddy simulation
(LES) methodology is employed to investigate the flow in a supersonic turbine
cascade with two different wall thermal boundary conditions. These include an
adiabatic and a cooled walls, where the wall to inlet temperature ratio is set
as $T_{w}/T_{\infty}=0.75$. First, the numerical methodology is described
including the grid details and flow configurations. Spanwise and time averaged
pressure and skin-friction coefficients, as well as the mean flow fields, are
presented to assess the effect of cooling on the size and form of the
separation bubble. Then, flow snapshots are analyzed to investigate the
features of the separation bubbles and the shear layer dynamics at different
instants of the SBLI. Finally, the effects of the wall thermal boundary
conditions on the turbulence activity are analyzed by assessing the Reynolds
stress distributions.

##  2 Numerical Methodology

The present wall-resolved large eddy simulations solve the compressible
Navier-Stokes equations in a curvilinear coordinate system. The fluid is
assumed to be a calorically perfect gas, where the molecular viscosity $\mu$
is considered to depend on the local temperature through the nondimensional
Sutherland’s law. The spatial discretization of the governing equations is
performed using a sixth-order accurate finite-difference compact scheme [10]
implemented on a staggered grid. A sixth-order compact interpolation scheme is
also used to obtain flow quantities on the different nodes of the staggered
grid configuration.

Two grids are employed in the present simulations: one is a body-fitted O-grid
block which surrounds the airfoil and the other is an H-grid block used to
enforce the pitchwise periodicity of the cascade. In the O-grid, the time
integration of the equations is carried out by the implicit second-order
scheme of Beam and Warming [11]. This method overcomes the stiffness problem
arising from the wall-resolving boundary layer mesh. In the background H-grid
block, a third-order Runge-Kutta scheme is used for time advancement of the
governing equations. A fourth-order Hermite interpolation scheme [12, 13] is
used to exchange information between grid blocks in an overlapping zone.
Further details about the numerical procedure can be found in [13].

Due to the non-dissipative characteristics of the compact finite-difference
schemes, numerical instabilities may arise from mesh non-curvature,
interpolation between the overset grids, and boundary conditions. To preserve
stability of the numerical simulations, the high wavenumber compact filter
presented by Lele [14] is applied in flow regions far away from solid
boundaries at each time step. A shock capturing scheme is also employed to
capture the shock waves forming in the present flows. In order to introduce
minimal numerical dissipation in the vicinity of the shocks, without damping
the small scales of turbulence, the localized artificial diffusivity (LAD)
method [15] is employed to compute the artificial bulk viscosity and thermal
conductivity. The approach LAD-D2-0 proposed by Kawai et al. [16] is employed
here with no artificial shear viscosity. In order to transition the boundary
layers, we apply a body forcing on the RHS of the Navier-Stokes equations, as
described by Sansica [17]. Here, an unsteady actuation with a random spanwise
treatment is assumed and the amplitude of the disturbances are chosen
experimentally in order to guarantee a bypass transition with minimal flow
disturbance. More details of the numerical procedure can be found in [5].

##  3 Flow and Mesh Configurations

This section shows details of the flow configuration studied and describes the
computational grid used in the LES calculations. Figure 1 (a) presents the
geometrical parameters and flow conditions. The inlet Mach number is set as
$M$ = 2.0 and the Reynolds number based on the inlet velocity $U_{\infty}$ and
axial blade chord is $Re$ = 200,000. The ratio of specific heats is chosen as
$\gamma=1.31$, the Prandlt number is $Pr=0.747$ and the ratio of the
Sutherland constant over inlet temperature is set as $S_{\mu}/T_{\infty}$ =
0.07182. These conditions are chosen based on previous studies [18, 4, 5].

\begin{overpic}[trim=48.36967pt 28.45274pt 42.67912pt
28.45274pt,clip,width=429.28616pt]{flow_configuration.png} \put(0.0,20.0){(a)}
\put(32.0,20.0){(b)} \end{overpic} Figure 1: Schematics of (a) flow
configuration and geometrical parameters, and (b) computational domain
skipping every $5$ grid points.

Figure 1 (b) displays a schematic of the overset grid employed in the LES
along with the implemented boundary conditions. The O-grid block has
$1200\times 280\times 144$ points and is embedded in the background Cartesian
grid block of size $960\times 280\times 72$. Therefore, the grid has
approximately $68,000,000$ points. Depending on the case, adiabatic or
isothermal boundary conditions are applied along the blade surface. For the
latter, the wall to inlet temperature ratio is $T_{w}/T_{\infty}=0.75$,
representing a cooled wall. Supersonic inflow boundary conditions are used to
set the inlet conditions. For the outflow, a boundary condition based on the
Navier-Stokes characteristic boundary condition (NSCBC) [19] is employed. A
damping sponge is also applied near the inflow and outflow boundaries to
minimize reflections of disturbances [20, 10]. Periodic boundary conditions
are used in the $y$-direction of the background grid, according to Fig. 1 (a),
in order to simulate a linear cascade of blades and periodic boundary
conditions are also applied in the spanwise direction, to enforce a
statistically homogeneous flow along the span.

For the adiabatic wall case, the grid resolution in terms of wall units is
kept in the range given by $6<\Delta s^{+}<25$, $0.1<\Delta n^{+}<0.3$, and
$3<\Delta z^{+}<9$, where $s$, $n$ and $z$ represent the streamwise, wall-
normal and spanwise flow coordinates. For the isothermal wall simulation, the
near-wall grid spacing ranges from $15<\Delta s^{+}<60$, $0.2<\Delta
n^{+}<0.6$, and $6<\Delta z^{+}<19$. These numbers are computed for regions
where the boundary layers are fully developed and in equilibrium, away from
the tripping and recirculation regions. It is worthwhile to mention that the
same computational grid is used for both cases, but higher values in terms of
wall units are obtained for the isothermal wall case due to a inherent
reduction of the viscous length scales caused by cooling.

The simulation is initialized with a uniform flow and statistics are computed
after the initial transients are discarded. In the simulations, a variable
time step is computed based on an inviscid CFL parameter of 0.8. The body-
force tripping is applied at $0.22<x/c_{ax}<0.27$ for the suction side, and at
$0.10<x/c_{ax}<0.15$ for the pressure side. The wall normal height of the
body-foce region is $\delta=0.001c_{ax}$ and the actuation changes every
$\Delta t\approx$ 0.003 in a spanwise-random fashion.

##  4 Results

This section presents results obtained by the LES computed for adiabatic and
isothermal (cooled) wall boundary conditions. Flow quantities are collected
for 4 flow through times, based on the inlet velocity and blade axial chord.
Figure 2 shows iso-surfaces of $Q$-criterion colored by the $u$-velocity
component together with a background view of density gradient magnitude,
$|\nabla\rho|$. The top and bottom rows present results for the adiabatic and
cooled wall cases, respectively.

\begin{overpic}[trim=2.84526pt 2.84526pt 2.84526pt
2.84526pt,clip,width=208.13574pt]{cascade_nsnap1864_adiabatic.png}
\put(1.0,58.0){(a)} \end{overpic}

\begin{overpic}[trim=2.84526pt 2.84526pt 2.84526pt
2.84526pt,clip,width=208.13574pt]{SBLI_view_nsnap1864_adiabatic.png}
\put(1.0,58.0){(b)} \end{overpic}

\begin{overpic}[trim=2.84526pt 2.84526pt 2.84526pt
2.84526pt,clip,width=208.13574pt]{cascade_nsnap1664_isothermal.png}
\put(1.0,58.0){(c)} \end{overpic}

\begin{overpic}[trim=2.84526pt 2.84526pt 2.84526pt
2.84526pt,clip,width=208.13574pt]{SBLI_view_nsnap1664_isothermal.png}
\put(1.0,58.0){(d)} \end{overpic}

Figure 2: Iso-surfaces of $Q$-criterion colored by $u$-velocity component for
the adiabatic (top) and cooled (bottom) wall cases. The background plane
displays the shock waves by visualizing the density gradient magnitude
$|\nabla\rho|$.

In Figs. 2 (a) and (c), we can observe the complex shock structure across the
turbine passage. The detached oblique shock waves generated at the airfoil
leading edges interact with the boundary layers of the neighboring blades and
are reflected across the cascade. On the pressure side, the incident shock
wave becomes normal to the wall and, then, a Mach reflection is formed, while
an oblique shock reflection is generated on the suction side. To highlight the
effect of cooling on the SBLI, a detailed view of the flow field can be seen
in Figs. 2 (b) and (d), where one can observe differences between the lengths
of the separation bubbles, especially on the suction side. For the cooled
wall, a smaller recirculation region is noticed.

(a) Skin friction coefficient $c_{f}$

(b) Pressure coefficient $c_{p}$.

Figure 3: Mean skin-friction and pressure coefficient distributions for the
adiabatic (black) and cooled (blue) wall cases. The distributions are shown
only along the suction side.

\begin{overpic}[trim=113.81102pt 5.69054pt 113.81102pt
5.69054pt,clip,width=212.47617pt]{mean_uvelocity_adiabatic.png}
\put(0.0,34.0){(a)} \end{overpic}

\begin{overpic}[trim=113.81102pt 5.69054pt 113.81102pt
5.69054pt,clip,width=212.47617pt]{mean_uvelocity_isothermal.png}
\put(0.0,34.0){(b)} \end{overpic}

\begin{overpic}[trim=113.81102pt 5.69054pt 113.81102pt
5.69054pt,clip,width=212.47617pt]{mean_temperature_adiabatic_v1.png}
\put(0.0,34.0){(c)} \end{overpic}

\begin{overpic}[trim=113.81102pt 5.69054pt 113.81102pt
5.69054pt,clip,width=212.47617pt]{mean_temperature_isothermal_v1.png}
\put(0.0,34.0){(d)} \end{overpic}

Figure 4: Time-averaged contours of normalized $u$-velocity (top) and
temperature (bottom) for the adiabatic (left) and cooled (right) wall cases.
The black lines display the shock waves visualized by pressure gradient
magnitude. The black dashed lines show the sonic line.

The mean skin-friction coefficient distribution
$c_{f}=\frac{\tau_{w}}{0.5\rho_{\infty}U_{\infty}^{2}}$ is provided in Fig.
3(a) for the blade suction side. This plot shows the presence of a separation
bubble characterized by locations where $c_{f}<0$, which is delimited by a
horizontal dashed line. The effect of cooling on the size of the recirculation
region is evident. For the isothermal case, one can observe a downstream
displacement of the separation region compared to the adiabatic wall setup. On
the other hand, the reattachment locations are similar for both cases. Hence,
the cooled wall depicts a smaller separation bubble. For the adiabatic wall
case, the time-averaged characteristic length of the separation bubble is
$\langle L_{SB}\rangle=0.16c_{ax}$ and it is observed along
$0.70<x/c_{ax}<0.86$. For the cooled wall, $\langle L_{SB}\rangle=0.10c_{ax}$
and it is formed on $0.75<x/c_{ax}<0.85$. After a small negative skin-friction
coefficient plateau, a similar recovery is observed downstream of the
reattachment location for both cases.

Figure 3 (b) plots the mean pressure coefficient
$c_{p}=\frac{p-p_{\infty}}{0.5\rho_{\infty}U_{\infty}^{2}}$ along the airfoil
chord. For both adiabatic and cold wall cases, it is possible to note two
pressure rises: the first occurs near the separation point due to the
compression waves formed upstream of the separation bubble, and the second
takes place near the reattachment location as a result of the incident shock
impingement and the turbulence amplification mechanism [21]. For the cooled
wall setup, a steeper variation of $c_{p}$ is observed, especially for the
second pressure rise.

To highlight the influence of the wall thermal boundary conditions on the size
and shape of the separation bubbles, the mean (spanwise and time averaged)
$u$-velocity contours are presented in Figs. 4 (a) and (b), for the adiabatic
and isothermal cases, respectively. Here, the velocity component is normalized
by the inlet speed of sound. These figures reinforce the findings observed in
the friction coefficient distributions. The main effect of wall cooling is to
reduce the viscous length scales near the wall [8, 9] which in turn affects
the shock penetration, as shown in 4 (a) and (b). One can see that the
impinging shock penetrates deeper in the boundary layer for the cooled wall
case due to the displacement of the sonic line (displayed as a dashed line)
towards the wall. This effect is responsible for the steeper variation in the
pressure coefficient observed in Fig. 3(b). One can also see that, for the
cooled wall, the incident shock reaches further downstream compared to the
adiabatic case.

Figures 4 (c) and (d) show the mean temperature fields for the the adiabatic
and cold wall boundary conditions, respectively. The values are presented
normalized by the inlet temperature. For the former case, one can observe that
a region of maximum temperature occurs within the separation bubble. On the
other hand, when cooling is applied, higher temperature values are observed in
the free shear layer, downstream the bubble. For the adiabatic wall, friction
from the shear stresses near the wall and around the bubble are converted into
heat which is transferred along the boundary layer and inside the bubble. This
causes the near-wall flow to reach higher temperatures. However, heat from the
flow is transferred to the blade in the isothermal case, which has a lower
temperature than the surrounding flow. For the cooled wall case, the maximum
temperature values are observed along the free shear layer, behind the bubble,
due to strong shearing effects that cause aerodynamic heating.

Figure 5: Temporal variation of the suction side separation bubble length
$L_{SB}$ for the adiabatic (top) and cooled (bottom) wall cases.

\begin{overpic}[trim=113.81102pt 56.9055pt 113.81102pt
142.26378pt,clip,width=208.13574pt]{z_vorticity_nsnap3629_adiabatic.png}
\put(10.0,52.0){(a)} \end{overpic}

\begin{overpic}[trim=113.81102pt 56.9055pt
113.81102pt,clip,width=208.13574pt]{z_vorticity_nsnap4845_adiabatic.png}
\put(10.0,52.0){(b)} \end{overpic}

\begin{overpic}[trim=113.81102pt 56.9055pt 113.81102pt
142.26378pt,clip,width=208.13574pt]{z_vorticity_nsnap5431_isothermal.png}
\put(10.0,52.0){(c)} \end{overpic}

\begin{overpic}[trim=113.81102pt 56.9055pt 113.81102pt
142.26378pt,clip,width=208.13574pt]{z_vorticity_nsnap4412_isothermal.png}
\put(10.0,52.0){(d)} \end{overpic}

Figure 6: Spanwise $z$-vorticity contours at different time instants for the
adiabatic (top) and cooled (bottom) wall cases. The green line delimits the
bubble while the black line shows the incident shock wave.

The temporal evolution of the separation bubble length $L_{SB}$ is shown in
Fig. 5 for the adiabatic and isothermal walls. The instantaneous length of the
bubble is defined as the distance between the instantaneous reattachment and
separation locations. One can observe that the separation region undergoes a
contraction/expansion motion for both cases. The excursions from the mean
appear to be similar for both cases. A spectral analysis of this signal should
provide further information on the frequency scales related to the bubble
motion and such analysis should be conducted in future work, once longer
signals are collected for statistical convergence of the lower frequencies of
interest.

To highlight the 2D structure of the suction side separation bubble and shear
layer at different time instants, snapshots of $z$-vorticity are displayed in
Fig. 6 for both thermal boundary conditions. These snapshots correspond to the
instants indicated by the letters “a-d” in Fig. 5. The region enclosed by the
green line shows the separation region and the black lines display the
impinging shocks. In addition, the mean separation and reattachment positions
are indicated by the orange and cyan squares, respectively. For both cases,
when the bubble suffers a contraction, the instantaneous separation
(reattachment) point moves downstream (upstream) with respect to its mean
value, as can be visualized in Figs. 6(a) and (c). On the other hand, when the
bubble undergoes an expansion, one can observe the upstream (downstream)
movement of the instantaneous separation (reattachment) point with respect to
its mean position. This indicates that the bubble has a breathing pattern, but
its central position does not have large excursions from the mean. Figure 6
also shows that the shear layer downstream of the bubble is more diffused for
the adiabatic case, while more concentrated vorticity values are observed when
cooling is applied. These findings corroborate the maximum temperature values
observed in Figs. 4(c) and (d). For example, in the adiabatic case, the shear
layer around the bubble creates a zone of intense heating.

\begin{overpic}[trim=113.81102pt 5.69054pt 113.81102pt
5.69054pt,clip,width=143.09538pt]{ut_ut_adiabatic.png} \put(10.0,54.0){(a)}
\end{overpic}

\begin{overpic}[trim=113.81102pt 5.69054pt 113.81102pt
5.69054pt,clip,width=143.09538pt]{un_un_adiabatic.png} \put(10.0,54.0){(b)}
\end{overpic}

\begin{overpic}[trim=113.81102pt 5.69054pt 113.81102pt
5.69054pt,clip,width=143.09538pt]{TKE_adiabatic.png} \put(10.0,54.0){(c)}
\end{overpic}

\begin{overpic}[trim=113.81102pt 5.69054pt 113.81102pt
5.69054pt,clip,width=143.09538pt]{ut_ut_isothermal.png} \put(10.0,54.0){(d)}
\end{overpic}

\begin{overpic}[trim=113.81102pt 5.69054pt 113.81102pt
5.69054pt,clip,width=143.09538pt]{un_un_isothermal.png} \put(10.0,54.0){(e)}
\end{overpic}

\begin{overpic}[trim=113.81102pt 5.69054pt 113.81102pt
5.69054pt,clip,width=143.09538pt]{TKE_isothermal.png} \put(10.0,54.0){(f)}
\end{overpic}

Figure 7: Turbulence quantities on the suction side for the adiabatic (top)
and cooled (bottom) cases: $\langle u_{t}u_{t}\rangle$ (left), $\langle
u_{n}u_{n}\rangle$ (middle) and TKE (right).

The effects of the thermal boundary conditions on the turbulence properties,
are investigated by the tangential and wall-normal Reynolds stresses, $\langle
u_{t}u_{t}\rangle$ and $\langle u_{n}u_{n}\rangle$, respectively, and the
turbulent kinetic energy (TKE) are presented in Fig. 7. In this figure, the
top and bottom rows display results for the adiabatic and isothermal walls,
respectively. In Figs. 7 (a) and (d), it can be seen that the highest
fluctuations of $\langle u_{t}u_{t}\rangle$ are observed just upstream of the
shock-bubble interaction for both cases, with similar fluctuation values. The
amplification of $\langle u_{t}u_{t}\rangle$ is associated with the
development of the shear layer [21]. The peak values of $\langle
u_{n}u_{n}\rangle$ are found along the free shear layer downstream of the
bubble. The magnitude of $\langle u_{n}u_{n}\rangle$ decreases when cooling is
applied. In Figs. 7 (c) and (f), one can observe that the turbulent kinetic
energy combines the trends observed from the $\langle u_{t}u_{t}\rangle$ and
$\langle u_{n}u_{n}\rangle$ components. In addition, before the SBLI, we can
notice a downstream displacement of the maximum turbulence amplification
location for the cooled wall case. This occurs due to the higher shock
penetration discussed previously.

##  5 Conclusions

Wall-resolved large eddy simulations are employed to investigate thermal
effects in a supersonic turbine cascade. Simulations are performed for
adiabatic and isothermal boundary conditions, where in the latter case the
blade is cooled. For the present flow configurations, oblique shock waves are
generated at the leading edges of the airfoils, and they interact with the
boundary layers of the neighboring blades. A study of the shock-boundary layer
interactions is presented for the blade suction side, where an incident
oblique shock reflects on the wall leading to the formation of a separation
bubble.

The impact of the thermal boundary conditions on the separation bubbles is
investigated. The distributions of mean skin-friction show that the separation
bubble is considerably smaller for the cooled wall compared to the adiabatic
case. Pressure coefficient distributions show that a steeper pressure rise
occurs downstream the incident shock wave for the cooled wall. For this case,
cooling induces the formation of a thinner boundary layer and the sonic line
forms closer to the wall. Results in terms of mean velocity contours reveal
that the more pronounced pressure rise occurs due to the higher penetration of
the incident shock in the isothermal (cooled) wall. Maximum temperature values
are observed along the bubble, for the adiabatic case, and the free shear
layer, for the cooled wall. In the former case, aerodynamic heating is
transferred to the bubble due to a surrounding shear layer. For the latter
case, intense shearing is observed along the free shear layer, behind the
bubble, and leads to high temperatures.

An analysis of the instantaneous separation and reattachment locations
demonstrates that the separation bubbles have a breathing pattern of
contractions and expansions. For the contraction motions, the instantaneous
separation point moves downstream while the reattachment point moves upstream.
The other way around is observed for the expansion motions. The tangential
Reynolds stress distributions reach maximum values just upstream the shock-
bubble interactions, being similar for both the adiabatic and isothermal
walls. However, due to the higher shock penetration of the isothermal wall,
the peaks appear more downstream along the blade chord. The wall-normal
Reynolds stresses reach maximum amplitudes downstream the SBLI and they are
more pronounced for the adiabatic wall. In future work, further analysis of
the SBLI dynamics will be provided for both suction and pressure side boundary
layers.

## Acknowledgments

The authors acknowledge the financial support received from Fundação de Amparo
à Pesquisa do Estado de São Paulo, FAPESP, under grants No. 2013/08293-7,
2019/26196-5 and 2021/06448-0. The authors also thank Conselho Nacional de
Desenvolvimento Científico e Tecnológico, CNPq, for supporting this research
under grants No. 407842/2018-7 and 308017/2021-8. This work was granted access
to the HPC resources of IDRIS under the allocation 2021-A0112A12067 made by
GENCI.

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