# Gravity from the Determinant of the Energy-Momentum:
Astrophysical Implications

Hemza Azri1<EMAIL_ADDRESS>Salah Nasri1,2<EMAIL_ADDRESS><EMAIL_ADDRESS>1Department of Physics, United Arab Emirates University,
Al Ain 15551 Abu Dhabi, UAE  2International Center for Theoretical Physics,
Trieste, Italy

###### Abstract

Determinants of the second-rank tensors stand useful in forming generally
invariant terms as in the case of the volume element of the gravitational
actions. Here, we extend the action of the matter fields by an arbitrary
function $f(\bm{D})$ of the determinants of their energy-momentum, and the
metric, $\bm{D}=|\textbf{det}.T|/|\textbf{det}.g|$. We derive the
gravitational field equations and examine the nonlinear terms induced by the
determinant, specifically, the inverse of the energy-momentum tensor. We also
show that these extensions require a nonzero stress-energy tensor for the
vacuum. We propose a scale-free model, $f(\bm{D})=\lambda\bm{D}^{1/4}$, and
show how it induces the familiar invariant terms formed by the trace of the
energy-momentum tensor by expanding the action around the stress-energy of the
vacuum. We study the hydrostatic equilibrium equations for a neutron star by
providing relevant values of the dimensionless constant $\lambda$. We show
that the differences from the predictions of general relativity, in the mass-
radius relations, which are sensitive to the equations of state, are
conspicuous for $\lambda\sim-10^{-2}$. We also show that the model does not
affect the predictions on the primordial nucleosynthesis when it is applied to
the early radiation era. This novel and unfamiliar type of gravity-matter
coupling can lead to a rich phenomenology in gravitational physics.

##  I Introduction

Despite the great success of General Relativity (GR) as a theory of gravity,
it is still facing various challenges. Theoretical problems such as spacetime
singularities and the incomplete description of the theory itself at the
quantum level, as well as from the observational data which brings to the
forefront the problems of the dark sector, may be an indication to the need
for going beyond GR observations ; ishak ; review_on_mog .

There have been many attempts to modify GR in the last few years. First of
all, gravity, as presently understood, is identified by the curvature of
spacetime which in turn results from the distribution of matter fields via
their energy-momentum tensor $T_{\mu\nu}$. Therefore, there are ways to modify
GR either from the pure geometric (curvature) or from the matter (energy-
momentum) parts.

One line of thought seeks to modify the geometric sector solely and consider
scalar curvature invariant terms added to the Einstein-Hilbert action. The
resulting theories such as the famous $f(R)$ gravity have been extensively
studied and considered for a possible explanation of various cosmological
problems f(R) . The other line of thought is to modify the matter sector. In
this way, the extension largely follows from the energy-momentum tensor of
matter fields by adding a general function of its trace $T$ or its square
$T_{\mu\nu}T^{\mu\nu}$. Among these, are the familiar
$f(R,T,T_{\mu\nu}T^{\mu\nu},\ldots)$ constructed from both spacetime curvature
$R$ and the stress-energy momentum tensor f(T) .

The shared property of all these terms is that they are formed by contracting
spacetime indices, a standard procedure that allows the covariant character of
the theories to manifest both generally and locally. However, it is known that
the determinant of the spacetime metric tensor is also crucial for the overall
actions to be generally invariant. It turns out that not only the metric but
any tensor of second rank can be considered. Thereby, the determinant of rank-
two tensors are supported by the principle of general invariance, and
therefore are allowed in the generalized gravitational actions unless
forbidden by a specific symmetry.

For instance, the square-root of the determinant of the Ricci tensor (usually
referred to Eddington action) leads to the same field equations which can be
derived from the standard Einstein-Hilbert action. Interestingly, the Ricci-
determinant can be combined with the metric-determinant to finally form a
scalar that is also relevant to gravity ricci_determinant ; padbanabham .
Motivated by these determinant-based constructions, it is worth considering
the matter sector as well by establishing a new type of matter-geometry
interaction through its stress-energy momentum tensor.

In this paper, we will consider the determinant of the energy-momentum tensor
$|\textbf{det}.T|$ for the possible extension of the matter sector. The common
practice is to build the volume element from the metric density,
$|\textbf{det}.g|^{1/2}$ as in GR, and therefore gravity from the determinant
of the energy momentum will be based on a general function $f(\bm{D})$ of the
scalar $\bm{D}=|\textbf{det}.T|/|\textbf{det}.g|$.

Besides the fundamental difference from the usual extensions, the presence of
the determinant will induce the inverse of the energy-momentum of matter
fields in the equations of motion. In the cosmological context where the
sources are approximated by perfect fluids, this contribution will in turn
induce the inverse of the equation of state of the species det_cosmology .
Generally speaking, the inverse of the energy-momentum tensor is defined only
when the determinant does not vanish everywhere, and therefore one has to
consider the nonzero vacuum energy (the cosmological constant) in the absence
of matter fields. Thus, at large scales, the solutions of these models are not
completely flat but (anti-) de Sitter spacetimes.

After setting up the general framework, we propose a specific model in which
the coupling constant is dimensionless, i.e.
$f(\bm{D})=\lambda\,\bm{D}^{1/4}$. We adapt the resulting field equation for a
static spherically symmetric background and study the corresponding stellar
structure. In order to constrain our coupling constant, we apply the model to
the famed mass-radius relation of a neutron star. The latter has been a viable
accommodation for testing gravity models at the strong field regime. However,
since the equation of state is not rigorously constrained, we will study our
model through various, mainly four different equations of state. We will solve
the stellar structure equations numerically, and based on the resulting mass-
radius relations we show how the predictions of our model become
distinguishable from those of GR. This obviously depends on the strength of
matter-gravity coupling induced by the determinant of the stress energy.
Guided by the maximum measured mass and radius of the neutron star, we deduce
the relevant constraints on the free parameter of the model. We will also
highlight the possible effects on primordial nucleosynthesis.

The rest of the article is organized as follows: in section II.1 we set up the
general formalism starting from the action principle that includes the
determinant of the energy-momentum. In section II.3, we propose the scale-
independent model and apply it to the perfect fluid sources. We then study the
stellar structure in III.2 and conclude in section IV.

##  II Gravity from the determinant of the energy-momentum

###  II.1 Gravitational action and the field equations

In this section, we will introduce our total gravitational action which
includes the standard GR action (Einstein-Hilbert) plus matter and extend it
with the determinant of the energy-momentum. We will derive the associated
field equations by varying the action with respect to the main field (the
metric tensor.) This can also be realized in the Palatini formalism if the GR
action is written in terms of both metric and a symmetric connection as
independent variables. In this paper, we will be interested only in the
former. The invariant action reads

$\displaystyle S=$ $\displaystyle\int
d^{4}x\sqrt{|\textbf{det}.g|}\,\left\\{\frac{1}{2\kappa}\left(R-2\Lambda_{0}\right)+L^{\text{m}}[g]\right\\}$
(1) $\displaystyle+\int d^{4}x\sqrt{|\textbf{det}.g|}\,f(\bm{D}),$

where $\kappa=8\pi G$, with $G$ being Newton’s constant. In the first line of
this action we have the Ricci scalar $R$, the cosmological constant
$\Lambda_{0}$, and the Lagrangian density of matter fields $L^{\text{m}}[g]$.
The latter describes the minimal coupling of matter to gravity, i.e. through
only the metric. The novel quantity in the above action is $f(\bm{D})$ which
is a general function of the scalar

$\displaystyle\bm{D}\equiv\frac{|\textbf{det}.T|}{|\textbf{det}.g|}~{}.$ (2)

where $\textbf{det}.T$ is the determinant of the energy-momentum tensor
$T_{\alpha\beta}$. This is given by its definition

$\displaystyle\textbf{det}.T=\frac{1}{4!}\epsilon^{\alpha\beta\gamma\rho}\epsilon^{\bar{\alpha}\bar{\beta}\bar{\gamma}\bar{\rho}}T_{\alpha\bar{\alpha}}T_{\beta\bar{\beta}}T_{\gamma\bar{\gamma}}T_{\rho\bar{\rho}},$
(3)

where $\epsilon^{\alpha\beta\gamma\rho}$ is the anti-symmetric Levi-Civita
(permutation) symbol. Like the determinant of any second rank tensor, the
quantity (3) transforms identically to $\textbf{det}.g$, thus, the quantity
$\bm{D}$ transforms like a scalar and then the total action is invariant under
general coordinate transformation.

In varying the action, one has to take into account that the energy-momentum
depends on the metric, hence, it is worth giving the expression for the
variation of its determinant. This reads

$\displaystyle\delta_{g}\textbf{det}.T=\textbf{det}.T\left(T^{\text{inv}}\right)^{\mu\nu}\delta_{g}T_{\mu\nu},$
(4)

where $\left(T^{\text{inv}}\right)^{\mu\nu}$ is the inverse of $T_{\mu\nu}$
such that
$\left(T^{\text{inv}}\right)^{\nu\alpha}T_{\mu\alpha}=\delta^{\nu}_{\,\mu}$,
and it is given by its definition as

$\displaystyle\left(T^{\text{inv}}\right)^{\mu\nu}=\frac{1}{3!}\frac{1}{\textbf{det}.T}\epsilon^{\mu\alpha\beta\gamma}\epsilon^{\nu\bar{\alpha}\bar{\beta}\bar{\gamma}}T_{\alpha\bar{\alpha}}T_{\beta\bar{\beta}}T_{\gamma\bar{\gamma}}.$
(5)

Therefore, by varying the total action (1), the gravitational field equations
are obtained as

$\displaystyle G_{\mu\nu}=-\Lambda_{0}g_{\mu\nu}+\kappa T_{\mu\nu}+\kappa
f(\bm{D})g_{\mu\nu}+2\kappa\bm{D}f^{\prime}(\bm{D})\mathcal{T}_{\mu\nu}$

where $G_{\mu\nu}$ is the Einstein tensor,
$T_{\mu\nu}=L^{\text{m}}g_{\mu\nu}-2\delta L^{\text{m}}/\delta g^{\mu\nu}$ is
the energy-momentum tensor of matter, and $f^{\prime}(\bm{D})=df/d\bm{D}$. The
last term is given by

$\displaystyle\mathcal{T}_{\mu\nu}=$ $\displaystyle-
g_{\mu\nu}+L^{\text{m}}\left(T^{\text{inv}}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T^{\text{inv}}\right)+\frac{1}{2}T^{\text{inv}}T_{\mu\nu}$
(7)
$\displaystyle+2(T^{\text{inv}})^{\alpha\beta}\frac{\delta^{2}L^{\text{m}}}{\delta
g^{\alpha\beta}\delta g^{\mu\nu}}.$

where $T^{\text{inv}}$ is the trace of $(T^{\text{inv}})^{\mu\nu}$ (not to be
confused with the inverse of the trace of $T_{\mu\nu}$), and
$T^{\text{inv}}_{\mu\nu}=g_{\alpha\mu}g_{\beta\nu}\left(T^{\text{inv}}\right)^{\alpha\beta}$.

Therefore, invoking the determinant of the stress-energy tensor in the
gravitational action modifies the Einstein’s field equations as follows:

  * •

First, the geometrical part (curvature terms) is not altered since matter
field are assumed to be coupled only to the metric. Then, the Bianchi
identity, $\nabla^{\mu}G_{\mu\nu}=0$, can be easily applied. However, this
will certainly lead to an extended energy-momentum conservation equation.

  * •

An important contribution to the modified matter sector originates from the
inverse of the energy-momentum tensor. Hence, the determinant of the energy-
momentum tensor should not vanish. This is guaranteed if one considers the
non-vanishing cosmological constant which is highly supported by observations.
Indeed, the total energy-momentum tensor contains, in addition to matter
fields’ sources, the stress-energy tensor of the vacuum.

$\displaystyle
T_{\mu\nu}^{(\text{tot})}=T_{\mu\nu}^{\text{(vac)}}+T_{\mu\nu}^{(\text{i})},$
(8)

where $T_{\mu\nu}^{(\text{i})}$ is the energy-momentum tensor of all fluid
types (matter and radiation) in the universe. The tensor
$T_{\mu\nu}^{\text{(vac)}}=\mathcal{E}g_{\mu\nu}$ is the stress-energy of the
vacuum where $\mathcal{E}$ being the vacuum energy density. The latter
contains the bare cosmological constant $\Lambda_{0}$ and all non-vanishing
contributions from the ground states of the quantum fields as well as from
phase transitions (see Refs ccp  for detailed reviews). Cosmological
observations imply a value around $10^{-10}\,\text{erg}/\text{cm}^{3}$ but not
zero for the vacuum energy density. Thus, even in the absence of matter (and
radiation) sources, i.e., when $T_{\mu\nu}^{(\text{i})}=0$, the energy-
momentum tensor and its determinant do not vanish thanks to the cosmological
constant term (the vacuum energy).

###  II.2 The case of perfect fluids

Before proposing any particular model $f(\bm{D})$, one can calculate and
examine the quantity $\bm{D}$ in terms of all sources of gravity at hand. If
the latter are described by perfect fluids then one writes

$\displaystyle
T_{\mu\nu}=\sum_{a}(\rho_{a}+P_{a})u_{\mu}u_{\nu}+\sum_{a}P_{a}g_{\mu\nu},$
(9)

which contains all the content of the universe, namely, nonrelativistic matter
and radiation as well as the vacuum (cosmological constant).

With all the contributions, one finds

$\displaystyle\bm{D}=\left|\left(\sum_{a}\rho_{a}\right)\left(\sum_{a}P_{a}\right)^{3}\right|.$
(10)

We notice here that unlike the case of models extended by functions of the
trace or the square of the energy-momentum, the contribution of the pressure
controls this new quantity at the level of the action. However, it is worth
noticing again that the inverse of both pressure and energy density will come
out due to the emergence of the inverse of the energy-momentum after the
variation of the action. That being said, we should emphasize again that, in
general, the total energy-momentum tensor involves a nonzero vacuum energy. In
fact, the last expression of the determinant can be written for matter and
radiation ($a=\text{i}$), and the vacuum ($a=\text{vac}$) as

$\displaystyle\bm{D}=\left|\left(\sum_{\text{i}}\rho_{\text{i}}+\mathcal{E}\right)\left(\sum_{\text{i}}P_{\text{i}}-\mathcal{E}\right)^{3}\right|.$
(11)

Hence, for matter (or radiation) such as the case of the interior of massive
objects, the cosmological constant is negligible, whilst in the vacuum (where
$\rho_{\text{i}}$ and $P_{\text{i}}$ vanish), the cosmological constant
becomes important, and in this case $\bm{D}=\mathcal{E}^{4}$ is nonzero.

In the subsequent section we will propose a scale free-model; the first model
of the energy-momentum-determinant gravity that we consider in practice.

###  II.3 Scale-free model

As in extended gravity theories, one may think of $\bm{D}^{n}$ models where
$n$ does not need to be an integer. In contrast, dimensional analysis favors
an inverse of an integer. However, the high dimension of the determinant
requires a high dimensional constant for the action to be dimensionless. With
a mass scale $M$, this requires a general model of the form

$\displaystyle f(\bm{D})=M^{4(1-4n)}\bm{D}^{n}.$ (12)

Powers of $\bm{D}$ are not the only interesting models one may consider. In
fact, from the field equations (II.1) we notice the presence of the term
$\bm{D}f^{\prime}(\bm{D})$ which can be absorbed by a logarithmic model, and
leads to a simple dynamics.

Nevertheless, as a first application of our setup based on the energy-momentum
determinant, we will consider the scale-independent model, $n=1/4$ and
$M^{0}\equiv\lambda$ (a dimensionless constant) which takes the form

$\displaystyle f(\bm{D})=\lambda\bm{D}^{1/4}.$ (13)

Based on the generic case we studied above, the model (13) is worth studying
because it is the only resulting case that does not require a higher (mass)
dimensional constant in the gravitational action (1). Additionally, with the
non-vanishing stress-energy of the vacuum (see the above discussion), this
particular model can lead to interesting implications. In fact, the
determinant structure (13) can induce the familiar invariant terms formed by
the trace of the energy-momentum tensor as an approximation. This can be
realized if we expand the energy-momentum tensor around the stress-energy of
the vacuum as

$\displaystyle T_{\mu\nu}\rightarrow\mathcal{E}g_{\mu\nu}+T_{\mu\nu},$ (14)

where $T_{\mu\nu}$ is a small perturbation.

This leads to

$\displaystyle\textbf{det}.T\simeq\mathcal{E}^{4}\times\textbf{det}.g\times\Bigg{\\{}1$
$\displaystyle+\frac{T_{\,\,\,\nu}^{\nu}}{\mathcal{E}}+\left(\frac{T_{\,\,\,\nu}^{\nu}}{2\mathcal{E}}\right)^{2}-\frac{T_{\,\,\,\beta}^{\nu}T_{\,\,\,\nu}^{\beta}}{2\mathcal{E}^{2}}$
(15)
$\displaystyle+\mathcal{O}\left(\frac{T_{\,\,\,\nu}^{\nu}}{\mathcal{E}}\right)^{3}\Bigg{\\}},$

and finally

$\displaystyle\bm{D}^{1/4}\simeq\mathcal{E}$
$\displaystyle+\frac{1}{4}\,T+\left(\frac{1}{16\mathcal{E}}\right)\,T^{2}-\left(\frac{1}{8\mathcal{E}}\right)\,T_{\mu\nu}T^{\mu\nu}$
(16) $\displaystyle+\mathcal{O}(T/\mathcal{E})^{3}.$

The first term in this expansion is the vacuum energy with a theoretical
estimation of $\mathcal{E}\sim\Lambda_{\text{UV}}^{4}$, where
$\Lambda_{\text{UV}}$ being an Ultra-Violet cutoff, defined as the energy
scale up to which one trusts quantum field theory. At the level of the action
(1), this term would simply contribute to the cosmological constant. The terms
proportional to the trace of the energy-momentum tensor have been considered
separately in the literature as possible models to be applied to cosmology and
astrophysics, such as the $f(R,T)$ and energy-momentum squared gravity f(T) .
These models seem to arise from the determinant structure we proposed here.

Next, we will explore the astrophysical implication of this model where we
will focus on the stellar structure for neutron stars. We will give a thorough
phenomenological study and examine the new effects by putting relevant
constraints on our free parameter.

##  III Astrophysical implication

In this section we restrict the study to the interior of a neutron star where
the cosmological term is ignored compared to matter. To simplify the
calculation, we simply use $\rho_{\text{i}}=\rho$ and $P_{\text{i}}=P$ for the
single fluid describing the star. Then, from (11), one writes $\bm{D}=\rho
P^{3}$ which does not vanish inside the star.

In this case, the gravitational field equations (II.1) read (taking
$\kappa=1$)

$\displaystyle G_{\mu\nu}=$
$\displaystyle\left(\rho+P\right)u_{\mu}u_{\nu}+Pg_{\mu\nu}$ (17)
$\displaystyle+\lambda\bm{D}^{1/4}\left\\{g_{\mu\nu}+\left(1+\frac{P}{4\rho}+\frac{3\rho}{4P}\right)u_{\mu}u_{\nu}\right\\}$

However, as stated above, outside the star the energy-momentum tensor will be
described in terms of the cosmological constant. In this vacuum case, the
Lagrangian $L^{\text{vac}}=P^{\text{vac}}=-\mathcal{E}$ and the energy-
momentum tensor $T^{\text{vac}}_{\mu\nu}=\mathcal{E}g_{\mu\nu}$. Hence, one
can easily show that the gravitational action (1) leads to the field equations
in vacuum

$\displaystyle G_{\mu\nu}=-\Lambda_{\text{eff}}\,g_{\mu\nu},$ (18)

where

$\displaystyle\Lambda_{\text{eff}}=\Lambda_{0}+\kappa(1-\lambda)\mathcal{E}$
(19)

is simply a nonzero effective cosmological term.

If the contribution to the vacuum energy comes only from the bare cosmological
constant $\Lambda_{0}$, i.e., when $\mathcal{E}=\Lambda_{0}/\kappa$, one would
simply have $\Lambda_{\text{eff}}=(1-\lambda)\Lambda_{0}$. Here, the factor of
two in $\Lambda_{0}$ that would arise from the second term in (19) is ignored
because in this case the (similar) contributions from the Lagrangian
$L^{\text{vac}}$ and the term $\Lambda_{0}$ in the action describe the same
source.

The nonzero effective cosmological constant (19) implies (anti-) de Sitter
spacetime solution.

###  III.1 Weak-field limit

The weak field limit (or the Newtonian regime) of the previous equations will
be derived now by considering small perturbations about all the quantities. We
expand the energy density and pressure around their background values
$\bar{\rho}$ and $\bar{P}$ as

$\displaystyle\rho\rightarrow\bar{\rho}+\rho,\quad P\rightarrow\bar{P}+P,$
(20)

and apply that for the perturbed spacetime metric in which the time-time
component is given by the Newtonian potential $|\Phi|\ll 1$ as
$g_{00}\simeq-1-2\Phi$.

In the weak field limit, one considers tiny stresses $T_{ij}$ (or pressure)
compared to the energy density $T_{00}$, i.e. $|T_{ij}|/T_{00}\ll 1$.
Therefore, we will neglect all the terms proportional to $P/\rho$ for both the
background and first order terms. To that end, at first order, the time-time
component of the gravitational equations (17) reads

$\displaystyle\nabla^{2}\Phi=4\pi
G_{N}\left(1+\frac{3\lambda}{4}\left(\frac{\bar{\rho}}{\bar{P}}\right)^{1/4}\right)\rho$
(21)

which describes the deviation from the standard Poisson equation when
$\lambda\neq 0$.

It is worth noting here that, unlike most of the familiar modified theories,
this deviation is proportional to the inverse of the pressure which results
from the inverse of the stress-energy tensor. This contribution is significant
in the case of small pressure (compared to the energy density) as in the case
of the Solar system which we consider here. The quantity $\bar{\rho}/\bar{P}$
in the second term can be constrained from the strength of the Newtonian
gravitational potential of the Solar system which is nowhere larger than
$10^{-5}$ will . In fact, the pressure to which the system is subjected is
comparable to $\rho|\Phi|$, hence, one has

$\displaystyle\frac{\bar{P}}{\bar{\rho}}\sim|\Phi|\sim 10^{-5},$ (22)

which leads to the modified Poisson equation

$\displaystyle\nabla^{2}\Phi\simeq 4\pi G_{N}\left(1+0.13\times
10^{2}\lambda\right)\rho.$ (23)

In the following section we study the stellar structure equations for a
neutron star based on the energy-momentum determinant gravity model.

###  III.2 Stellar structure equations for a neutron star

In what follows we apply our field equations (17) to a static spherically
symmetric background

$ds^{2}=-{\rm e}^{2\Phi(r)}\,dt^{2}+{\rm
e}^{2\Psi(r)}\,dr^{2}+r^{2}\,d\theta^{2}+r^{2}\sin^{2}\theta\,d\phi^{2}$ (24)

where $\Phi(r)$ and $\Psi(r)$ are functions of the radial coordinate.

The following main equations can be easily derived by the same method used in
GR mtw . The main differences are in the right hand side of (17) which
includes the nonlinear terms in the energy density and pressure. Therefore, we
find it unnecessary to put here all the details of the derivation.

Using the field equations (17), one finds the equation for the potential
$\Phi(r)$ as

$\displaystyle\frac{d\Phi}{dr}=\frac{m+4\pi
r^{3}P}{r(r-2m)}+\lambda\bm{D}^{1/4}\left(\frac{4\pi r^{3}}{r(r-2m)}\right),$
(25)

where we have introduced the mass $m(r)$ of the sphere of radius $r$.

As we have seen previously, outside the star (at large $r$), the spacetime
approaches not the flat but (anti) de Sitter solution due to the presence of
the cosmological constant. Hence, the exterior vacuum solution implies

$\displaystyle{\rm
e}^{2\Psi(r)}=\left(1-\frac{2M}{r}+\frac{\Lambda_{\text{eff}}}{3}r^{2}\right)^{-1}.$
(26)

Needless to say, although the term $\Lambda_{\text{eff}}$ is required in our
theories in order to prevent the inverse of the energy-momentum from going
singular, its effect however is purely cosmological, thus it must not affect
the stellar structure.

On the other hand, the radial component of (17) gives the equation for the
mass as

$\frac{dm}{dr}=4\pi
r^{2}\rho\left\\{1+\frac{\lambda}{4}\bm{D}^{1/4}\left(\frac{P}{\rho^{2}}+\frac{1}{P}\right)\right\\}.$
(27)

The extended TOV equation is derived by applying the Bianchi identity on (17).
This in turn leads to a generalized continuity equation from which one gets
the equation for the pressure

$\displaystyle\frac{dP}{dr}=-$ $\displaystyle\left\\{\frac{m+4\pi
r^{3}P}{r(r-2m)}+\lambda\bm{D}^{1/4}\frac{4\pi
r^{3}}{r(r-2m)}\right\\}(\rho+P)$ (28)
$\displaystyle\times\left\\{1+\frac{\lambda}{4}\bm{D}^{1/4}\left(\frac{1}{\rho}+\frac{3}{P}\right)\right\\}$
$\displaystyle\times\left\\{1+\frac{\lambda}{4}\bm{D}^{1/4}\left(\frac{1}{c_{s}^{2}\rho}+\frac{3}{P}\right)\right\\}^{-1},$

where $c_{s}^{2}=dP/d\rho$ is the speed of the sound wave.

The set of equations (27)-(28) describes the stellar structure where their GR-
limit is clearly understood for $\lambda=0$. We notice that the new
contributions, induced by the determinant of the stress-energy of matter, are
nonlinear in the energy density and pressure. Thus, analytical expressions of
these quantities cannot be found easily. This necessitates a numerical
solution which we present below.

###  III.3 Constraints from neutron stars

In this section we will put a constraint on the coupling constant $\lambda$ by
studying the effects of the scale-free model (13) on the mass-radius relation
of the neutron stars. The latter have been used extensively in constraining
various models of gravity in strong field regimes ns_constraints .

Figure 1: Mass vs. radius of neutron stars from model (13). Each panel
corresponds to a different equation of state. The gray shaded region
corresponds to $R<2GM/c^{2}$ (Black hole). The maximum measured mass
$M=2.14_{-0.09}^{+0.10}$ for a neutron star max_mr  is shown by the horizontal
line. The measured radius $R=12.39_{-0.98}^{1.30}\,{\rm km}$ max_mr  is
illustrated by the cyan colored shaded region. Departure from GR is clear for
values around $\lambda\sim-10^{-2}$ for all the cases. However, the model is
compatible with the maximum measured mass and radius in the case of MPA1.

Here, we briefly describe the numerical method used to solve the hydrostatic
equilibrium equations (27)-(28). First, one has to close the system of
equations by providing an equation of state relating pressure to the energy
density, $P=P(\rho)$. In the case of neutron stars, the equations of state of
the dense matter is not accurately constrained by the nucleon scattering
experiments. Nevertheless, with various assumptions on the compositions and
the nucleon-nucleon interactions, several equations of state are provided eos
. In our case, we will solve the main equations for four different equations
of state, namely, AP4 AP4 , SLY4 SLY , MPA1 MPA1 , and MS1 MS1 , by inspecting
the effects of the novel contributions for each case.

For the numerical solution, we use the second order Runge-Kutta method
(midpoint method). We also use the adaptive radial step-sizes BPS

$\Delta
r=0.01\left(\frac{1}{m}\frac{dm}{dr}-\frac{1}{P}\frac{dP}{dr}\right)^{-1},$
(29)

which is adapted via the local mass and pressure gradients.

This technique permits for acceptable radial resolution at the regions of high
pressure gradients. Furthermore, we keep the steps up to $10^{3}\,{\rm cm}$,
but not larger. First, we choose a central pressure $P_{\rm c}$ (at the origin
$r=0$) and $m(0)=0$ as a boundary condition for our system. We then integrate
outwards up to the surface of the star where the pressure takes a very small
value. We take for the surface pressure $P\sim
10^{10}\,\text{dyne}/\text{cm}^{2}$ which can be considered negligible
compared to the pressure at the center, but yet is non-vanishing.

This point corresponds to the radius of the star $R$ and its total mass $M$.
Last but not least, we obtain both mass and radius by varying the central
pressure from $3\times 10^{33}$ to $9\times 10^{36}\,{\rm dyne\,cm^{-2}}$.
Since our equations include a free parameter, we repeat the above process for
various values of $\lambda$, and the above equations of state.

An important condition on $\lambda$ can be found from the requirement that the
mass $m(r)$ should increase while integrating outwards starting from the
center. Therefore, for general energy density and pressure, $dm/dr>0$ implies

$\displaystyle\left\\{1-\frac{\rho+P}{2\rho}\left(1+\frac{3\rho-P}{2P}\right)\right\\}\left(\frac{P}{\rho}\right)^{\frac{3}{4}}\lambda<1.$
(30)

At the center of a typical neutron star one has $P/\rho\sim 0.2$ which leads
to $\lambda>-0.8$. Therefore, the constraints on $\lambda$ from the
measurements of the mass-radius relation (see below) must not fall below this
negative value.

In Figure 1, we have depicted the mass of the neutron star versus its radius
for both GR and the model (13) for different values of $\lambda$ and for four
different equations of state. In all four cases, we notice that the departure
from GR becomes conspicuous starting from values $\lambda\lesssim-10^{-2}$.
However, this deviation from GR falls within the region of the maximum
measured mass and radius of a neutron star PSR J0740+6620,
$M=2.14_{-0.09}^{+0.10}$ and $R=12.39_{-0.98}^{1.30}\,{\rm km}$ max_mr , for
the case MPA1.

###  III.4 Effects on the big bang nucleosynthesis

One of the most important predictions of the standard hot big bang cosmology
is the determination of the abundances of the light elements formed in the
early universe. Below we discuss briefly the possible deviations from the
standard big bang-predicted abundances for $\mathrm{D}$ and
${}^{4}\mathrm{He}$.

The possible deviations can be described by the expansion rate parameter
$S\equiv H/H_{\text{SBBN}}$, where $H$ is the Hubble parameter associated to
the present model and $H_{\text{SBBN}}$ for the standard big bang
nucleosynthesis review_bbn . These parameters are calculated during the early
radiation era where the energy density of the universe is dominated by
relativistic particles.

From the gravitational field equations (17), one can show that the energy
density of the radiation evolves with the redshift $z$ as det_cosmology

$\displaystyle\rho_{\text{r}}=\rho_{\text{r}0}(1+z)^{\frac{4+10\times(1/3)^{3/4}\lambda}{1+7\times(1/3)^{7/4}\lambda}},$
(31)

whereas the expansion rate is given by

$\displaystyle
3H^{2}=\kappa\left(1+\frac{7}{3}\left(\frac{1}{3}\right)^{4/3}\lambda\right)\rho_{\text{r}}.$
(32)

The changes from the standard big bang nucleosynthesis arise when $S\neq 1$
where

$\displaystyle S=\frac{2\left(9+7\times
3^{1/4}\lambda\right)}{3\left(6+5\times 3^{1/4}\lambda\right)}.$ (33)

In constraint_on_s , a limited range on the baryon density and the parameter
$S$ are provided as $4\lesssim\eta_{10}\lesssim 8$ and $0.85\lesssim S\lesssim
1.15$, respectively. Interestingly, the values of $\lambda$ predicted from the
mass-radius relations of the neutron star (see Figure 1 above) are compatible
with this range. Hence, the scale-free model in (13) does not alter the
cosmological constraints on the abundances of the light elements although it
leads to deviations from the standard hot big bang nucleosynthesis. We leave
the thourough study of more cosmological implications of this model to a
separate work det_cosmology .

##  IV Conclusion

In this paper, we have considered a new type of matter-gravity coupling which
is based on the determinant of the energy-momentum of matter fields. The main
motivation is inherited from the fact that, like scalar invariant terms which
are formed by contracting the covariant indices, the determinantal actions
(mainly second rank tensors) also stands useful in curved spactimes.

In a general framework, we have extended the standard matter Lagrangian by an
arbitrary function $f(\bm{D})$ that involves the determinant of the energy-
momentum and the metric so that the overall quantity,
$\bm{D}\equiv|\textbf{det}.T|/|\textbf{det}.g|$, transforms as a scalar under
the general coordinate transformations. We have derived the most general field
equations from the action principle and examined the effects of the new terms
induced by the determinant. Unlike the familiar extensions of gravity, we
noticed the emergence of the inverse of the stress-energy tensor. We have
discussed how the theory coincides with general relativity plus an effective
cosmological constant in the case where the energy-momentum involves only the
vacuum energy.

To apply the work to an astrophysical phenomenon, we have proposed a model in
which the free parameter is a dimensionless constant, i.e. a scale-independent
model where $f(\bm{D})=\lambda\bm{D}^{1/4}$. When matter fields are
approximated to perfect fluids, we have seen that, as expected, the novel
contribution are nonlinear in the energy density and pressure, and involve
their inverse. We derived the hydrostatic equilibrium equations and solved
them numerically by assuming various equations of state for a neutron star.
Then, we have presented our predictions in plots showing the important mass-
radius relation for different values of the coupling constant $\lambda$. In
this respect, we have focused on values that bring out noticeable departure
from the predictions of general relativity, namely, values around
$\lambda\sim-10^{-2}$.

We have also shed light on the possible effects of the scale-free model on the
predictions of primordial nucleosynthesis. In this regard, we have shown that
although the model must bring out slight deviations from the abundances of
light elements predicted by the standard big bang nucleosynthesis, it still
fits with observation. A separate work will be devoted to a more detailed
study of the cosmological implications of this model det_cosmology .

Gravity models where matter fields enter the action principle through its
determinant can lead to more interesting phenomenology. Therefore, various
scale-dependent models different than the one presented here must be explored
in various theoretical and phenomenological contexts. On the other hand, the
rapid growth of the gravitational wave astronomy via LIGO, LISA, and other
experiments is expected to probe precisely the gravitational physics
gravitational_wave_astronomy . This will provide a test for the validity of
various extended theories of gravity including the stress-energy determinant
gravity we explored in the present work gravitational_wave_astronomy_review .

## acknowledgments

HA is indebted to Yavuz Ekşi for the help with the numerical solutions, and to
Daniela Doneva for useful comments. He also thanks Glenn Starkman, Fabian
Schmidt, and Albert Stebbins for fruitful discussions. This work is supported
by the UAEU under UPAR Grant No. 12S004.

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Extracting Norms from Contracts Via ChatGPT: Opportunities and Challenges

Extracting Norms from Contracts Via ChatGPT

Amanul Haque Munindar P. Singh

North Carolina State University

<EMAIL_ADDRESS><EMAIL_ADDRESS>
We investigate the effectiveness of ChatGPT in extracting norms from contracts.

Norms provide a natural way to engineer multiagent systems by capturing how to govern the interactions between two or more autonomous parties.
We extract norms of commitment, prohibition, authorization, and power, along with associated norm elements (the parties involved, antecedents, and consequents) from contracts.
Our investigation reveals ChatGPT's effectiveness and limitations in norm extraction from contracts.
ChatGPT demonstrates promising performance in norm extraction without requiring training or fine-tuning, thus obviating the need for annotated data, which is not generally available in this domain.
However, we found some limitations of ChatGPT in extracting these norms that lead to incorrect norm extractions.
The limitations include oversight of crucial details, hallucination, incorrect parsing of conjunctions, and empty norm elements.
Enhanced norm extraction from contracts can foster the development of more transparent and trustworthy formal agent interaction specifications, thereby contributing to the improvement of multiagent systems.

§ INTRODUCTION

Normative relationships—norms, for short—typically refer to a social construct that governs the behavior of individuals in a given social context.
The behaviors could range from etiquette and social expectations to moral and ethical standards, depending on the context.
These norms help capture the essential aspects of interactions between individuals independent of low-level implementation details.
Recent approaches have shown how we might engineer a multiagent system starting from specifying norms between member agents <cit.>.

Contracts are a rich source of norms.
Although contracts are expressed in natural language, it is well recognized that they can be understood as capturing the norms between the contracting parties <cit.>.
Norms in contracts define the terms of the agreement between the contracting parties and their expected behaviors.

This paper shows how we might begin from existing annotated contracts to identify relevant norms.
Even if we don't fully formalize the norms in this way, progress made in identifying the norms can help in producing valid norm specifications for further implementation.

We use ChatGPT, a pretrained Large Language Model (LLM), to extract norms from contracts.
LLMs are sophisticated AI systems capable of understanding language and have pushed the state-of-the-art on many standard language benchmark tasks <cit.>.
We use a model of norms based on Custard <cit.> and its variants, e.g., <cit.>, and extract norms such as commitments, prohibitions, authorizations, and powers and conduct a qualitative analysis of the extracted norms.

Whereas ChatGPT demonstrates notable effectiveness in norm extraction, our analysis uncovers recurring challenges.
Some recurring problems include overlooking crucial details, hallucinations, incorrect parsing of conjunctions, and empty norm elements in the extracted norms.

Additionally, our investigation emphasizes the scarcity of high-quality annotated datasets crucial for furthering norm extraction from contracts and, hence, advancement toward automated contract comprehension.
Existing datasets are marred by limitations, including a small number of manually annotated contracts, lack of diversity in types of contracts and clauses, and poor manual annotations.

Norms offer a principled basis for engineering flexible, autonomous multiagent systems for business and other cross-organizational settings.
Automating norm extraction from unstructured text documents can help us design more robust and efficient decentralized autonomous multiagent systems.
A clear path to engineering a multiagent system based on norms derived from contracts can improve the comprehensibility of the resulting specifications and clearer accountability and, thus, trust.
Our efforts are a step forward in that direction.
The insights from this study can guide future research aimed at overcoming barriers to effective automated norm extraction from natural text.

The rest of this paper is organized as follows.
Section <ref> discusses related work.
Section <ref> describes the methodology (including the definitions, dataset, and the ChatGPT Prompt used).
Section <ref> discusses the findings of our qualitative analysis, followed by a conclusion in Section <ref>.

§ RELATED WORK

Automating comprehension of legal documents has long been a topic of investigation.
Many prior works have automated the extraction of useful information from unstructured legal documents.
This includes extracting software requirements <cit.>, unfair or risky clauses <cit.>, legal norms <cit.>, conflicting norms <cit.>, and business events and their temporal constraints <cit.>.

However, despite considerable progress, the goal of achieving comprehensive automation in contract understanding is far from complete.

Automating contract execution from unstructured text documents involves two fundamental steps.
First, identifying the contract clauses and then inferring the relationship between the identified clauses.
This involves identifying relevant text spans in the document (i.e., context) and extracting relationships between different entities from the context.
Second, translating the identified relationships into formal specifications suitable for computational processing, such as Hercule <cit.> and Symboleo <cit.>.
In this work, we focus on the first task, more precisely, extracting norms (normative relationships) from unstructured legal contracts.

A major roadblock in automating contract understanding is the lack of manually annotated gold-standard datasets.
The manual annotation of contracts has long posed a challenge essentially due to greater complexity and intricacy of language in contract documents that require experts for annotations.
Additionally, contract documents are extensive in length (a contract may be a few pages to hundreds and sometimes even thousands of pages long).
Although some previous works have created manually annotated contract datasets, they are often small (i.e., contain few manually annotated contracts), lack coverage (i.e., identify a limited number of different types of clauses and contracts), and exhibit poor quality human annotations.

<cit.> annotate 179 lease agreements for named entities and red flags.

Although their dataset is useful for automating contract comprehension, it focuses exclusively on one type of contract (leases) and a small number of types of clauses.
<cit.> create ContractNLI, a dataset for Natural Language Inference (NLI) on contracts.
Given a contract and a set of hypotheses, they annotate each hypothesis as entailed by, contradicting, or not mentioned by (neutral to) the contract.
They also identify evidence for the annotation as text spans in the contract.
Although ContractNLI offers more manually annotated contracts ($\approx$600) than most datasets containing annotated contracts, its annotations are less granular.
For instance, ContractNLI doesn't identify different contract clauses, which is essential for automated contract understanding.

For our analysis, we chose CUAD (Contract Understanding Atticus Dataset)
<cit.>, a Contract Clause Extraction (CCE) dataset containing more than 500 contracts manually annotated by experts (lawyers and law students).
CUAD proposes the task of contract comprehension as a question-answering task and contains text spans of the contract clauses manually identified as belonging to one of 41 categories.
CUAD is more diverse than previous datasets on contracts (i.e., contains contracts from different domains), has a higher number of annotated contracts, and is annotated by experts.
Further, the ontology is developed for the given dataset (as opposed to adapting an ontology from another work that may or may not be suitable for the current dataset).

Prior research has proposed several methodologies for extracting essential clauses from unstructured legal documents, including Case-Based Reasoning (CBR) <cit.>, grammars, word lists, and heuristics <cit.>, rule-based NLP <cit.>, hand-crafted rules and machine learning <cit.>, neural models such as hierarchal BiLSTM with attention and CNN (Convolution Neural Networks) <cit.>, and fine-tuned transformer-based models, such as Legal-BERT <cit.>.
Although some of the prior approaches are effective, they usually require a lot of manual effort, either in hand-crafting rules or annotating large amounts of data for training.

Recent advances in NLI and NLU (Natural Language Understanding) present promising avenues for alleviating the burden of manual effort in automated contract understanding.
<cit.> proposed Transformers, a deep learning model architecture based on multihead self-attention, which pushed the state-of-the-art performance for multiple NLI and NLU benchmarks, including the General Language Understanding Evaluation (GLUE) <cit.>, Stanford Question Answering (SQuAD) <cit.>, and Multigenre NLI (MNLI) <cit.>.
We use ChatGPT (model GPT-3.5-turbo), an instance of the Generative Pretrained Transformer (GPT), to extract norms from contracts.

ChatGPT has pushed the state of the art on multiple benchmarking language tasks.
ChatGPT showcases capabilities that closely mirror human-level performance on language understanding and inference and occasionally exceed human proficiency on specific tasks <cit.>.
ChatGPT has been used in several domains including healthcare <cit.>, business and finance <cit.>, creative content generation <cit.>, writing code <cit.>, and education and training <cit.>.

Although transformer-based models show promising results on a variety of NLU and NLI tasks, evidence suggests that most LLMs perform poorly in contract understanding <cit.>.
Further, there is a lack of qualitative analysis of how well the LLMs' perform in contract understanding.
Such analysis is essential to devise reliable automated contract understanding systems that could streamline the process of contract analysis and review, saving valuable time and resources for legal professionals and organizations.
To facilitate such efforts, we qualitatively evaluate the effectiveness of ChatGPT in understanding contracts via norm extraction.

§ METHODOLOGY

We extract norms (normative relationships), for short—from contracts (unstructured text documents) to facilitate the engineering of agents and multiagent systems that respect stakeholder needs as captured in contracts.

In abstract terms, a norm creates a legitimate expectation of one autonomous party over another autonomous party.
The sense of norm used here is the one established in the legal literature <cit.> and is formal and prescriptive.
A norm can be viewed as an element of a contract. Or, conversely, a contract is (primarily) a set of norms.
Norms in contracts serve multiple purposes and fall into a variety of types of clauses <cit.>.

Norms provide a natural way to engineer multiagent systems because they provide a natural basis (i.e., one that is close to stakeholders' conception) for governance <cit.> and accountability <cit.> in settings involving two or more autonomous entities.
The benefits of using norms include improvements in trustworthiness and resilience of sociotechnical systems <cit.>.

We adopt gao-2014AAMAS:extracting_norms_from_contracts approach to define normative relationships to extract from a contract.
A norm is directed from a subject to an object and is constructed as a conditional relationship involving an antecedent (which brings the norm into force, i.e., the condition on which the action of the subject depends) and a consequent (which brings the norm to satisfaction, i.e., the outcomes of the action).

A norm can be one of the following four types:

A commitment means that its subject commits to its object to ensure the consequent if the antecedent holds.

A prohibition means that its subject is forbidden by its object from bringing about the consequent if the antecedent holds.

An authorization means that its object authorizes its subject to bring about the consequent if the antecedent holds.

A power means that its subject is empowered by its object to bring about the consequent if the antecedent holds.

The norm types of authorization and power involve similar language; however, some differences exist <cit.>, which we can use to differentiate them.
An authorization is when the subject is allowed to bring about the consequent of the norm.
A power is when the subject brings about the consequent as a way to change an existing normative relationship.
Consider a manufacturing contract.
The manufacturer may use some equipment provided by the purchaser under the stated conditions.
No norms change as a result of such usage.
Therefore, the use of equipment maps most naturally to an authorization.
The purchaser may cancel an order with prior notice, i.e., it can terminate a commitment at will, thereby changing the normative relationship between itself and the manufacturer.
Therefore, termination in this case (and usually) is mapped to power.

§.§ Dataset

There is a lack of good-quality gold standard datasets for contract understanding.
This is partly because of the complexity of the task itself; annotating contracts often requires expert judgment.
Further, there is no limit to the variety of clauses that may be present in a contract depending on its purpose.
For instance, a manufacturing contract may have clauses that are not found in a lease agreement, and vice versa.
Similarly, for other types of contracts.
Not surprisingly, prior works focus on a few clauses to annotate depending on the task.

For our analysis, we chose CUAD (Contract Understanding Atticus Dataset) that contains 510 manually annotated contracts.
To ensure diversity in the dataset, CUAD includes 25 types of contracts, such as manufacturing, service, and leasing.
Contracts in CUAD have varying lengths, ranging from a few pages to over one hundred pages.
Each contract in CUAD is annotated manually by legal experts by identifying essential clauses (i.e., text spans in the contracts) that facilitate contract comprehension.
CUAD defines 41 types of clauses (categories), including temporal clauses such as Effective Date, Renewal Date, and Expiration Date, service-related clauses such as Governing Laws, Revenue-Profit Sharing, and Price Restrictions, and termination-related clauses such as Termination for Convenience and Post Termination Services.
CUAD contains 13000 of such clauses identified manually by experts.
In addition to identifying the relevant spans, CUAD contains a short answer (derived from the relevant text spans) to a question for each clause, such as `on what date will the contract's initial term expire?' (Expiration Date), or `Is one party required to share revenue or profit with the counter party for any technology, goods, or services?' (Revenue/Profit Sharing).

Although CUAD annotations are claimed to be reliable and double-checked by experts, we found instances where the annotations were lacking.
We found some annotations leave out contexts (text spans from the contract) that may help answer the clause related questions more accurately.
We manually verified some contracts and found termination-related clauses, such as Post-Termination Services and Termination For Convenience, often overlooked relevant text spans casting doubt on the quality of these annotations.

This shows the inherent difficulties of the task that is challenging even for experts.
This may be due to long and complex sentence structures common in contracts or information relevant to one clause often being spread across the document, with each contract document's length ranging from a few pages to hundreds of pages.

Some clause types in CUAD are easy to identify.
For instance, temporal clauses such as agreement date and expiration dates have a distinctive syntax.
Likewise, the parties involved in a contract are easy to identify as they are explicitly mentioned at the beginning of the contract and repeated multiple times across the document.
For simplicity, we focus our analysis on clause types related to the termination of a contract, such as Termination For Convenience, Right of first refusal (or offer or negotiation) (Rofr/Rofo/Rofn), and Post-Termination Services, and clause types related to the change of control of a party to the contract, such as Change of Control.
These clause types are a good venue to look for norms (normative relationships).
Further, these are some of the toughest clause types to identify, as evidenced by the fact that three of these four clauses types were in the top six worst-performing clause types benchmarked using a Transformer model (DeBERTa) <cit.>.
We pick clauses from the CUAD dataset of the above-mentioned types and analyze the norms identified by ChatGPT.
We randomly sample 100 clauses to evaluate.
Further, we identify 50 more clauses that potentially contain more challenging logical forms to extract, such as conditional actions and multiple conjunctions (OR, AND).
We use keywords such as `and', `or', `upon', `unless', and so on to identify such challenging clauses.
We use ChatGPT to extract norms from these selected clauses.

§.§ Extracting Norms via ChatGPT

ChatGPT is a state-of-the-art LLM based on the GPT (Generative Pretrained Transformer) architecture.

ChatGPT pushed the state-of-the-art for multiple NLP tasks such as language translation, text summarization, sentiment analysis, and question-answering <cit.>.
A major advantage of using ChatGPT is that it requires no training data.
ChatGPT can be instructed via prompts to perform the norm extraction.

We craft prompts to extract the norms.
We incrementally refine the prompt by manually inspecting the extracted norms against the expected norms for a few sample clauses from CUAD.
We found that whereas ChatGPT isn't too sensitive to small changes in a prompt (such as rephrasing or substituting with synonyms), it performs better when given more detailed prompts, i.e., when they contain more information about the concepts to be extracted.
For instance, we initially did not specify the direction of the relationship between the subject and object for different norm types in the prompt.
The norms extracted via such prompts often involve the correct parties, i.e., objects and subjects, but in incorrect roles.
This may be because the roles (directionality) differ depending on the norm type.
For example, in the case of commitments, the subject commits to the object, whereas in the case of prohibitions, the object prohibits the subject.
Including the directions explicitly in the definitions for the subject and object in the prompt improved the results by reducing such mismatches.

As input, we give ChatGPT the prompt followed by the clause from the contract (from which the norm needs to be extracted).
Below is the prompt we use to extract the norms given a clause.

ChatGPT Prompt to Extract Norms from Contracts

Given a sentence from a contract (between two parties), extract norms for better contract understanding.
A norm specifies the expectations from parties involved in the contract in terms of their behavior and actions and is represented by four elements subject, object, antecedent, and consequent. A norm is directed from a subject (the party on whom the norm applies) to an object (the party with respect to whom the norm applies) and is constructed as a conditional relationship involving an antecedent (which brings the norm into force, i.e., the condition on which the action of the subject depends) and a consequent (which brings the norm to satisfaction, i.e., the outcomes of the action).
A norm can be one of the following four types:

(a) A commitment means that its subject commits to its object to ensure the consequent if the antecedent holds.

(b) A prohibition means that its subject is forbidden by its object from bringing about the consequent if the antecedent holds.

(c) An authorization means that its object authorizes its subject to bring about the consequent if the antecedent holds.

(d) A power means that its subject is empowered by its object to bring about the consequent if the antecedent holds.

The norm types of authorization and power are similar; however, some differences exist. Authorization allows the subject to bring about the consequent (provided it doesn't violate some other norm), whereas power empowers the subject to bring about the consequent when that causes a change in the existing norms.

In cases where more than one norm exists, extract all norms. Identify the following given a contract sentence.

1. Norm type: norm type can be `authorization,' `power,' `commitment,' or `prohibition'

2. Subject: the party on whom the norm applies

3. Object: the party with respect to whom the norm applies

4. Antecedent: which brings the norm into force

5. Consequent: which brings the norm to satisfaction

The contract sentence is as follows:

We run our experiments on the “GPT-3.5-Turbo” version of ChatGPT.
Since the ChatGPT model is nondeterministic by default, we set the temperature equal to 0.0 to make it semideterministic and improve reproducibility across runs.

§ FINDINGS

Example 1 (below) shows the results for successfully extracting correct norms from a sample contract clause using ChatGPT with the above prompt.
ChatGPT extracts an authorization norm and correctly infers all norm elements.

Example 1: Norm Extraction via ChatGPT

Notwithstanding any other provision of this Agreement, Rogers may terminate this Agreement, at any time, upon sixty (60) days prior written notice to Licensor.

Norm 1: Authorization

Subject: Rogers

Object: Licensor

Antecedent: Rogers providing sixty (60) days' prior written notice

Consequent: termination of the Agreement

ChatGPT did a good job in extracting norms from contract clauses in many cases.
However, the extractions are not without limitations.
In some cases, ChatGPT failed to extract a norm that existed in the contract clause, and in other cases, it extracted a norm that did not exist in the clause.
Further, we find that the extracted norms were incorrect sometimes due to incorrect norm type, incorrect norm elements and missing crucial details.

We notice contract clauses containing redactions lead to more incorrect extracted norms.
This implies that contract clauses with redacted information are more challenging for ChatGPT than clauses without redactions.
This could be critical in situations where some of the information in the contract is redacted due to confidentiality reasons, such as contracts involving national security, and intellectual property.
We leave further anlaysis of norm extraction from clauses containing redactions to future work and only discuss norm extraction from clauses that do not contain any redactions.

Below we discuss some of the recurring issues in ChatGPT’s norm extractions with examples.

§.§.§ Incorrect Norm Types

In some cases, ChatGPT extracts incorrect norm types.
ChatGPT often extracts a commitment norm (in addition to the authorization norm) from clauses that contain only an authorization clause.
Example 2 shows a norm extraction by ChatGPT that inferred an incorrect norm type.
ChatGPT extracts two norms, both have the same norm elements except for the antecedent.
Whereas norm 1 (authorization) is correct, norm 2 (commitment) is incorrect.
This is because the original clause uses `may' to describe the potential action making the action optional (has the authority to) but not mandatory (i.e., commitment).
Hence, the extracted commitment norm is incorrect.
We notice that ChatGPT often confuses what must be done with what can be done, leading to incorrect norm extraction.

Example 2: Incorrect Norm Type

A Party may terminate the JSMA immediately upon the delivery of written notice to the other Party if there has been a Change in Control.

Norm 1: Authorization

Subject: A Party

Object: the other Party

Antecedent: Change in Control

Consequent: Termination of the JSMA

Norm 2: Commitment

Subject: A Party

Object: the other Party

Antecedent: Change in Control

Consequent: Termination of the JSMA

§.§.§ Incorrect Norm Elements

In some of the extracted norms, the norm elements are incorrect.
Example 3 shows one such example where multiple extracted norm elements are incorrect.
Note that similar to the previous example (Example 2), ChatGPT extracts an incorrect norm type, i.e., norm 2 (commitment) is incorrect as the original clause contains only authorization.
In norm 1, the consequent and object are incorrect, and the antecedent is only partially captured.

Example 3: Incorrect Norm Elements

Notwithstanding the foregoing, each Party may assign the rights and obligations under this Agreement in whole, without consent of the other Party, to a Third Party or Affiliate in connection with the transfer or sale of all or substantially all of its business or in the event of a merger, consolidation or change in control provided that the assignee assumes in writing and becomes directly obligated to the other Party to perform all of the obligations of assignor under this Agreement.

Norm 1: Authorization

Subject: Each party

Object: Third Party or Affiliate

Antecedent: Assignment of rights and obligations under the Agreement in connection with the transfer or sale of business, merger, consolidation, or change in control

Consequent: The assignee assumes in writing and becomes directly obligated to perform all obligations of the assignor under the Agreement

Norm 2: Commitment

Subject: Assignee

Object: Other Party

Antecedent: Assignment of rights and obligations under the Agreement in connection with the transfer or sale of business, merger, consolidation, or change in control

Consequent: The assignee becomes directly obligated to perform all obligations of the assignor under the Agreement

§.§.§ Overlooking Crucial Details

ChatGPT sometimes overlooks crucial details while extracting norms.
For instance, in Example 4, the detail that the agreement will be in effect until March 18th is not captured by any norm (the date is overlooked).
Further, the detail that 30 days written notice is needed for termination is also lacking from all extracted norms.

Example 4: Overlooking Crucial Details

This Agreement shall be in effect until March 18. 2021, unless sooner terminated by either party upon (30) days written notice, without cause.

Norm 1: Commitment

Subject: Either party

Object: Either party

Antecedent: Written notice of termination

Consequent: Termination of the Agreement

Norm 2: Authorization

Subject: Either party

Object: Either party

Antecedent: Written notice of termination

Consequent: Termination of the Agreement

Norm 3: Power

Subject: Either party

Object: Either party

Antecedent: Termination of the Agreement

Consequent: End of the Agreement period

§.§.§ Hallucinaton

Hallucination refers to a phenomenon where a model generates outputs that are not grounded in the input data.
Hallucination is a known issue in generative models such as ChatGPT and has been found in downstream tasks such as abstractive text summarization.
Hallucination can be highly problematic for norm extraction from contracts that rely heavily on the accuracy of the extracted information, and even minor hallucinations could change the meaning and applicability of clauses.

Example 5 shows a norm extraction by ChatGPT that demonstrates hallucination.
The subject is inferred as VS (Vendor Service), and although the acronym (VS) is correct (i.e., the entity mentioned in the clauses), its expansion `Vendor Service' is incorrect (the correct expansion is VirtualScopics).
Interestingly, the input clause doesn't contain the expansion of the acronym, but ChatGPT hallucinated it (the word `Service' occurs in the input clause but not `Vendor').
Hallucination could lead to incorrect inferences based on the extracted norms.

Additionally, the norm extraction in Example 5 identifies two norms types for the same extracted norm which is incorrect.
The commitment norm type is correct and prohibition incorrect.

Example 5: Hallucination

VS shall not refer that opportunity to any other third party provider unless PPD declines to bid on such opportunity or does not offer the service in question.

Norm types:
- Prohibition, Commitment

Subject: VS (Vendor Services)

Object: Any other third party provider

Antecedent: PPD declines to bid on such opportunity or does not offer the service in question

Consequent: VS shall not refer that opportunity to any other third party provider

§.§.§ Incorrect Parsing of Conjunctions.

Clauses with conjunctions, such as AND and OR, are sometimes not parsed correctly, leading to incorrect norm extractions.
Example 6 shows one such incorrect extraction.
The antecedent (based on the input clause) has two parts joined by a conjunction (`and'), which is incorrectly broken down into antecedent (the first half) and consequent (the second half).
Hence, the incorrect parsing of conjunction led to the incorrect extraction of norm elements (antecedent and consequent for norm 1 in Example 6).
Further, extracted norm 2 is incorrect as it fails to identify any antecedent, and the extracted consequent is incorrect.

Whereas in some cases, conjunctions lead to incorrect norm extractions.
In other cases, conjunctions could lead to long and complex antecedents and consequents (when the conjunctions are kept as it is (without rephrasing)).
To address this, we could include instructions in the prompt to break down the conjunctions appropriately.
However, this complicates the prompt for all cases (even for clauses with no conjunction).
A potential solution could be to have a separate prompt that first breaks down the logical form based on the conjunctions before extracting the norms.
We leave this to future work as the focus of this study is to identify the challenges and opportunities in norm extraction.

Example 6: Incorrect Parsing of Conjunctions

Nothwithstanding the foregoing, this Agreement may be assigned without the AS Parties' consent by MusclePharm in connection with a change of control transaction; provided that the acquirer of MusclePharm shall have financial resources substantially similar or greater than MusclePharm and shall specifically assume the obligations of MusclePharm under this Agreement in writing prior to the consummation of the change of control transaction.

Norm 1: Authorization

Subject: MusclePharm

Object: AS Parties

Antecedent: The acquirer of MusclePharm shall have financial resources substantially similar or greater than MusclePharm

Consequent: The acquirer of MusclePharm shall specifically assume the obligations of MusclePharm under this Agreement in writing prior to the consummation of the change of control transaction.

Norm 2: Commitment

Subject: MusclePharm

Object: AS Parties

Antecedent: None explicitly stated

Consequent: MusclePharm commits to ensuring that the acquirer of MusclePharm specifically assumes the obligations of MusclePharm under this Agreement in writing prior to the consummation of the change of control transaction.

§.§.§ Empty Norm Elements.
For some extracted norms, ChatGPT leaves some norm elements empty (i.e., `', N/A, NULL, or Not Specified).
This is more common for subjects and objects, followed by antecedents.
Consequent is rarely left empty.
Although an empty element in an extracted norm may indicate poor extraction in some cases, it may be the correct response in others.
For instance, when no precondition (or trigger) exists for a consequent, it implies an empty antecedent.
Similarly, an empty subject (or object) could mean either no party or all parties subject to the contract.

However, in many cases, empty norm elements imply incorrect norm extraction.
In Example 6, norm 2 of the extracted norms has no antecedent (`None explicitly stated'), implying that the stated consequent would hold without any precondition/trigger, which is incorrect.

§ CONCLUSION

This study delves into the use of LLMs such as ChatGPT to extract norms from contracts, shedding light on the array of opportunities and challenges inherent in automating contract comprehension.
We uncover critical insights through a qualitative analysis of the norms extracted by ChatGPT.

We find that the extraction is influenced by the prompts used.
A well-crafted prompt that includes definitions of the formal terms used in the prompt yields much better results than less detailed prompts.
Although a comprehensive prompt consistently yields superior outcomes, the results don't seem too sensitive to minor changes in the prompt (such as rephrasing and substitution with synonyms).

Despite ChatGPT's overall efficacy in norm extraction, our scrutiny unveils recurrent issues.
Overlooking crucial information, hallucination, incorrect processing of conjunctions (AND and OR), incorrect norm types, and incorrect norm elements in the extracted norms emerge as persistent challenges.
Moreover, ChatGPT occasionally identifies norms with empty norm elements, long consequents (that include the subject, object, and antecedents), and incorrect objects and subjects.

Furthermore, our exploration highlights the dearth of high-quality annotated datasets essential for advancing automated contract understanding.
Existing datasets suffer from limitations such as small size, limited diversity, and subpar annotations, hindering progress in this domain.

Despite these challenges, our findings underscore the promise of using ChatGPT for norm extraction from unstructured text documents.
Particularly compelling is the fact that no training or fine-tuning is necessitated, positioning ChatGPT (and similar LLMs) as a preferred model for tasks like contract understanding that lack high-quality annotated datasets.

In conclusion, this research not only provides valuable insights into the efficacy and challenges of employing LLMs for norm extraction from contracts but also underscores their potential as powerful tools in the realm of automated contract understanding.

§ ACKNOWLEDGMENTS

Thanks to the US NSF (grant IIS-1908374) for support for this research.