We adopt the relativistic mean field (RMF) theory to derive all the physical quantities of dense matter at finite temperature. The RMF theory is one of nuclear many-body frameworks based on an effective Lagrangian, which is relativistically covariant, composed of nucleon and meson degrees of freedom. We refer to the review article by Serot & Walecka (1986) for details concerning with the relativistic many-body framework. All the details on the relativistic EOS for neutron stars and supernovae in the RMF theory can be found in recent papers (Sumiyoshi & Toki 1994; Sumiyoshi et al. 1995).

We start with the phenomenological Lagrangian composed of nucleon field and , and meson fields and solve the relativistic many-body problem within the mean field approximation. The adopted Lagrangian also contains non-linear and meson terms (Sugahara & Toki 1994), which are motivated by the recent success of the relativistic Brückner Hartree Fock (RBHF) theory (Brockmann & Machleidt 1990). It should be noted that the RBHF theory can reproduce the saturation property of nuclear matter starting from the elementary nucleon-nucleon interaction. RMF has been used very successfully as an effective theory of RBHF to describe dense matter and nuclear structure (Sugahara & Toki 1994).

The Lagrangian contains the parameters: meson mass, coupling constants between nucleon and mesons and self-coupling constants of mesons. We adopt the best parameter set called TMA, which has been determined by fitting the experimental data of masses and radii of many nuclei in the periodic table (Sugahara 1995; Hirata et al. 1997). The symmetry energy of EOS using TMA is 30.68 MeV and the incompressibility is 318 MeV. The RMF theory with TMA has been applied to studying the ground state properties of about 2000 even-even nuclei covering all the region of the nuclear chart. It has also proved very successful in reproducing the experimental data of stable nuclei as well as unstable nuclei (Hirata et al. 1997). The prediction of nuclear structure in the RMF theory with TMA is being checked in experiments on radioactive nuclear beam facilities in the world, so we can constrain the EOS for astrophysics at the same time.

The extension of the RMF theory to finite temperature is straightforward (Serot & Walecka 1986; Sumiyoshi & Toki 1994). Another field theoretical approach to provide EOS at finite temperature has been worked out (Diaz Alonso et al. 1989) and has been applied to studying warm cores in neutron stars (Martí et al. 1988; Romero et al. 1992). All the physical quantities of dense matter under various conditions of chemical composition, density and temperature are calculated in the RMF theory with TMA and are prepared in the form of a table of numerical data, which is available for numerical simulations of protoneutron star cooling and supernova explosion (Sumiyoshi & Toki 1994).

In the present study, we assume that the dense matter in protoneutron
stars is composed of neutrons, protons, electrons,
neutrinos, their anti-particles and photons, and satisfies the chemical
equilibrium and the charge neutrality. We characterize the dense
matter by the lepton fraction and the entropy per baryon
*s*, which we assume to be constant inside stars.
We calculate the EOS of the dense matter under the above
condition by adding lepton contributions to nuclear contributions
and use them to calculate the equilibrium configuration
of slowly rotating protoneutron stars. As for the EOS of dense
matter at low densities below , we adopt the
EOS by Lattimer & Swesty (1991) with the addition of lepton
contributions and the constraints on and *s* and connect
smoothly with the EOS by the RMF theory with TMA at high density.

The properties of the relativistic EOS in the RMF theory and
the profiles of *non-rotating* neutron stars and protoneutron
stars using the relativistic EOS have been reported in detail
by Sumiyoshi & Toki (1994) and Sumiyoshi et al. (1995a). It is to be noted here that the relativistic
EOS has different characteristics compared to the
conventional EOS in non-relativistic frameworks. For example,
the proton fraction inside the neutron stars using the
relativistic EOS is remarkably large
(see also Sumiyoshi et al. 1995b).
The current study is the first result of the application of
relativistic EOS, taking into account the conditions
on and *s*, in the RMF theory to *rotating*
protoneutron stars.
It would be of great interest to explore the characteristics
of rotating protoneutron stars by comparing them with non-rotating
ones and other studies with conventional EOSs.

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