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Gravitons are described by the propagator in teleparallel gravity in nearly flat space-time. Finite temperature is introduced by using Thermofield Dynamics formalism. The gravitational Casimir effect and Stefan-Boltzmann law are calculated as a function of temperature. Then an equation of state for gravitons is determined.

The teleparallel theory of gravity is similar to the general theory of gravity due to Einstein [

There are several approaches to introduce finite temperature formulation in a theory that can be quantized. Such a method involves doubling the physical space, Thermofield Dynamics (TFD) [

The objective of the paper is to use teleparallel theory of gravity and formulation of TFD to calculate the gravitational Casimir effect and Stefan-Boltzmann law. Originally the Casimir effect was observed by H. Casimir for parallel plates [

This paper is organized as follows. In Section

Teleparallel gravity has unique features such as a well-defined expression for the gravitational energy-momentum tensor. This alternative theory is constructed in the framework of Weitzenböck geometry that is described by torsion and a vanishing curvature. Thus a Weitzenböckian manifold is endowed with a Cartan connection [

It is important to note that the Cartan connection is curvature free, but it has the torsion tensor

If a derivative of the Lagrangian density with respect to the tetrad field is performed, it yields

This section includes brief details of Thermofield Dynamics (TFD) [

The creation (

A doublet notation is introduced by

In this section the Stefan-Boltzmann law and the Casimir effect at zero and finite temperature are calculated in the teleparallel gravity framework. The free Lagrangian of the teleparallel gravity is

Now it is necessary to calculate the mean value of

In the TFD formalism, using the tilde conjugation rules, the vacuum average of the gravitational energy-momentum tensor is

Following the Casimir prescription, the physical energy-momentum tensor is given by

Now some applications are considered for different choices of the

A first application is the case

In the framework of TFD formalism the Casimir effect at zero temperature is determined when

The effect of temperature in the Casimir effect is introduced by taking

The graviton propagator is calculated using teleparallel gravity and this leads to a Green function. Details of TFD formalism are given to introduce temperature. Stefan-Boltzmann law and Casimir effect are calculated at finite temperature. This leads to finding pressure as a function of temperature. These results are obtained for a nearly flat space-time. These results play an important role in comparison with experimental results obtained for systems in outer space. An extension of this program for different space-time points will be carried out later. In addition the first law of thermodynamics is used to establish the dependency of the gravitational pressure on the temperature. The equation of state is found identical to that obtained for photons.

No data were used to support this study. Our study is completely theoretical.

F. C. Khanna is Professor Emeritus, Physics Department, Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, Canada.

The authors declare that they have no conflicts of interest.

This work by A. F. Santos is supported by CNPq projects 308611/2017-9 and 430194/2018-8.