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Based on the physical metric proposed by the author, temperature distribution for compact objects, neutron stars and black holes, has been explained. Outside the extended horizon, the temperature is positive and approaches infinity at the extended horizon boundary. Inside the extended horizon, the temperature is negative which implies higher temperature than outside the horizon. This outcome is the result of the repulsive nature of gravity inside the extended horizon in the authorâ€™s physical metric. Overall, the physical metric explains temperature structure of compact objects more completely than the Schwarzschild metric, and is supported by the emerging evidence of X-ray data collected from neutron stars and black holes (AGN).

The physical metric has been introduced by the author [

In the spherical symmetric and static (SSS) metric of general relativity,

the physical metric is defined as

In other words, the speed of light in the spherical direction that is perpendicular to the radial direction of gravity is unchanged from the vacuum value. It is remarkable that such a condition is required to fit the experi- mental data for general relativity test. In the original formulation of Schwarzschild metric [

In order to get the physical metric, one has to get the Schwarzschild metric after the coordinate transformation

which implies that

where

is the Schwarzschild radius for mass M. Then one gets

For positive value of

and we restrict the range of

From Equation (7), one gets

and

In order to extend the solution for the physical metric beyond Equation (8), one uses the non-asymptotic solution,

for

Then one gets

The continuity of Equation (7) and Equation (14) at the boundary yields

Differentiating Equation (14), one gets

Then one gets

Choosing the parameter space in Equation (15)

all metric functions are positive definite. With the choice of these parameter ranges, the speed of light are well defined throughout all the space time points. This is very different from the Schwarzschild metric. It is remark- able that only such a natural metric fits the experimental data, as was stated earlier.

The distance r can be reached at zero when

One may note that there is an undecided one parameter which can be fixed from the physics inside the dis- tance at

At

and from Equation (10)

and hence

From

The temperature of compact objects can be calculated as the extension of the Hawking temperature

for the outside of the extended horizon. This has the same structure as the Hawking temperature, since it is inversely proportional to the distance. The only difference is that the coefficient becomes infinitely large for

and

where

and

Using Equation (7), one gets

and the expansion of Equation (24) yields

For

and then

the temperature becoms infinitely large towards the extended horizon.

Using the conversion formula (for

the 1 keV temperature corresponds to

then the location of 1 keV is

Then, for the value of Equation (23), one gets

or

and hence

This small value of the high temperature thickness makes the observation from this thin layer very unlikely, even by using high density of neutron star (~10^{39} neutrons/cm^{3}).

Inside the extended horizon, the expression for the temperature becomes

where Equation (16) has been used. Since

and

the temperature in Equation (39) is negative definite. In other words, the temperature inside the extended horizon is negative. A negative temperature is a temperature higher than any positive temperature, since high energy states are more abundant than lower energy states [

The matter inside the extended horizon is in a higher temperature environment. However any phenomena in a deep inside region is not exposed to outside observers. However the matter near the extended horizon is subject to an oscillation between the attractive force outside the extended horizon and the repulsive force inside the extended horizon. One notices that the outside layer is thin, since the attractive force at the extended horizon is infinitely large. Any phenomena of the oscillating matter through the extended horizon is subject to the observation of the outside observers. The X-ray emission from highly ionized atoms of a neutron density in 1 cm thickness can be observed from the edge of the Milky Way.

For AGN or massive black holes with higher mass, the density is reduced by 1/M^{2}, but the surface area is increased by M^{2}. Hence the number of events are independent on mass. Only the difference is the distance to the events from the observers on the Earth. So the events from the nearer AGN are more likely observable. Of course, the radiation of lower frequency can be more observable, since the depth of the radiation is increased with the wave length. The recent report [

The metric for a mass with rotation is expressed as [

where the metric functions,

in order to accommodate to the physical metric in the limit of no rotation. For the asymptotic region

or outside of the extended horizon, one gets

and

where a is the angular momentum per mass. Here,

in Equation (18). In the limit of

these metric functions coincide with the physical metric in Section 2.

In order to find the gravitational force implied by this Kerr metric, let us compute

Here

becomes positive infinity at the extended horizon. For a uniform distribution of the density,

and

Using

and

one gets

For

or

Notice that the fastest rotation observed [

Expanding all quantities in this section for a small value of a and making the angular averages

and

or more generally

one gets

For Equation (52), using the expansion, Equation (25) and Equation (26), one can get

which reproduces the temperature structure of non-rotating compact objects for outside the extended horizon, with small variation of the rotation effects.

For the temperature structure inside the extended horizon, one may use the Kerr metric for the internal solution of the physical metric,

where

and

The coordinate transformation to the physical netric inside the extended horizon defines

as in Equation (14), and the continuity of the

with constraints

for positive definite metric functions.

From Equation (70) one gets

The expansion in a parameter a gives

Near the surface of the extended horizon, the

This constraint is well satisfied for all the observed neutron stars, as is seen from Equation (61).

If one may replaces

in Equation (52), one gets the same expressions for Equation (69) and Equation (78) with different coefficients for the

In conclusion, the temperature structure for rotating compact objects near the extended horizon is the same as those for non-rotating compact objects, as long as the condition of Equation (79) is satisfied. Namely, the temperature outside the extended horizon is positive approaching infinitely large value at the extended horizon, and the temperature inside the extended horizon is negative in the neighborhood of the extended horizon.

The active observation of X-ray spectra from compact objects is going on using X-ray satellites. Many emission lines in the range of 10 - 30 Å ngstrom indicate the presence of multiplly ionized atoms [

These are the indication of high temperature nature of the surface of compact objects. Since the physical metric suggests a definite value of gravitational redshift on the surface of the extended horizon of the compact objects, a consistent description of gravitational redshift will be tested in the future.

From the introduction of the physical metric, one encounters a revolutionary change in the feature of compact objects, black holes and neutron stars. The extended horizon, 2.60 times of that of the Schwarzschild radius, is the size of compact objects. The gravitational red shift on the surface of compact objects is the universal value of

It is a great pleasure to thank Peter K. Tomozawa for reading the manuscript.

YukioTomozawa, (2015) Temperature Structure of Compact Objects. Journal of Modern Physics,06,1412-1420. doi: 10.4236/jmp.2015.610146