Thermodynamical Metrics in General Relativity Theory

Preprint

Article

Thermodynamical Metrics in General Relativity Theory

Altmetrics

Downloads

Views

Comments

Espen Haug^*

This version is not peer-reviewed

Submitted:

16 March 2024

Posted:

18 March 2024

You are already at the latest version

Alerts

Abstract

We demonstrate that multiple well-known metrics, such as the Schwarzschild metric and the extremal solution of the Reissner-Nordstr ̈om metric, can be expressed in what we will call thermodynamic form. These formulations appear to be valid for all Rs = ct growing black holes and, therefore, also within the framework of Rh = ct growing black hole cosmology. However, the metrics can also be applied to non-black hole gravitational objects as long as one calculates the CMB and Hawking temperature of the hypothetical black hole from the gravitational mass.

Keywords:

Subject: Physical Sciences - Theoretical Physics

1. Background on the Hawking Temperature and the New CMB Temperature Formula

In 1974, Hawking [1,2] introduced what is now known as the Hawking temperature of a black hole:

T_{H w} = \frac{ℏ c^{3}}{k_{b} 8 π G M} .

(1)

Where

k_{b}

is the Boltzmann constant, and ℏ is the reduced Planck constant, also known as the Dirac constant (

ℏ = \frac{h}{2 π}

). The Hawking temperature formula plays a central role in many papers analyzing black holes. Additionally, Tatum et. al [3] introduced a similar formula for the CMB temperature heuristically in 2015:

T_{c m b} = \frac{ℏ c^{3}}{k_{b} 8 π G \sqrt{M m_{p}}} = \frac{ℏ c}{k_{b} 4 π \sqrt{R_{h} 2 l_{p}}} = 2 . 7276_{- 0.0743}^{+ 0.0723} K

(2)

where

m_{p}

is the Planck mass and

l_{p}

is the Planck length, M is the black hole mass (critical Friedmann mass), and

R_{h} = \frac{c}{H_{0}}

is the Hubble radius. The CMB temperature value given is calculated based on using the Sneppen et al. [4], which reports

H_{0} = 67 \pm 3.6, km / s / Mpc

. Additionally, we have included the NIST CODATA (2018) uncertainty in the Planck length. The other constants are exact according to the NIST CODATA (2018) standard.

This formula, in more general terms for a black hole, is given by:

T_{c m b} = \frac{ℏ c^{3}}{k_{b} 8 π G \sqrt{M_{B H} m_{p}}} = \frac{ℏ c}{k_{b} 4 π \sqrt{R_{s} 2 l_{p}}}

(3)

where

M_{B H}

is the mass of a Schwarzschild black hole,

M_{B H} = \frac{c^{2} R_{s}}{2 G}

where

R_{s}

is the Schwarzschild radius

R_{s} = \frac{2 G M_{B H}}{c^{2}}

. Recently, Haug and Wojnow have demonstrated that the formula can be derived from the Stefan-Boltzmann law [5,6]. Haug and Tatum [7] have also demonstrated that the formula can be derived as a geometric mean temperature of the maximum and minimum temperatures related to a black hole. CMB, or cosmic microwave background, temperature is a bit of a misnomer, as it is only for a Hubble sphere-sized black hole that the geometric mean internal temperature in the black hole correspond to microwave wavelength, for smaller black holes this temperature increase and the corresponding wavelength is shortened. A perhaps better word for this temperature would simply be the geometric mean internal black hole temperature.

To measure the CMB temperature (black hole geometric mean internal temperature) one would need to be inside a black hole. According to black hole cosmology we are inside a black hole. This idea that the Hubble sphere could be a black hole was likely first suggested by Pathria [8] in 1972 and the idea that the universe we live in could be a black hole is actively discussed to this day, see for example [9,10,11,12,13,14,15,16,17].

Haug [5] has also demonstrated that we must have:

\frac{T_{c m b}^{2}}{T_{H w}^{2}} = \frac{l_{p}}{\bar{λ}}

(4)

Furthermore, Haug [5,18] has demonstrated that Einstein’s [19] field equation can be rewritten as:

\begin{matrix} R_{μ ν} - \frac{1}{2} R g_{μ ν} & = & \frac{8 π G}{c^{4}} T_{μ ν} \\ R_{μ ν} - \frac{1}{2} R g_{μ ν} & = & \frac{8 π l_{p}^{2}}{c ℏ} T_{μ ν} \end{matrix}

(5)

where

l_{p}

is the Planck length [20,21]:

l_{p} = \sqrt{\frac{G ℏ}{c^{3}}}

. That is, we have simply solved the Planck length formula for G, which gives

G = \frac{l_{p}^{2} c^{3}}{ℏ}

and replaced G in Einstein’s field equation with this. Solving the Planck units for G was suggested at least as early as 1984 by Cahill [22,23]. However, Cohen [24] in 1987 correctly pointed out that this leads to a circular problem, as no one had found a way to determine the Planck length independent of first knowing G. This view has been reiterated at least up to 2016; see, for example, [25]. However, in 2017, Haug [26] was able to demonstrate that one could find the Planck length independently of knowing G, and further, without knowledge of G and ℏ; see [27,28]. Recently, it has also been demonstrated how one can find the Planck length from the Hubble constant and the observed cosmological redshift by the formula:

l_{p} = \frac{H_{0}}{T_{c m b}^{2}} \frac{ℏ^{2} c}{k_{b} 32 π^{2}}

(6)

Haug [29] has recently demonstrated how one can also extract the Planck length from the observed redshift of 580 supernovas in the Union2 database. Additionally, it is important to note that any kilogram mass can be expressed by simply solving the Compton [30] wavelength formula:

\bar{λ} = \frac{h}{m c}

. With respect to M, this gives:

M = \frac{h}{λ} \frac{1}{c} = \frac{ℏ}{λ} \frac{1}{c}

(7)

where

\bar{λ}

is the reduced Compton wavelength. Some may claim that such a formula can only be used for electrons; however, this is not correct. The formula can be applied to any kilogram mass, and the reduced Compton wavelength can be determined for any mass. What is true is that likely only elementary particles have a physical Compton wavelength. The Compton wavelength of composite masses, however, can be seen as an aggregate of the Compton wavelengths of all the elementary particles making up the composite mass, as discussed, for example, in [31], see also [32,33].

By simply replacing G with

G = \frac{l_{p}^{2} c^{3}}{ℏ}

and M with

M = \frac{ℏ}{\bar{λ}} \frac{1}{c}

the Schwarzschild [34] metric can be rewritten as:

\begin{matrix} d s^{2} & = & - (1 - \frac{2 G M}{c^{2} r}) c^{2} d t^{2} + {(1 - \frac{2 G M}{r c^{2}})}^{- 1} d r^{2} - r^{2} d Ω^{2} \\ d s^{2} & = & - (1 - \frac{2 l_{p}}{r} \frac{l_{p}}{{\bar{λ}}_{M}}) c^{2} d t^{2} + {(1 - \frac{2 l_{p}}{r} \frac{l_{p}}{{\bar{λ}}_{M}})}^{- 1} d r^{2} - r^{2} d Ω^{2} \end{matrix}

(8)

where the term

\frac{l_{p}}{{\bar{λ}}_{M}}

represents the reduced Compton frequency per Planck time. This, in our view, is the genuine quantization of matter and gravity, a notion supported by recent research indicating that matter indeed ’ticks’ at the Compton frequency [35,36]. .

Since we have

\frac{T_{c m b}^{2}}{T_{H w}^{2}} = \frac{l_{p}}{\bar{λ}}

this implies that the Schwarzschild metric can also be expressed as:

\begin{matrix} d s^{2} & = & - (1 - \frac{2 l_{p}}{r} \frac{T_{c m b}^{2}}{T_{H w}^{2}}) c^{2} d t^{2} + {(1 - \frac{2 l_{p}}{r} \frac{T_{c m b}^{2}}{T_{H w}^{2}})}^{- 1} d r^{2} - r^{2} d Ω^{2} \end{matrix}

(9)

This is what we will call the thermodynamical version of the Schwarzschild metric. In the extreme solution of the Reissner-Nordström [37,38] (RN), as well as in the minimal solution of the Haug-Spavieri [39] metric, which is mathematically identical to the RN extremal solution, one can rewrite the metric as:

\begin{matrix} d s^{2} & = & - (1 - \frac{2 G M}{c^{2} r} + \frac{G^{2} M^{2}}{c^{4} r^{2}}) c^{2} d t^{2} + {(1 - \frac{2 G M}{r c^{2}} + \frac{G^{2} M^{2}}{c^{4} r^{2}})}^{- 1} d r^{2} - r^{2} d Ω^{2} \\ d s^{2} & = & - (1 - \frac{2 l_{p}}{r} \frac{l_{p}}{{\bar{λ}}_{M}} + \frac{l_{p}^{2}}{r^{2}} \frac{l_{p}^{2}}{{\bar{λ}}_{M}^{2}}) c^{2} d t^{2} + {(1 - \frac{2 l_{p}}{r} \frac{l_{p}}{{\bar{λ}}_{M}} + \frac{l_{p}^{2}}{r^{2}} \frac{l_{p}^{2}}{{\bar{λ}}_{M}^{2}})}^{- 1} d r^{2} - r^{2} d Ω^{2} \end{matrix}

(10)

This suggests that the extremal solution of the Reissner-Nordström metric and the minimal solution of the Haug-Spavieri metric can be expressed in the following thermodynamic form:

\begin{matrix} d s^{2} & = & - (1 - \frac{2 l_{p}}{r} \frac{T_{c m b}^{2}}{T_{H w}^{2}} + \frac{l_{p}^{2}}{r^{2}} \frac{T_{c m b}^{4}}{T_{H w}^{4}}) c^{2} d t^{2} \\ + {(1 - \frac{2 l_{p}}{r} \frac{T_{c m b}^{2}}{T_{H w}^{2}} + \frac{l_{p}^{2}}{r^{2}} \frac{T_{c m b}^{4}}{T_{H W}^{4}})}^{- 1} d r^{2} - r^{2} d Ω^{2} \end{matrix}

(11)

Table 1 shows predictions derived from the Schwarzschild metric written in their standard form as well as in their thermodynamical form. The thermodynamical form is valid as long as the Hawking temperature and CMB temperature are calculated using the hypothetical CMB and Hawking temperature of the equivalent black hole of the gravitational mass, so in this way it can be used for any gravitational mass.

Table 2 provides a deeper level of the gravity formulas in their quantized form. Once more, the term

\frac{l_{p}}{\bar{λ}}

represents the reduced Compton frequency per Planck time.

2. Conclusion

We have demonstrated that metrics such as the Schwarzschild metric, the extremal solution of the Reissner-Nordström metric, and the minimal solution of the Haug-Spavieri metric can be expressed in what we can call thermodynamic forms. The term

\frac{T_{CMB}^{2}}{T_{Hw}^{2}}

, which represents the ratio of the squared CMB temperature to the squared Hawking temperature (or the geometric mean internal temperature of a black hole), gives the reduced Compton frequency per Planck time in the gravitational object. This, in our view, is a central element for quantization in gravity.

References

S. Hawking. Black hole explosions. Nature, 248, 1974. [CrossRef]
S. Hawking. Black holes and thermodynamics. Physical Review D, 13(2):191, 1976. [CrossRef]
E. T. Tatum, U. V. S. Seshavatharam, and S. Lakshminarayana. The basics of flat space cosmology. International Journal of Astronomy and Astrophysics, 5:116, 2015. 2015. [CrossRef]
A. Sneppen, D. Watson, D. Poznanski, O. Just, A. Bauswein, and R. Wojtak. Measuring the Hubble constant with kilonovae using the expanding photosphere method. Astronomy and Astrophysics, 678, 2023. 2023. [CrossRef]
E. G. Haug. CMB, Hawking, Planck, and Hubble scale relations consistent with recent quantization of general relativity theory. International Journal of Theoretical Physics, Nature-Springer, 63(57), 2024a. [CrossRef]
E. G. Haug and S. Wojnow. The blackbody CMB temperature, luminosity, and their relation to black hole cosmology, compared to Bekenstein-Hawking luminosity. Hal archive, 2024.
E. G. Haug and E. T. Tatum. The Hawking Hubble temperature as a minimum temperature, the Planck temperature as a maximum temperature and the CMB temperature as their geometric mean temperature. Hal archive, 2023.
R. K. Pathria. The universe as a black hole. Nature, 240:298, 1972. [CrossRef]
P. Christillin. The Machian origin of linear inertial forces from our gravitationally radiating black hole universe. The European Physical Journal Plus, 129:175, 2014. [CrossRef]
T. X. Zhang and C. Frederick. Acceleration of black hole universe. Astrophysics and Space Science, 349:567, 2014. [CrossRef]
N. Popławski. The universe in a black hole in Einstein–Cartan gravity. The Astrophysical Journal, 832:96, 2016. [CrossRef]
T. X. Zhang. The principles and laws of black hole universe. Journal of Modern Physics, 9:1838, 2018. [CrossRef]
D. A. Easson and R. H. Brandenberger. Universe generation from black hole interiors. Journal of High Energy Physics, 2001, 2001. [CrossRef]
E. Gaztanaga. The black hole universe, part i. Symmetry, 2022:1849, 2022. [CrossRef]
Z. Roupas. Detectable universes inside regular black holes. The European Physical Journal C, 82:255, 2022. [CrossRef]
E. Siegel. Are we living in a baby universe that looks like a black hole to outsiders? Hard Science, Big Think, January 27, 2022.
C. H. Lineweaver and V. M. Patel. All objects and some questions. American Journal of Physics, 91(819), 2023.
E. G. Haug. Different mass definitions and their pluses and minuses related to gravity. Foundations, 3:199–219., 2023. [CrossRef]
A. Einstein. Näherungsweise integration der feldgleichungen der gravitation. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin, 1916.
M. Planck. Natuerliche Masseinheiten. Der Königlich Preussischen Akademie Der Wissenschaften: Berlin, Germany, 1899. URL https://www.biodiversitylibrary.org/item/93034#page/7/mode/1up. /.
M. Planck. Vorlesungen über die Theorie der Wärmestrahlung. Leipzig: J.A. Barth, p. 163, see also the English translation “The Theory of Radiation" (1959) Dover, 1906.
K. Cahill. The gravitational constant. Lettere al Nuovo Cimento, 39:181, 1984a. [CrossRef]
K. Cahill. Tetrads, broken symmetries, and the gravitational constant. Zeitschrift Für Physik C Particles and Fields, 23:353, 1984b. [CrossRef]
E. R. Cohen. Fundamental Physical Constants, in the book Gravitational Measurements, Fundamental Metrology and Constants. Edited by Sabbata, and Melniko, V. N., Netherland, Amsterdam, Kluwer Academic Publishers, p 59, 1987.
M. E. McCulloch. Quantised inertia from relativity and the uncertainty principle. Europhysics Letters (EPL), 115(6):69001, 2016. [CrossRef]
E. G. Haug. Can the Planck length be found independent of big g ? Applied Physics Research, 9(6):58, 2017. [CrossRef]
E. G. Haug. Finding the Planck length multiplied by the speed of light without any knowledge of g, c, or h, using a newton force spring. Journal Physics Communication, 4:075001, 2020. [CrossRef]
E. G. Haug. Planck units measured totally independently of big g. Open Journal of Microphysics, 12:55, 2022a. [CrossRef]
E. G. Haug. Extracting the Planck units as well as the Compton frequency from 580 type ia supernovae redshifts consistent with quantized general relativity theory. Research gate March 9 version, 2024b. 9 March. [CrossRef]
A. H. Compton. A quantum theory of the scattering of x-rays by light elements. Physical Review, 21(5):483, 1923. [CrossRef]
E. G. Haug. Progress in the composite view of the newton gravitational constant and its link to the Planck scale. Universe, 8(454), 2022b. [CrossRef]
L.S. Levitt. The proton Compton wavelength as the `quantum’ of length. Experientia, 14:233, 1958. [CrossRef]
O. L. Trinhammer and H. G. Bohr. On proton charge radius definition. EPL, 128:21001, 2019. [CrossRef]
K. Schwarzschild. über das gravitationsfeld einer kugel aus inkompressibler flussigkeit nach der einsteinschen theorie. Sitzungsberichte der Deutschen Akademie der Wissenschaften zu Berlin, Klasse fur Mathematik, Physik, und Technik, page 424, 1916.
S. Lan, P. Kuan, B. Estey, D. English, J. M. Brown, M. A. Hohensee, and Müller. A clock directly linking time to a particle’s mass. Science, 339:554, 2013. [CrossRef]
D. Dolce and A. Perali. On the Compton clock and the undulatory nature of particle mass in graphene systems. The European Physical Journal Plus, 130(41):41, 2015. [CrossRef]
H. Reissner. Über die eigengravitation des elektrischen feldes nach der einsteinschen theorie. Annalen der Physics, 355:106, 1916. [CrossRef]
G. Nordström. On the energy of the gravitation field in Einstein’s theory. Koninklijke Nederlandsche Akademie van Wetenschappen Proceedings, 20:1238, 1918. 1918.
E. G. Haug and G. Spavieri. Mass-charge metric in curved spacetime. International Journal of Theoretical Physics, 62:248, 2023. [CrossRef]

Table 1. The table displays a series of gravity predictions derived from general relativity theory, presented in both their conventional form and their thermodynamic form. The thermodynamic form is applicable when the gravitational object is a Schwarzschild black hole.

Prediction	Formula:
Gravity acceleration	$g = \frac{G M}{R^{2}} = \frac{c^{2} l_{p}}{R^{2}} \frac{T_{c m b}^{2}}{T_{H w}^{2}}$
Orbital velocity	$v_{o} = \sqrt{\frac{G M}{R}} = c \frac{T_{c m b}}{T_{H w}} \sqrt{\frac{l_{p}}{R}}$
Orbital time	$T = \frac{2 π R}{\sqrt{\frac{G M}{R}}} = \frac{2 π R}{\frac{T_{c m b}}{T_{H w}} c \sqrt{\frac{l_{p}}{R}}}$
Gravitational red shift	$z = \frac{\sqrt{1 - \frac{2 G M}{R_{1} c^{2}}}}{\sqrt{1 - \frac{2 G M}{R_{2} c^{2}}}} - 1 = \frac{\sqrt{1 - \frac{2 l_{p}}{R_{1}} \frac{T_{c m b}^{2}}{T_{H w}^{2}}}}{\sqrt{1 - \frac{2 l_{p}}{R_{2}} \frac{T_{c m b}^{2}}{T_{H w}^{2}}}} - 1$
Time dilation	$T_{R} = T_{f} \sqrt{1 - \frac{2 G M}{R c^{2}}} = T_{f} \sqrt{1 - \frac{2 l_{p}}{R} \frac{T_{c m b}^{2}}{T_{H w}^{2}}}$
Gravitational deflection	$θ = \frac{4 G M}{c^{2} R} = 4 \frac{l_{p}}{R} \frac{T_{c m b}^{2}}{T_{H w}^{2}}$
Schwarzschild radius	$R_{s} = \frac{2 G M}{c^{2}} = 2 l_{p} \frac{T_{c m b}^{2}}{T_{H w}^{2}}$

Table 2. The table displays a series of gravity predictions derived from general relativity theory, presented in their conventional form, as well as in the new quantized formulation of the field equation. Each formula includes the term

\frac{l_{p}}{\bar{λ}}

, which represents the reduced Compton frequency in the mass M per Planck time.

\frac{l_{p}}{\bar{λ}}

, which represents the reduced Compton frequency in the mass M per Planck time.

Prediction	Formula:
Gravity acceleration	$g = \frac{G M}{R^{2}} = \frac{c^{2} l_{p}}{R^{2}} \frac{l_{p}}{{\bar{λ}}_{M}}$
Orbital velocity	$v_{o} = \sqrt{\frac{G M}{R}} = c \sqrt{\frac{l_{p}}{R} \frac{l_{p}}{{\bar{λ}}_{M}}}$
Orbital time	$T = \frac{2 π R}{\sqrt{\frac{G M}{R}}} = \frac{2 π R}{c \sqrt{\frac{}{}} l_{p} R \frac{l_{p}}{{\bar{λ}}_{M}}}$
Gravitational red shift	$z = \frac{\sqrt{1 - \frac{2 G M}{R_{1} c^{2}}}}{\sqrt{1 - \frac{2 G M}{R_{2} c^{2}}}} - 1 = \frac{\sqrt{1 - \frac{2 l_{p}}{R_{1}} \frac{l_{p}}{{\bar{λ}}_{M}}}}{\sqrt{1 - \frac{2 l_{p}}{R_{2}} \frac{l_{p}}{{\bar{λ}}_{M}}}} - 1$
Time dilation	$T_{R} = T_{f} \sqrt{1 - \frac{2 G M}{R c^{2}}} = T_{f} \sqrt{1 - \frac{2 l_{p}}{R} \frac{l_{p}}{{\bar{λ}}_{M}}}$
Gravitational deflection	$θ = \frac{4 G M}{c^{2} R} = 4 \frac{l_{p}}{R} \frac{l_{p}}{{\bar{λ}}_{M}}$
Schwarzschild radius	$R_{s} = \frac{2 G M}{c^{2}} = 2 l_{p} \frac{l_{p}}{\bar{λ}}$

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

MDPI Initiatives

Important Links

Choose an area of interest and we will send you notifications of new preprints at your preferred frequency.

Disclaimer

Thermodynamical Metrics in General Relativity Theory

Abstract

1. Background on the Hawking Temperature and the New CMB Temperature Formula

2. Conclusion

References

MDPI Initiatives

Important Links

Subscribe