Ruppeiner geometry

<h2 id="application-to-black-hole-systems">Application to black hole systems</h2>
This geometry has been applied to <a href="/facts/Black_hole_thermodynamics/LghBnFWD">black hole thermodynamics</a>, with some physically relevant results. The most physically significant case is for the <a href="/facts/Kerr_black_hole/SjL2K6R0">Kerr black hole</a> in higher dimensions, where the curvature singularity signals thermodynamic instability, as found earlier by conventional methods.
The entropy of a black hole is given by the well-known <a href="/facts/Bekenstein%25E2%2580%2593Hawking_formula/LghBnFWD">Bekenstein–Hawking formula</a>

S
        =
        
          
            
              
                k
                
                  B
                
              
              
                c
                
                  3
                
              
              A
            
            
              4
              G
              ℏ
            
          
        
      
    
    {\displaystyle S={\frac {k_{\text{B}}c^{3}A}{4G\hbar }}}

where 
 
 
 
 
 k
 
 B
 
 
 
 
 {\displaystyle k_{\text{B}}}
 
 is the <a href="/facts/Boltzmann_constant/5DIejyR3">Boltzmann constant</a>, 
 
 
 
 c
 
 
 {\displaystyle c}
 
 is the <a href="/facts/Speed_of_light/sKSQwdNx">speed of light</a>, 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is the <a href="/facts/Newtonian_constant_of_gravitation/GdQ8tcBD">Newtonian constant of gravitation</a> and 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is the area of the <a href="/facts/Event_horizon/KXH2ptnI">event horizon</a> of the black hole. Calculating the Ruppeiner geometry of the black hole's entropy is, in principle, straightforward, but it is important that the entropy should be written in terms of extensive parameters,

S
        =
        S
        (
        M
        ,
        
          N
          
            a
          
        
        )
      
    
    {\displaystyle S=S(M,N^{a})}

where 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is <a href="/facts/ADM_mass/48tVaQfr">ADM mass</a> of the black hole and Na are the conserved charges and a runs from 1 to n. The signature of the metric reflects the sign of the hole's <a href="/facts/Specific_heat/9P0U6YgI">specific heat</a>. For a <a href="/facts/Reissner%25E2%2580%2593Nordstr%25C3%25B6m_metric/K3FeAYEV">Reissner–Nordström black hole</a>, the Ruppeiner metric has a Lorentzian signature which corresponds to the negative <a href="/facts/Heat_capacity/I63nzSyp">heat capacity</a> it possess, while for the <a href="/facts/BTZ_black_hole/9u2gYja8">BTZ black hole</a>, we have a <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean</a> signature. This calculation cannot be done for the Schwarzschild black hole, because its entropy is

S
        =
        S
        (
        M
        )
      
    
    {\displaystyle S=S(M)}

which renders the metric degenerate.

<ul><li>Ruppeiner, George (1995). "Riemannian geometry in thermodynamic fluctuation theory". Reviews of Modern Physics. 67 (3): 605–659. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1995RvMP...67..605R">1995RvMP...67..605R</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2FRevModPhys.67.605">10.1103/RevModPhys.67.605</a>..</li>
<li>Åman, John E.; Bengtsson, Ingemar; Pidokrajt, Narit; Ward, John (2008). "Thermodynamic Geometries of Black Holes". The Eleventh Marcel Grossmann Meeting. pp. 1511–1513. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1142%2F9789812834300_0182">10.1142/9789812834300_0182</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-283-426-3.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Crooks, Gavin E. (2007). "Measuring Thermodynamic Length". Phys. Rev. Lett. 99 (10): 100602. arXiv:0706.0559. Bibcode:2007PhRvL..99j0602C. doi:10.1103/PhysRevLett.99.100602. PMID 17930381. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Ruppeiner geometry open-in-new

Ruppeiner geometry