Tolman–Oppenheimer–Volkoff equation

{{short description|Equation explaining structure of a spherical body of isotropic material}} {{general relativity}}

In astrophysics, the '''Tolman–Oppenheimer–Volkoff''' ('''TOV''') '''equation''' constrains the structure of a spherically symmetric body of isotropic material which is in static gravitational equilibrium, as modeled by general relativity. The equation<ref name="ov"> {{cite journal |first1=J. R. |last1=Oppenheimer |first2=G. M. |last2=Volkoff |date=1939 |title=On Massive Neutron Cores |journal=Physical Review |volume=55 |issue=4 |pages=374–381 |doi=10.1103/PhysRev.55.374 |bibcode = 1939PhRv...55..374O }}</ref> is

:<math>\frac{dP}{dr}=-\frac{G m}{r^2}\rho\left(1+\frac{P}{\rho c^2}\right)\left(1+\frac{4\pi r^3P}{mc^2}\right)\left(1-\frac{2Gm}{rc^2}\right)^{-1}</math>

Here, <math display="inline">r</math> is a radial coordinate, and <math display="inline">\rho(r)</math> and <math display="inline">P(r)</math> are the density and pressure, respectively, of the material at radius <math display="inline">r</math>. The quantity <math display="inline">m(r)</math>, the total mass within <math display="inline">r</math>, is discussed below.

The equation is derived by solving the Einstein equations for a general time-invariant, spherically symmetric metric. For a solution to the Tolman–Oppenheimer–Volkoff equation, this metric will take the form<ref name="ov" />

:<math>ds^2=e^{\nu} c^2 \,dt^2 - \left(1-\frac{2Gm}{rc^2}\right)^{-1} \,dr^2 - r^2\left(d\theta^2 + \sin^2 \theta \,d\phi^2\right) </math>

where <math display="inline">\nu (r)</math> is determined by the constraint<ref name="ov" />

:<math>\frac{d\nu}{dr}=- \left(\frac{2}{P+\rho c^2} \right) \frac{dP}{dr} </math>

When supplemented with an equation of state, <math display="inline">F(\rho,P)=0</math>, which relates density to pressure, the Tolman–Oppenheimer–Volkoff equation completely determines the structure of a spherically symmetric body of isotropic material in equilibrium. If terms of order <math display="inline">1/c^2</math> are neglected, the Tolman–Oppenheimer–Volkoff equation becomes the Newtonian hydrostatic equation, used to find the equilibrium structure of a spherically symmetric body of isotropic material when general-relativistic corrections are not important.

If the equation is used to model a bounded sphere of material in a vacuum, the zero-pressure condition <math display="inline">P(r)=0</math> and the condition <math display="inline">e^{\nu} = 1 - 2 G m/c^2 r</math> should be imposed at the boundary. The second boundary condition is imposed so that the metric at the boundary is continuous with the unique static spherically symmetric solution to the vacuum field equations, the Schwarzschild metric:

:<math>ds^2=\left(1-\frac{2GM}{rc^2}\right) c^2 \,dt^2 - \left(1-\frac{2GM}{rc^2}\right)^{-1} \,dr^2 - r^2(d\theta^2 + \sin^2 \theta \,d\phi^2) </math>

==Total mass== Let <math display="inline">m(r)</math> be the total mass contained inside radius <math display="inline">r</math>, as measured by the gravitational field felt by a distant observer. It satisfies <math display="inline">m(0) = 0</math>.<ref name="ov" />

:<math>\frac{dm}{dr}=4 \pi r^2 \rho </math>

Here, <math display="inline">M</math> is the total mass of the object, again, as measured by the gravitational field felt by a distant observer. If the boundary is at <math display="inline">r = R</math>, continuity of the metric and the definition of <math display="inline">m(r)</math> require that

:<math>M=m(R)=\int_0^{R} 4\pi r^2 \rho \, dr</math>

Computing the mass by integrating the density of the object over its volume, on the other hand, will yield the larger value

:<math>M_1=\int_0^{R} \frac{4\pi r^2 \rho}{\sqrt{1-\frac{2Gm}{rc^2}}} \, dr</math>

The difference between these two quantities,

:<math>\Delta M=\int_0^{R} 4\pi r^2 \rho \left(1-\frac{1}\sqrt{1-\frac{2Gm}{rc^2}}\right) \, dr,</math>

will be the gravitational binding energy of the object divided by <math display="inline">c^2</math> and it is negative.

==Derivation from general relativity==

Let us assume a static, spherically symmetric perfect fluid. The metric components are similar to those for the Schwarzschild metric:<ref>{{cite book |first1=Charles W. |last1=Misner |first2=Kip S. |last2=Thorne |first3=John Archibald |last3=Wheeler |chapter=Coordinates and Metric for a Static, Spherical System |title=Gravitation |publisher=Princeton University Press |year=2017 |pages=594–595 |isbn=978-0-691-17779-3 |chapter-url=https://books.google.com/books?id=zAAuDwAAQBAJ&pg=PA594 |title-link=Gravitation (book) }}

</ref> :<math>c^2 \,d\tau^2 = g_{\mu\nu} \,dx^\mu \,dx^\nu = e^{\nu} c^2 \,dt^2 - e^{\lambda} \,dr^2 - r^2 \,d\theta^2 - r^2 \sin^2 \theta \,d\phi^2 </math>

By the perfect fluid assumption, the stress-energy tensor is diagonal (in the central spherical coordinate system), with eigenvalues of energy density and pressure: :<math>T_0^0 = \rho c^2</math> and :<math>T_i^j = - P \delta_i^j </math> Where <math display="inline">\rho(r)</math> is the fluid density and <math display="inline">P(r)</math> is the fluid pressure.

To proceed further, we solve Einstein's field equations: :<math>\frac{8 \pi G}{c^4} T_{\mu\nu} = G_{\mu\nu} </math>

Let us first consider the <math display="inline">G_{00}</math> component: :<math>\frac{8 \pi G}{c^4} \rho c^2 e^\nu = \frac{e^\nu}{r^2} \left(1 - \frac{d}{dr} [r e^{-\lambda}] \right) </math>

Integrating this expression from 0 to <math display="inline">r</math>, we obtain

:<math>e^{-\lambda} = 1 - \frac{2 Gm}{r c^2}</math>

where <math display="inline">m(r)</math> is as defined in the previous section.

Next, consider the <math display="inline">G_{11}</math> component. Explicitly, we have :<math>- \frac{8 \pi G}{c^4} P e^{\lambda} = \frac{- r \nu' + e^{\lambda} - 1}{r^2} </math> which we can simplify (using our expression for <math display="inline">e^{\lambda}</math>) to

:<math> \frac{d \nu}{d r} = \frac{1}{r}\left(1 - \frac{2 G m}{c^2 r}\right)^{-1} \left(\frac{2 G m}{c^2 r} + \frac{8 \pi G}{c^4} r^2 P\right) </math>

We obtain a second equation by demanding continuity of the stress-energy tensor: <math display="inline">\nabla_{\mu} T^{\mu}_{\,\nu} = 0</math>. Observing that <math display="inline">\partial_t \rho = \partial_t P = 0</math> (since the configuration is assumed to be static) and that <math display="inline">\partial_{\phi} P = \partial_{\theta} P = 0</math> (since the configuration is also isotropic), we obtain in particular <ref>{{cite book|last=Tolman |first=R. C. |date=1934 |title=Relativity Thermodynamics and Cosmology |publisher=Oxford Press |pages=243–244}}</ref>

:<math>0 = \nabla_\mu T^\mu_1 = - \frac{d P}{d r} - \frac12 \left(P + \rho c^2\right) \frac{d\nu}{d r} \;</math>

Rearranging terms yields:

:<math>\frac{dP}{dr} = - \left( \frac{\rho c^2 + P}{2} \right) \frac{d\nu}{dr} \;</math>

This gives us two expressions, both containing <math display="inline">d\nu/dr</math>. Eliminating <math display="inline">d\nu/dr</math>, we obtain:

:<math>\frac{dP}{dr} = - \frac{1}{r} \left( \frac{\rho c^2 + P}{2} \right) \left(\frac{2 G m}{c^2 r} + \frac{8 \pi G}{c^4} r^2 P\right) \left(1 - \frac{2 G m}{c^2 r}\right)^{-1} </math>

Pulling out a factor of <math display="inline">G/r</math> and rearranging factors of 2 and <math display="inline">c^2</math> results in the Tolman–Oppenheimer–Volkoff equation: :{|cellpadding="2" style="border:2px solid #ccccff" |<math>\frac{dP}{dr} = - \frac{G}{r^2} \left( \rho + \frac{P}{c^2} \right) \left(m + 4 \pi r^3 \frac{P}{c^2} \right) \left( 1 - \frac{2 G m}{c^2 r} \right)^{-1} </math> |}

==History== Richard C. Tolman analyzed spherically symmetric metrics in 1934 and 1939.<ref> {{cite journal |first=R. C. |last=Tolman |date=1934 |title=Effect of Inhomogeneity on Cosmological Models |journal=Proceedings of the National Academy of Sciences |volume=20 |issue=3 |pages=169–176 |doi=10.1073/pnas.20.3.169 |pmid=16587869 |pmc=1076370 |bibcode = 1934PNAS...20..169T |url=http://authors.library.caltech.edu/9466/1/TOLpnas34c.pdf|doi-access=free }}</ref><ref> {{cite journal |first=R. C. |last=Tolman |date=1939 |title=Static Solutions of Einstein's Field Equations for Spheres of Fluid |journal=Physical Review |volume=55 |issue=4 |pages=364–373 |doi=10.1103/PhysRev.55.364 |bibcode = 1939PhRv...55..364T |url=https://authors.library.caltech.edu/4362/1/TOLpr39.pdf }}</ref> The form of the equation given here was derived by J. Robert Oppenheimer and George Volkoff in their 1939 paper, "On Massive Neutron Cores".<ref name="ov" /> In this paper, the equation of state for a degenerate Fermi gas of neutrons was used to calculate an upper limit of ~0.7 solar masses for the gravitational mass of a neutron star. Since this equation of state is not realistic for a neutron star, this limiting mass is likewise incorrect. Using gravitational wave observations from binary neutron star mergers (like GW170817) and the subsequent information from electromagnetic radiation (kilonova), the data suggest that the maximum mass limit is close to 2.17 solar masses.<ref name="Margalit2017">{{cite journal|last1= Margalit|first1= B.|last2= Metzger|first2=B. D.|title= Constraining the Maximum Mass of Neutron Stars from Multi-messenger Observations of GW170817|journal= The Astrophysical Journal|volume= 850|issue= 2|date= 2017-12-01|page= L19|doi= 10.3847/2041-8213/aa991c|arxiv= 1710.05938|bibcode= 2017ApJ...850L..19M|s2cid= 119342447|doi-access= free}}</ref><ref name="Shibata2017">{{cite journal|last1= Shibata|first1= M.|last2= Fujibayashi|first2= S.|last3= Hotokezaka|first3= K.|last4= Kiuchi|first4= K.|last5= Kyutoku|first5= K.|last6= Sekiguchi|first6= Y.|last7= Tanaka|first7= M.|title= Modeling GW170817 based on numerical relativity and its implications|journal= Physical Review D|volume= 96|issue= 12|article-number= 123012|date= 2017-12-22|doi= 10.1103/PhysRevD.96.123012|arxiv= 1710.07579|bibcode= 2017PhRvD..96l3012S|s2cid= 119206732}}</ref><ref name="Ruiz2018">{{cite journal|last1= Ruiz|first1= M.|last2= Shapiro|first2=S. L.|last3= Tsokaros|first3= A.|title= GW170817, general relativistic magnetohydrodynamic simulations, and the neutron star maximum mass|journal= Physical Review D|volume= 97|issue= 2|article-number= 021501|date= 2018-01-11|doi= 10.1103/PhysRevD.97.021501|pmid= 30003183|pmc= 6036631|arxiv= 1711.00473|bibcode= 2018PhRvD..97b1501R}}</ref><ref name="Rezzolla2018">{{cite journal|last1= Rezzolla|first1= L.|last2= Most|first2=E. R.|last3= Weih|first3=L. R.|title= Using Gravitational-wave Observations and Quasi-universal Relations to Constrain the Maximum Mass of Neutron Stars|journal= Astrophysical Journal|volume= 852|issue= 2|date= 2018-01-09|pages= L25|doi= 10.3847/2041-8213/aaa401|arxiv= 1711.00314|bibcode= 2018ApJ...852L..25R|s2cid= 119359694|doi-access= free}}</ref><ref>{{cite news|title=How massive can neutron star be?|url=http://www.goethe-university-frankfurt.de/69949580/003|publisher=Goethe University Frankfurt|date=15 January 2018|access-date=19 February 2018}}</ref> Earlier estimates for this limit range from 1.5 to 3.0 solar masses.<ref> {{cite journal |first=I. |last=Bombaci |date=1996 |title=The Maximum Mass of a Neutron Star |journal=Astronomy and Astrophysics |volume=305 | pages=871–877 |bibcode=1996A&A...305..871B }}</ref>

==Post-Newtonian approximation== In the post-Newtonian approximation, i.e., gravitational fields that slightly deviates from Newtonian field, the equation can be expanded in powers of <math display="inline">1/c^2</math>. In other words, we have

:<math>\frac{dP}{dr}=-\frac{G m}{r^2}\rho\left(1+\frac{P}{\rho c^2}+\frac{4\pi r^3P}{mc^2}+\frac{2Gm}{rc^2}\right) + O(c^{-4}).</math>

==References== {{Reflist|30em}}

{{DEFAULTSORT:Tolman-Oppenheimer-Volkoff equation}} Category:Astrophysics Category:Exact solutions in general relativity Category:J. Robert Oppenheimer