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Derivation

Tolman-Oppenheimer-Volkoff Equation

D-336 Home PU-308 Threads fields · force · matter · energy Depends on einstein-field-equations, schwarzschild-metric, Equation of Stellar Hydrostatic Equilibrium, stress-energy-perfect-fluid
Statement

For a static, spherically symmetric star made of a perfect fluid, solving the Einstein field equations \(G_{\mu\nu}=\dfrac{8\pi G}{c^4}T_{\mu\nu}\) with the interior metric \(ds^2=-e^{2\Phi(r)}c^2dt^2+e^{2\Lambda(r)}dr^2+r^2d\Omega^2\) reduces to the Tolman–Oppenheimer–Volkoff (TOV) equation for hydrostatic equilibrium, \(\dfrac{dP}{dr}=-\dfrac{G\left(\rho+P/c^2\right)\left(m+4\pi r^3 P/c^2\right)}{r^2\left(1-\dfrac{2Gm}{rc^2}\right)}\), together with the mass-continuity relation \(\dfrac{dm}{dr}=4\pi r^2\rho\), where \(\rho(r)\) is the total mass–energy density, \(P(r)\) the isotropic pressure, and \(m(r)\) the mass–energy enclosed within areal radius \(r\). It is the relativistic generalisation of Newtonian stellar hydrostatic equilibrium and governs the structure of white dwarfs, neutron stars and other compact objects.

Why it matters

The TOV equation is the master equation of relativistic stellar structure. Every calculation of a neutron-star mass–radius relation, of the maximum mass a cold dense star can support before collapsing to a black hole, and of the constraints that gravitational-wave and X-ray observations place on the ultra-dense equation of state, begins by integrating this single first-order system from the centre outward.

Physically it exposes why gravity in general relativity is self-strengthening: pressure, being a form of energy, itself gravitates. All three relativistic corrections in the equation — the terms \(P/c^2\), \(4\pi r^3P/c^2\) and the metric factor \((1-2Gm/rc^2)^{-1}\) — act to increase the inward pull relative to Newton. Pressure can no longer be raised without limit to fight gravity, and this is precisely what produces a maximum mass and the inevitability of gravitational collapse.

Assumptions
Static and spherically symmetric: all fields depend only on the areal radius \(r\), with no time dependence and no rotation, so the metric takes the form \(ds^2=-e^{2\Phi(r)}c^2dt^2+e^{2\Lambda(r)}dr^2+r^2d\Omega^2\).If dropped, the metric acquires \(g_{t\phi}\) (rotation) or time dependence (pulsation/collapse); frame dragging and gravitational radiation appear, and equilibrium is described by the far harder stationary-axisymmetric or fully dynamical Einstein equations, not a single radial ODE.
Perfect fluid: the stress–energy tensor is \(T^{\mu\nu}=\left(\rho+P/c^2\right)u^\mu u^\nu+P g^{\mu\nu}\) with isotropic pressure and no shear, viscosity or heat flux.If dropped, anisotropic stresses \(P_r\neq P_\perp\) (from strong magnetic fields, solid crusts or superfluid tensions) add a term \(2(P_\perp-P_r)/r\) to the balance, and the diagonal-\(T^{\mu\nu}\) reduction fails.
A barotropic equation of state closes the system: \(P=P(\rho)\) is supplied by microphysics.If dropped, the two structure equations contain three unknowns \((\rho,P,m)\) and cannot be integrated; the entire mass–radius relation is a functional of the assumed \(P(\rho)\).
The enclosed mass is defined by the areal radius via \(e^{-2\Lambda}=1-2Gm(r)/rc^2\): this identifies \(m(r)\) as the Misner–Sharp / Schwarzschild mass that matches continuously onto the exterior Schwarzschild solution at the surface.If dropped (e.g. using proper volume \(\int 4\pi r^2 e^{\Lambda}dr\) instead), \(m(r)\) no longer equals the gravitating mass seen from infinity and the exterior match at \(r=R\) is inconsistent; the difference \(M-\int\rho\,dV_{\text{proper}}\) is the gravitational binding energy.
Local energy conditions hold and \(\rho,P\ge0\) with \(1-2Gm/rc^2>0\) throughout: the fluid is non-exotic and no horizon forms inside the star.If dropped, the metric factor changes sign (a trapped surface / horizon forms), \(dP/dr\) diverges, and no static solution exists — the configuration must collapse.
Derivation
1
\[ ds^2=-e^{2\Phi(r)}c^2dt^2+e^{2\Lambda(r)}dr^2+r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right). \]
The most general static, spherically symmetric line element, written with two unknown radial functions \(\Phi(r)\) and \(\Lambda(r)\); the angular part is fixed to \(r^2d\Omega^2\) by defining \(r\) as the areal radius (a 2-sphere at \(r\) has area \(4\pi r^2\)). This is the interior counterpart of the prior result schwarzschild-metric. A
2
\[ u^\mu=\left(c\,e^{-\Phi},0,0,0\right),\qquad T^{\mu}{}_{\nu}=\mathrm{diag}\!\left(-\rho c^2,\;P,\;P,\;P\right). \]
For a fluid at rest the four-velocity has only a time component, normalised by \(g_{\mu\nu}u^\mu u^\nu=-c^2\Rightarrow u^t=c\,e^{-\Phi}\). Inserting this into the perfect-fluid tensor (prior result stress-energy-perfect-fluid) gives the diagonal mixed components: energy density \(\rho c^2\) and isotropic pressure \(P\). B
3
\[ G^{t}{}_{t}=-\frac{1}{r^2}\frac{d}{dr}\!\left[r\left(1-e^{-2\Lambda}\right)\right]=\frac{8\pi G}{c^4}T^{t}{}_{t}=-\frac{8\pi G}{c^2}\rho. \]
The \(t\,t\)-component of the Einstein tensor for this metric depends only on \(\Lambda\). Equating it to \(\tfrac{8\pi G}{c^4}T^{t}{}_{t}\) (the prior result einstein-field-equations) isolates the mass function; note the \(t\,t\) equation does not involve \(\Phi\) at all. C
4
\[ \text{Define }\;e^{-2\Lambda}\equiv1-\frac{2Gm(r)}{rc^2}\;\Longrightarrow\; \frac{d}{dr}\!\left[\frac{2Gm}{c^2}\right]=\frac{8\pi G}{c^2}\rho\,r^2\;\Longrightarrow\;\boxed{\ \frac{dm}{dr}=4\pi r^2\rho\ } \]
Substituting the definition into Step 3 turns the \(t\,t\)-equation into the mass-continuity relation. This choice makes \(m(r)\) the mass–energy enclosed within \(r\) and forces continuity with the exterior Schwarzschild metric, where \(e^{-2\Lambda}=1-2GM/rc^2\). B
5
\[ G^{r}{}_{r}=\frac{1}{r^2}\left(2r\,e^{-2\Lambda}\,\Phi'-\left(1-e^{-2\Lambda}\right)\right)\cdot\frac{1}{r}=\frac{8\pi G}{c^4}P \;\Longrightarrow\; \frac{d\Phi}{dr}=\frac{G\left(m+4\pi r^3P/c^2\right)}{r^2c^2\left(1-\dfrac{2Gm}{rc^2}\right)}. \]
The \(r\,r\)-component of Einstein's equations relates \(\Phi'\) to \(P\) and \(m\). Solving for \(\Phi'\) and eliminating \(e^{-2\Lambda}\) with Step 4 gives the "effective gravitational field" of GR: the Newtonian \(Gm/r^2c^2\) is augmented by the pressure source \(4\pi r^3P/c^2\) and divided by the metric redshift factor. C
6
\[ \nabla_\mu T^{\mu r}=0\;\Longrightarrow\; \frac{dP}{dr}=-\left(\rho c^2+P\right)\frac{d\Phi}{dr}=-\left(\rho+\frac{P}{c^2}\right)c^2\,\frac{d\Phi}{dr}. \]
Local energy–momentum conservation (\(\nabla_\mu T^{\mu\nu}=0\), guaranteed by the contracted Bianchi identity applied to the field equations) gives the relativistic Euler equation for a static fluid. The inertia of the pressure appears as the enthalpy density \(\rho c^2+P\) rather than \(\rho c^2\) alone — this is the relativistic statement that pressure has weight. C
7
\[ \frac{dP}{dr}=-\left(\rho+\frac{P}{c^2}\right)c^2\cdot\frac{G\left(m+4\pi r^3P/c^2\right)}{r^2c^2\left(1-\dfrac{2Gm}{rc^2}\right)}=-\frac{G\left(\rho+\dfrac{P}{c^2}\right)\left(m+\dfrac{4\pi r^3P}{c^2}\right)}{r^2\left(1-\dfrac{2Gm}{rc^2}\right)}. \]
Substitute \(\Phi'\) from Step 5 into the Euler equation of Step 6; the factors of \(c^2\) cancel. This is the TOV equation — symbols only, no numbers yet. Together with \(dm/dr=4\pi r^2\rho\) and an equation of state \(P(\rho)\), it forms a closed first-order system. B
Result
\[ \frac{dP}{dr}=-\frac{G\left(\rho+\dfrac{P}{c^2}\right)\left(m+\dfrac{4\pi r^3P}{c^2}\right)}{r^2\left(1-\dfrac{2Gm}{rc^2}\right)},\qquad \frac{dm}{dr}=4\pi r^2\rho. \]

Reading. The left side is the pressure gradient needed to hold the star up against gravity; the right side is that gravitational pull. Three factors distinguish it from Newton's \(-Gm\rho/r^2\), each making gravity stronger: (i) \(\rho\to\rho+P/c^2\) — pressure contributes inertia and weight; (ii) \(m\to m+4\pi r^3P/c^2\) — pressure integrated over the interior acts as an additional gravitating source; (iii) the denominator \((1-2Gm/rc^2)^{-1}>1\) — spacetime curvature amplifies the field. Integrating inward from the surface (\(P(R)=0\)) with a chosen central density fixes the star's radius \(R\) and total mass \(M=m(R)\); scanning central densities traces the mass–radius curve.

Units check. \(dP/dr\) has SI units \(\mathrm{Pa\,m^{-1}}=\mathrm{kg\,m^{-2}s^{-2}}\). On the right, \(P/c^2=\mathrm{Pa}/(\mathrm{m^2s^{-2}})=\mathrm{kg\,m^{-3}}\) adds correctly to \(\rho\); \(4\pi r^3P/c^2=\mathrm{m^3}\cdot\mathrm{kg\,m^{-3}}=\mathrm{kg}\) adds to \(m\); \(2Gm/rc^2\) is dimensionless. The whole right side reads \(\dfrac{[\mathrm{m^3kg^{-1}s^{-2}}]\,[\mathrm{kg\,m^{-3}}]\,[\mathrm{kg}]}{[\mathrm{m^2}]}=\mathrm{kg\,m^{-2}s^{-2}}\), matching the left. Consistent.

Limiting cases
  • Newtonian limit \(P\ll\rho c^2\), \(4\pi r^3P\ll mc^2\), \(2Gm\ll rc^2\): all three corrections \(\to1\) and TOV collapses to the classical hydrostatic-equilibrium equation \(\dfrac{dP}{dr}=-\dfrac{Gm\rho}{r^2}\) (prior result hydrostatic-equilibrium-stellar).
  • Weak-field, leading relativistic correction: \(\dfrac{dP}{dr}\approx-\dfrac{Gm\rho}{r^2}\left(1+\dfrac{P}{\rho c^2}\right)\left(1+\dfrac{4\pi r^3P}{mc^2}\right)\left(1+\dfrac{2Gm}{rc^2}\right)\) — the three multiplicative enhancements to Newtonian gravity, each of order the compactness \(2GM/Rc^2\).
  • Uniform density (Schwarzschild interior): \(\rho=\text{const}\) integrates exactly to the central pressure \(P_c=\rho c^2\,\dfrac{1-\sqrt{1-2GM/Rc^2}}{3\sqrt{1-2GM/Rc^2}-1}\), which diverges as the compactness approaches \(8/9\).
  • Photon gas / ultrarelativistic core: with \(P=\tfrac13\rho c^2\) the enthalpy factor becomes \(\rho+P/c^2=\tfrac43\rho\), further stiffening the effective source and lowering the maximum mass relative to a naive \(P\)-neglecting estimate.
  • Buchdahl / near-collapse: as \(2GM/Rc^2\to8/9\) any static perfect-fluid star (regardless of equation of state, for \(\rho\) non-increasing outward) needs infinite central pressure — the Buchdahl bound \(R>\tfrac{9}{4}\dfrac{GM}{c^2}\).
Breaks when
  • Rotation or magnetic stress is significant. Millisecond pulsars and magnetars carry angular momentum and \(\sim10^{15}\,\mathrm{G}\) fields; these break spherical symmetry and isotropy, so the star must be modelled with the stationary-axisymmetric Einstein equations, not the radial TOV ODE.
  • The compactness reaches the Buchdahl limit \(2GM/Rc^2\to8/9\). The metric factor \((1-2Gm/rc^2)\) and the pressure gradient force \(P_c\to\infty\); no static solution exists and the object must undergo gravitational collapse to a black hole.
  • Time dependence: collapse, oscillation, or accretion. During supernova core collapse, stellar pulsation, or rapid accretion the assumption \(\partial_t=0\) fails; one needs the full dynamical Einstein equations and hydrodynamics, and gravitational waves can be radiated.
  • Anisotropic or non-fluid matter. A solid crust, superfluid vortices, or a strongly quantised field give \(P_r\neq P_\perp\); the balance acquires a \(2(P_\perp-P_r)/r\) term and the perfect-fluid TOV equation no longer applies.
Failure modes
  • Using rest-mass (baryon) density for \(\rho\). In TOV \(\rho\) is the total mass–energy density (rest mass plus internal energy); using only baryon rest mass omits the internal-energy contribution to gravity and mis-sizes the star.
  • Dropping one of the three relativistic terms. Keeping \((1-2Gm/rc^2)^{-1}\) but discarding \(4\pi r^3P/c^2\), or vice versa, is inconsistent — all three enter at the same order in compactness and each strengthens gravity.
  • Treating \(m(r)\) as \(\int\rho\,dV_{\text{proper}}\). The TOV mass uses the flat coordinate volume \(4\pi r^2dr\), not the proper volume \(4\pi r^2e^{\Lambda}dr\); the difference is the (negative) gravitational binding energy, so conflating them double-counts curvature.
  • Integrating outward from \(r=0\) without the regular boundary conditions. One must impose \(m(0)=0\) and \(dP/dr\to0\) at the centre; starting with \(m\neq0\) at \(r=0\) produces a spurious central singularity in \(dP/dr\propto1/r^2\).
  • Assuming pressure always helps support the star. Because pressure gravitates, adding pressure past a point increases the right-hand side faster than the left — the origin of the maximum mass, not merely a stiffer support.
  • Forgetting the surface condition. The stellar radius \(R\) is defined by \(P(R)=0\); stopping the integration at fixed \(r\) or at \(\rho=0\)-only, without tracking where pressure vanishes, gives the wrong \(R\) and \(M\).
Discussion

The deepest lesson of the TOV equation is that in general relativity pressure weighs. In Newtonian gravity only mass density sources the field, and one can in principle always increase pressure to hold up any amount of matter. Relativity closes this escape route three times over: pressure adds to the inertia of each fluid element (\(\rho\to\rho+P/c^2\)), pressure integrated over the interior adds to the gravitating mass (\(m\to m+4\pi r^3P/c^2\)), and the curvature of space near a dense body amplifies the pull (\((1-2Gm/rc^2)^{-1}\)). The feedback is self-reinforcing, and it is the physical origin of the maximum neutron-star mass and of gravitational collapse: past a critical central density, stiffening the equation of state cannot win the race against the gravity that the extra pressure itself creates.

Structurally, the TOV system is the general-relativistic descendant of Newtonian stellar structure. Setting all \(c^{-2}\) terms to zero recovers \(dP/dr=-Gm\rho/r^2\) and \(dm/dr=4\pi r^2\rho\) exactly, so a white dwarf — where relativistic gravity corrections are tiny even though the electrons are relativistic — is well described by the Newtonian equations plus a relativistic equation of state, while a neutron star, with surface compactness \(2GM/Rc^2\sim0.3\!-\!0.5\), genuinely requires the full TOV treatment. The distinction between "relativistic matter" and "relativistic gravity" is essential and often blurred.

The quantity \(m(r)\) defined by \(e^{-2\Lambda}=1-2Gm/rc^2\) is the Misner–Sharp mass, and its relation to the locally measured energy density is subtle: \(M=\int_0^R4\pi r^2\rho\,dr\) uses the coordinate volume, whereas the proper volume is larger by the factor \(e^{\Lambda}=(1-2Gm/rc^2)^{-1/2}\). The difference \(M-\int 4\pi r^2\rho\,e^{\Lambda}dr<0\) is precisely the gravitational binding energy released in forming the star — for a neutron star of order \(0.1Mc^2\), comparable to the energy radiated in a supernova and its neutrino burst. The exterior match at \(r=R\), where the interior metric must join the Schwarzschild solution with mass \(M=m(R)\), enforces continuity of \(\Phi\) via \(e^{2\Phi(R)}=1-2GM/Rc^2\), fixing the surface redshift that observers measure in neutron-star spectra.

Common misconceptions. First, the TOV equation is not simply "Newtonian hydrostatic equilibrium with a relativistic gas" — the corrections are to gravity itself, independent of what makes the pressure. Second, the metric factor \((1-2Gm/rc^2)\) being positive inside the star does not mean an event horizon "almost" forms at \(r=2Gm/c^2\); inside matter \(m(r)<M\) grows with \(r\) and the surface always sits outside \(2GM/c^2\) (indeed outside \(\tfrac{9}{4}GM/c^2\) by Buchdahl). Third, a larger central density does not always give a more massive star: beyond the maximum-mass turning point, higher central density yields lower mass and unstable configurations that collapse.

Worked examples

Example 1 — Surface compactness of a canonical neutron star. Estimate the dimensionless metric factor \(2GM/Rc^2\) at the surface of a neutron star of mass \(M=1.4\,M_\odot\) and radius \(R=12\,\mathrm{km}\), and hence gauge how far it departs from Newtonian gravity. Take \(G=6.674\times10^{-11}\,\mathrm{m^3kg^{-1}s^{-2}}\), \(c=2.998\times10^{8}\,\mathrm{m\,s^{-1}}\), \(M_\odot=1.989\times10^{30}\,\mathrm{kg}\).

1
\[ \chi\equiv\frac{2GM}{Rc^2}\qquad(\text{surface compactness, dimensionless}). \]
Define the compactness — the coefficient appearing in the TOV denominator evaluated at the surface, and the natural measure of "how relativistic" the star is. Symbol first. A
2
\[ M=1.4\times1.989\times10^{30}=2.785\times10^{30}\,\mathrm{kg},\qquad c^2=8.988\times10^{16}\,\mathrm{m^2s^{-2}}. \]
Convert the given quantities to SI. A
3
\[ \chi=\frac{2(6.674\times10^{-11})(2.785\times10^{30})}{(1.2\times10^{4})(8.988\times10^{16})}=\frac{3.717\times10^{20}}{1.079\times10^{21}}=0.345. \]
Substitute numbers with \(R=1.2\times10^4\,\mathrm{m}\). The denominator TOV factor is \(1-\chi=0.655\), so \((1-\chi)^{-1}=1.53\). A
\[ \frac{2GM}{Rc^2}\approx0.35,\qquad\left(1-\frac{2GM}{Rc^2}\right)^{-1}\approx1.5. \]

Reading. Gravity at the surface is roughly 50% stronger than the naive Newtonian value from the curvature factor alone — before even counting the pressure terms. A neutron star is unambiguously in the strong-field regime; Newtonian structure would be badly wrong. Units check. \(\dfrac{[\mathrm{m^3kg^{-1}s^{-2}}][\mathrm{kg}]}{[\mathrm{m}][\mathrm{m^2s^{-2}}]}=\dfrac{\mathrm{m^3\,s^{-2}}}{\mathrm{m^3\,s^{-2}}}=1\), dimensionless as required.

Example 2 — Central pressure of a uniform-density star (Schwarzschild interior solution). For a constant-density sphere the TOV equation integrates in closed form. Using the compactness \(\chi=0.345\) of Example 1 and the corresponding mean density, estimate the central pressure of the same neutron star. Use \(P_c=\rho c^2\dfrac{1-\sqrt{1-\chi}}{3\sqrt{1-\chi}-1}\).

1
\[ \rho=\frac{M}{\tfrac43\pi R^3},\qquad P_c=\rho c^2\,\frac{1-\sqrt{1-\chi}}{3\sqrt{1-\chi}-1}. \]
Mean mass–energy density and the closed-form central pressure obtained by integrating TOV with \(\rho=\text{const}\); symbols first. The formula follows from Step 7 with constant \(\rho\), \(m=\tfrac43\pi r^3\rho\). B
2
\[ \rho=\frac{2.785\times10^{30}}{\tfrac43\pi(1.2\times10^{4})^3}=\frac{2.785\times10^{30}}{7.238\times10^{12}}=3.85\times10^{17}\,\mathrm{kg\,m^{-3}}. \]
Evaluate the mean density with \(R^3=1.728\times10^{12}\,\mathrm{m^3}\). This is a few times nuclear saturation density \(\sim2.3\times10^{17}\,\mathrm{kg\,m^{-3}}\), as expected. A
3
\[ \sqrt{1-\chi}=\sqrt{0.655}=0.809,\qquad \frac{1-0.809}{3(0.809)-1}=\frac{0.191}{1.428}=0.134. \]
Compute the dimensionless bracket; note the denominator \(3\sqrt{1-\chi}-1>0\) confirms the star is below the Buchdahl limit (which would give a vanishing denominator, \(P_c\to\infty\)). B
4
\[ \rho c^2=(3.85\times10^{17})(8.988\times10^{16})=3.46\times10^{34}\,\mathrm{Pa},\quad P_c=0.134\times3.46\times10^{34}=4.6\times10^{33}\,\mathrm{Pa}. \]
Multiply the energy density by the bracket. A
\[ P_c\approx4.6\times10^{33}\,\mathrm{Pa}\quad\left(\approx0.13\,\rho c^2\right). \]

Reading. The central pressure is a substantial fraction (\(\sim13\%\)) of the central energy density — pressure is emphatically not a small correction here, confirming that the \(P/c^2\) and \(4\pi r^3P/c^2\) terms in TOV are essential. The order of magnitude \(10^{33}\!-\!10^{34}\,\mathrm{Pa}\) matches realistic neutron-star models. Units check. \(\rho c^2=[\mathrm{kg\,m^{-3}}][\mathrm{m^2s^{-2}}]=\mathrm{kg\,m^{-1}s^{-2}}=\mathrm{Pa}\); the bracket is dimensionless, so \(P_c\) is in \(\mathrm{Pa}\).

Problems
  1. Starting from the TOV equation, show explicitly that in the simultaneous limits \(P\ll\rho c^2\), \(4\pi r^3P/c^2\ll m\) and \(2Gm/rc^2\ll1\) it reduces to the Newtonian hydrostatic-equilibrium equation. Identify which physical effect each discarded term represents.
    Solution Write TOV as \(\dfrac{dP}{dr}=-\dfrac{Gm\rho}{r^2}\left(1+\dfrac{P}{\rho c^2}\right)\left(1+\dfrac{4\pi r^3P}{mc^2}\right)\left(1-\dfrac{2Gm}{rc^2}\right)^{-1}\). Under the three stated inequalities each parenthesis \(\to1\), leaving \(\dfrac{dP}{dr}=-\dfrac{Gm\rho}{r^2}\), the classical result. The discarded terms are: \(P/\rho c^2\) — pressure contributing to inertia/weight (relativistic enthalpy); \(4\pi r^3P/mc^2\) — pressure acting as an additional gravitating mass; \(2Gm/rc^2\) — spacetime curvature amplifying the field. All three strengthen gravity, so the Newtonian equation always underestimates the required pressure gradient.
  2. A neutron star has \(M=2.0\,M_\odot\) and \(R=11\,\mathrm{km}\). Compute its surface compactness \(2GM/Rc^2\) and the surface gravitational redshift \(z=(1-2GM/Rc^2)^{-1/2}-1\). (\(G=6.674\times10^{-11}\), \(c^2=8.988\times10^{16}\), \(M_\odot=1.989\times10^{30}\,\mathrm{kg}\).)
    Solution \(M=2.0\times1.989\times10^{30}=3.978\times10^{30}\,\mathrm{kg}\). \(2GM=2(6.674\times10^{-11})(3.978\times10^{30})=5.310\times10^{20}\). \(Rc^2=(1.1\times10^4)(8.988\times10^{16})=9.887\times10^{20}\). So \(2GM/Rc^2=0.537\). Then \(1-0.537=0.463\), \((0.463)^{-1/2}=1.470\), giving \(z=0.470\). A photon leaving the surface is redshifted by 47%, a large, potentially observable effect in spectral lines.
  3. For the uniform-density (Schwarzschild interior) star, the central pressure diverges at the Buchdahl limit. Using \(P_c=\rho c^2\dfrac{1-\sqrt{1-\chi}}{3\sqrt{1-\chi}-1}\), find the critical compactness \(\chi_{\max}\) and the corresponding minimum radius \(R_{\min}\) in terms of \(GM/c^2\).
    Solution \(P_c\to\infty\) when the denominator vanishes: \(3\sqrt{1-\chi}-1=0\Rightarrow\sqrt{1-\chi}=\tfrac13\Rightarrow1-\chi=\tfrac19\Rightarrow\chi_{\max}=\tfrac89\). Since \(\chi=2GM/Rc^2\), \(\dfrac{2GM}{R_{\min}c^2}=\dfrac89\Rightarrow R_{\min}=\dfrac{9}{4}\dfrac{GM}{c^2}\). This is the Buchdahl bound: no static perfect-fluid star (with density non-increasing outward) can be more compact than \(R=\tfrac94\,GM/c^2\), which lies safely outside the Schwarzschild radius \(2GM/c^2\).
  4. Estimate the fractional relativistic correction to the central pressure gradient for the Sun, treating it near its centre. Take a central density \(\rho_c\approx1.5\times10^{5}\,\mathrm{kg\,m^{-3}}\) and central pressure \(P_c\approx2.5\times10^{16}\,\mathrm{Pa}\). Evaluate the enthalpy correction \(P_c/\rho_c c^2\) and comment on whether the Sun needs TOV.
    Solution \(\rho_c c^2=(1.5\times10^5)(8.988\times10^{16})=1.35\times10^{22}\,\mathrm{Pa}\). Then \(P_c/\rho_c c^2=(2.5\times10^{16})/(1.35\times10^{22})=1.9\times10^{-6}\). The surface compactness is \(2GM_\odot/R_\odot c^2\approx4.2\times10^{-6}\). Both relativistic corrections are of order \(10^{-6}\), utterly negligible: the Sun is described to one part in a million by Newtonian hydrostatic equilibrium. TOV is needed only for compact objects (white dwarfs marginally, neutron stars essentially).
  5. Show that a photon-gas core with \(P=\tfrac13\rho c^2\) makes the relativistic enthalpy factor \(\rho+P/c^2\) equal to \(\tfrac43\rho\), and discuss qualitatively how this stiffening of the gravitational source (relative to neglecting \(P\)) affects the maximum mass a star can support.
    Solution With \(P=\tfrac13\rho c^2\), \(P/c^2=\tfrac13\rho\), so \(\rho+P/c^2=\rho+\tfrac13\rho=\tfrac43\rho\). Likewise the mass source \(m+4\pi r^3P/c^2=m+\tfrac13(4\pi r^3\rho)\) is enhanced. Both raise the right-hand side of TOV above the Newtonian value, meaning gravity is stronger than a pressure-neglecting estimate would suggest. Physically, a highly relativistic (soft-in-this-sense radiation-like) core cannot supply proportionally more support because its own pressure adds to the gravitating mass; this feedback caps the pressure's ability to resist collapse and lowers the maximum mass. It is the same mechanism (pressure gravitates) that produces the Oppenheimer–Volkoff maximum mass \(\sim0.7\,M_\odot\) for a non-interacting neutron gas and \(\sim2\!-\!2.5\,M_\odot\) for realistic stiff equations of state.