© Nature Publishing Group 1977 728 matters arising General relativistic incompressibility CooPERSTOCK and Sarracino 1 have at- tempted to redefine the concept of incom- pressibility in general relativity. Their re- sult, if correct, would lead to a higher allowable maximum redshift from the sur- face of a bound object than is usually said to be permitted 2 . We point out here a serious physical difficulty associated with their work. In calculating the mass of an equili- brium configuration in general relativity, the local mass density (a scalar quantity) can always be related to the pressure through an equation of state established in an inertial reference frame, without regard to the local gravitational potential. Nonetheless, Cooperstock and Sarracino choose to define a constant proper m ass density Pproper = p(r)/ grr 112 = constant = a (say). They then compute a stellar mass from the relation r 0 M = IP 4nr 2 u 112 dr (I) proper Orr ·o where grr is the radial component of the metric tensor. This definition of the mass is correct, being simply the general relativistic expression with Pprope,grr 112 in place of p (r). As p(r) is already a scalar, however , Pprope, has no well defined physical significance. The division of the integral for mass into a product of a 'proper' volume and 'proper ' density is arbitrary and mis- leading. Cooperstock and Sarracino , in asserting that the global contribution of gravitation to the overall mass-energy of a star will affect the local stress tensor, have missed the point of the equivalence prin- ciple, the very foundation of general re- lativity. The validity of this proposition guarantees that p itself(not p/ grr 112 ) is what one would measure when applying an 'er- gometer' to a small piece of matter, whether it was inside a neutron star or in empty space. In principle, the so urce stress tensorTμ,· appearing on the right hand side of the field equations Gμ.- = kT 1 ,,. should contain all sources of mass-energy except gravitation. In practice, the gravitational binding energy of two neutrons, whether I0 28 or Io- 13 cm apart is negligible. There- fore p i_ s the physical quantity relevant for local dynamical effects in neutron stars. Nonetheless, following the procedure of Tolman 3 , one can treat the relat ion p(r) = agrr 112 as a defining equation for the radial variation of p(r) and then deduce the equa- tion of state p = p(p). In that case, the equation of hydro s tatic equilibrium is still given by 4 dp = -GM(r)p(r)[, +__!J_r) _ ]x dr r 2 p(r)c 2 x [1 +!_m _ ·,p_(_, 2 ·)] [1 -2GM( r)rc 2 ] \2) M(r)c Substituting agrr 112 for p(r) and noting that in a physically allowable object p must be positive, all the terms on the right hand side of equation (2) mu st be positive, so that the pressure decreases with radius , that is dp/dr < 0. Now the numerical results of Cooperstock and Sarracino 1 show thatp(r) decreases with increasing g" 112 , that is dp/d(g,, I ;2) < 0 But since p(r) = agrr 1 12 must be a positive quantity , dp/dp < 0. This is an unstable and unphysical situation. In or- der to have microscopic stability4, one requires dp/dp > 0. Furthermore , dividing dp / dr by dp/dp, one finds dp /dr >0. It is hard to imagine that a physically realistic star with p increasing outward can be made . Though the resulting configuration is in equilibrium since it is a solution to the general relativistic hydro stat ic equilibrium equa- tion , it is unstable . Therefore , the resulting star constructed from Cooperstock and Sarracino's definition of incompressibility in general relativity is physically unrealis- able. It seems that the redshift limitz = 2 set by Bondi 2 remains the largest allowable su rface redshift co nsistent with both gen- eral relativity and microscopic stability. K. BRECH ER Department of Physics, Massachusetts institute of Technolo gy, Cambridge, Massachusetts 02139 I. WASS ERMA N Department of Physics, Harvard University, Cambridge, Massachusetts02138 1 Coopcrstock, F. l. & Sarracino, R. S. Nu111re 264, 529 ( l 976). 1 Bondi , H . Proc. R. Soc. A282, JO) ( 1964): /,eNures on General Relarivity, Brandeis Summer Institute i11 Th,•or(!{ical PhJ' S- ics, 1964 (eds Deser, S. & l'ord, K. W.) (Prent;ce-Hall. Englewood, l 965). J Tolm<HL R. C. Fin.\ . Rn. 55, 3(,4 ( !919). Weinberg. S. Gra~·iw11011 and Cosnw/og_1· (Wiky. New York. 1972) COOPERSTOCK AND SARRACINO REPLY- For Brechcr and Wasserman, pp,ov<• would have "well defined physical significance" only if it were completely invariant. In earlier correspondence, wc had directed them to the equivalence principle, noting that this Nature Vol. 269 20 October 1977 principle renders such invariance a priori unattainable . The equivalence principle im- plies that the gravi tational field can be locally transformed away by free-fall. That is not the point. The point is that with respect to a frame at rest relative to a spherically-symmetric body, /1,m,p« cer- tainly is well defined. It assumes the form pg" - 1 ' 2 in Schwarzschild coordinates. This form has been justified by Misner and Sharp1. 2 from dynamical consider ations. We have justified the form from static considerations, and the extension has been made to charged fluid spheres (F.l.C. and R.S.S., in preparation . also. F.1.C. a nd V. de la Cruz, in preparation). In another coordinate system, say isotropic coor- dinates, it would not assume this form, but rather be determined by the integrand of the energy integral of the body over proper volume f r M(r = d V ) Pprnpi.:r prnp i.:r " All energy. including gravitational energy. contributes to th e total mass of th e body and hence p 1 ,,opm which includes all energy. is the physically relevant quantity in general relativity. The failure to recognise its ce ntral role has been perpetuated by Newtonian conditioning. On the one hand. Brechcr and Wasser- man assert that the validity of the equ ival- ence principle guarantees that p and not f/proper is what their "ergometer" will meas- ure. On the other hand , they then say that the gravitational binding energy of two neutrons , even Io- 1 3 cm apart , is very sma ll at any rate. If the energy is not there in principle, why worry about it in practice') We feel that Brecher and Wasserman mi ss the point again . Certainly the gravi- tational binding fort wo neutron s is neglig- ible. But. we arc not co ncerned here about two neutrons nor indeed, necessarily a bout neutrons. We are concerned about con- ditions where large amounts of matter are being compressed towards their limit and gravitational energy is very significant in- deed. This significance is made quite evident in the distinction between the bodies which satisfy the p = constant and p 1 ,...,p,, = constant equations of state . We arc not "asserting that the global contribution of gravitation to the overall mass-energy of a star will affect the local stress tensor (T,,.) ...... We are asse rting that the local con tribution of gravitational energy, in addition to Tμ,, determines the physically relevant proper total energy density . Without entering the controversy regard-