14). Since the vector field ((WA)) is unique on CR, the idempotent field

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1 VOL. 51, 1964 PHYSICS: D. G. B. EDELEN 367 when we introduce the pressure p. This is the usual expression for the speed of a sonic disturbance, i.e., the quantity G - u. in the left member of (21). In particular, if u. = 0, the equation (21) yields the well-known expression vs~7/ for the velocity of propagation of a sonic disturbance into a gas at rest. * Supported in part by the National Science Foundation through grant NSF GP 1679 to Indiana University. 1 Thomas, T. Y., "The fundamental hydrodynamical equations and shock conditions for gases," Math. Mag., 22, 169 (1949). 2Edelen, D. G. B., "On the foundations of relativistic energy mechanics," Nuovo Cimento, 30, 292 (1963); also, "Material momentum-energy tensors and the calculus of variations," these PROCEEDINGS, 51, 367 (1964). 3 Thomas, T. Y., "Space-time coordinates in the general theory of relativity," Tensor (New Series), in press. MATERIAL MOMENTUM-ENERGY TENSORS AND THE CALCULUS OF VARIATIONS* BY DOMINIC G. B. EDELEN THE RAND CORPORATION, SANTA MONICA, CALIFORNIA Communicated by T. Y. Thomas, January 16, 1964 The purpose of this note is to demonstrate certain intrinsic limitations imposed on the Einstein theory of general relativity by the requirement that the field equations obtain from a homogeneous variational principle. These limitations result in significant constraint as to the nature of physical processes that can be examined. In particular, generalized irreversible processes are disallowed. Thus, if one attempts to develop relativistic fluid mechanics from a variational statement, any idea of entropy is highly artificial. On the other hand, if a variational statement is not required, the existence of intrinsic quantities with the properties of temperature and entropy can be rigorously established. (1) The basis for our considerations was presented in a previous communication.' We shall briefly summarize the salient results of the discussion, referring the reader to reference 1 for the details. Let 8 denote a four-dimensional hyperbolic-normal metric space in which the Einstein field equations hold, and let denote the components of the momentum-energy tensor associated with 8. A four-dimensional region CR of 8 is said to be material with a material momentumenergy tensor if and only if the momentum-energy tensor admits a unique time-like eigenvector whose associated eigenvalue is nonzero: TABWA = MWB, WAWA = 1, Y $ 0, (AB = 0,1,2,3). (1.1) It is sufficient for our present purposes to assume that the momentum-energy tensor of CR is of class one (the class of TAB is equal to the multiplicity of the eigenvalue 14). Since the vector field ((WA)) is unique on CR, the idempotent field 7A = A WAWB TAB

2 368 PHYSICS: D. G. E. EDELEN PROC. N. A. S. is unique on 6R and may be used to define a projection operation ir over the set of all geometric objects on 61 by (IB:: :) = C *B ID:: (1.2) Similarly,, the Lie derivative with respect to WA, is a well-defined operator on (R. It is then easily shown that the most general tensor of class one is given by material momentum-energy TAB = IwAwB + aab (1.3) where the symmetric tensor ((0-AB)) is such that r(aab) = GAB and aa VB =,V (1.4) if and only if VB = 0. In addition, we have 9 (h1) = ha EAB, h (-det(hab))"', (1.5) where EAR = 7r(W(AB)) = -7( hab) (1.6) is the Born rate of strain tensor. It follows that the dynamics of the region 61 can always be viewed as the flow of rest energy, irrespective of what actual physical processes occur, and that the quantities,u, WA, and cyab can be given consistent energetic interpretations. (2) A central point in the analysis is that the generalized stress tensor (((C.AB)) can be decomposed in a very particular fashion (see sect. 13 and the appendix of ref. 1). In all cases we may write IT = L + M + N, (2.1) where ((LAB)) and ((MAB)) are those parts of ((aab)) for which there exist scalar densities P and 2 and a scalar function 0($ 0) such that (P) = 2hLABEAB, (2) = 2hGMABEAB (2.2) are identities in CAB, and NAB is the residue. It then follows that and LAB = h-17(p,,ab), MAB = (ho)17r(22hab) (2.3) K1 + 2W(A*B)(1-7r)(PhAB) =0 (2.4) K2+ 2W(AiWB)(1-7r)(2,hA) 0, where K1 and K2 are scalar densities defined by the relations (P) = P2hABfhAB + K1, (2) = ZthABfhAB + K2 (2.5) and I'VA = WA;BWB. It thus follows that a = h17r(p,hab) + (h#)17r(22hab) + NAB (2.6) Physically, the density P can be interpreted as a stress potential; the density 2

3 VOL. 51, 1964 PHYSICS: D. G. B. EDELEN 369 can be interpreted as the intrinsic entropy (reducing to what is usually meant by entropy in the case of a generalized gas (see ref. 1, p. 316); and 0 can be interpreted as the reciprocal of the intrinsic temperature. If we substitute (2.1) into (1.5) and use (2.3), we obtain (ha - P/2) = 0-'1 (2/2) + hnab AB. (2.7) Since hma is the density of rest energy, (2.7) is a fundamental statement concerning the balance of energy in the presence of irreversible processes. The terms on the right-hand side of (2.7) represent the effects of such processes. It is also evident that if MAB = NAB = 0, the dynamical processes are intrinsically reversible, for we would then have the conservation law ((ha - P/2)WA),A = 0. (2.8) (3) We can now consider those cases in which the Einstein field equations are derived from a homogeneous variational principle. According to the prescription originally laid down by Einstein,2 one has to stationarize the functional 4 = f (hr + Q) dv4, where R denotes the scalar curvature of g, and Q is a scalar density that depends on hab and a collection of physical field variables, but not on the derivatives of hab. (Since 54 = 0, the variational principle is called homogeneous, in contrast to an inhomogeneous principle &t = f A5X dv4, where X denotes the arguments of the integrand.) As is well known, the functional 4 is stationary under functional variations that vanish on the boundary if and only if the quantities hab are such that RAB - 1RhAB = TAB 2 where the quantities TAB are defined by TAB = h 1(QXhAB)* (3.1) If we confine our attention to a region 11 of g that is material, the quantity Q must be such that there exists a unique vector field ((WA)) and a nonzero scalar s such that h '(Q,hAB) WA = MWB. (3.2) If we use the operator ir to decompose (3.1), the resulting form of the momentumenergy tensor is TAB = h 1(1-7r)(Q, hab) + h17r(q, hab). (3.3) From the definition of 7r and (3.2) we have (1-7r) (QhAB) = hs WAWB, (3.4) and hence (3.3) is equivalent to TAB = AWAWB + h-1r(q,^h). (3.5) It thus follows on comparing (3.5) and (1.3) that aab = h l r(q,aab). (3.6)

4 370 PHYSICS: D. G. B. EDELEN PROC. N. A. S. Now, (3.6) has exactly the same form that (2.3) had for LAB. It thus follows that MAB = NAB = 0, provided Q is identified with P and the first of (2.4) is satisfied. However, in view of (3.4), the first of (2.4) becomes K1 = 0, and this condition is satisfied since the first of (2.5) shows that K1 = 0 constitutes the conditions for the stationarization of the functional 4 with respect to the dynamical variables other than hab. We have thus established the following basic result. If the Einstein field equations are derived from the usual variational principle, then in any fourdimensional region GR in which the momentum-energy tensor is of class one, the generalized stresses admit a potential P = Q. Hence the dynamical processes in ar are intrinsically reversible and we have the conservation law ((ha - Q/2)WA),A = 0. (4) It may be instructive to consider several examples. Let a denote a scalar function that is functionally independent of hab, and let the scalar density Q that appears as an integrand in the definition of the functional 4 be defined by Q = 2h(a + XI), XI = XAXBhAB. (4.1) In this case, (3.1) gives htab = 2hXAXB + h(a + X2)hAB. (4.2) A trivial inspection of (4.2) shows that ((,A)) is a time-like eigenvector of TAB. We may thus define a unit time-like eigenvector by the relations XA = XWA and subsequently consider the quantity X as a scalar function which is hab-independent. In light of our previous considerations, we are then led to the following results: TAB = 2X2WAWB + (a + X2)hAB, (4.3) = a +32 (4.4) K1 =(a +X2)=, (4.5) and (2hX2WA),A = ((hg - Q/2)WA),A = 0. (4.6) (5) If we introduce the parameters p and p by the relations XI= p/2, a = -p - p/2, (5.1) the momentum-energy tensor (4.3) assumes the form considered by Thomas in his demonstration of the geodesic hypothesis,3 i.e., TAB = pwawb phab _ (5.2) Since we have derived (5.2) from a variational principle, we can conclude from (4.5), (4.6), and (5.1) that (p) = 0 (5.3) and (PWA);A = 0. (5.4) It thus follows that if the momentum-energy tensor considered by Thomas results from a variational principle, one does not have to assume (5.4); it is implied by the Einstein theory. Hence we have the following modification of Thomas' result: The tube representing the motion in the gravitational field of an isolated material body

5 VOL. 51, 1964 PHYSICS: D. G. B. EDELEN 371 whose momentum-energy (5.2) is derived from a variational principle contains a geodesic world line in its interior. In view of our present considerations, this most important result of Thomas' may be viewed as a direct statement of the intrinsic reversibility of the dynamical processes. (6) If we use a larger number of parameters, as in the substitution 2 = p0(l + E)/2 + p/2, -a = 3p/2 + p0(1 + E)/2, (6.1) the momentum-energy tensor (4.3) assumes the form considered by Taub :4 Since Taub derived (6.2) TAB = pa(l + E + p1p ) WAWB - pha (6.2) from a variational principle, the results stated in paragraph 4 are applicable. obtain Substituting (6.1) into (4.4) through (4.6), we thereby A = p0(l + E), (p) = 0 (6.3) and ((p0 + poe + p) WA); A = 0. (6.4) Thus, without assuming the constraint (powa);a = (po) + po0 = 0 (6.5) imposed by Taub, and without introducing any new parameters (such as "temperature" T and "entropy" S), we have obtained Taub's momentum-energy tensor and, in addition, the two conservation laws given by the second of (6.3) and (6.4). (7) Let us see just what happens when we make the additional assumptions that Taub introduces in his considerations. If we assume that (6.5) holds, then (6.4) together with (p) = 0 gives p0 (E) + pe = 0. (7.1) Introducing the temperature and entropy according to Taub's prescription, we have (E) = Ti(S) - pf(1/p0). (7.2) Eliminating (po) between (7.2) and (6.5), we have (E) = T9 (S) - pe/p0. (7.3) Combining (7.1) and (7.3), we thus obtain the relation Tp0 (S) = 0. (7.4) Taub obtains a similar relation (p0g(s) = 0) after considerable manipulation, but he fails to note that (7.4) eliminates irreversible processes: (7.4) shows that for T # 0 only mechanical changes affect the free energy E when the actual motion is considered. It is indeed difficult to envision how one can arrive at a description of basic thermodynamic processes by introducing additional parameters and equations in the variational process, for a variational derivation invariably leads to processes that are intrinsically reversible. If, on the other hand, one does not insist on deriving the momentum-energy tensor from a variational principle, the generalized stresses take the more general form given by (2.6). In this instance, even with NAB = 0, the examples given in reference 1 show that well defined generalized fluid mechanical models are obtained in which entropy and temperature

6 372 GENETICS: KAPLAN ET AL. PROC. N. A. S. are intrinsically defined, and the opportunity to consider viscosity effects and general equations of state is not lost. * This research is sponsored by the United States Air Force under Project RAND-contract no. AF 49(638)-700 monitored by the Directorate of Development Planning, Deputy Chief of Staff, Research and Development, Hq USAF. Views or conclusions contained in this Memorandum should not be interpreted as representing the official opinion or policy of the United States Air Force. I Edelen, D. G. B., "On the foundations of relativistic energy mechanics," Nuovo Cimento, 30, 292 (1963). 2 Einstein, A., "Hamiltonsches princip und allgemeine relativititstheorie," Sitzber. Pruessischen Akad. Wis&. (1916). 3 Thomas, T. Y., "On the geodesic hypothesis in the theory of gravitation," these PROCEEDINGS, 48, 1567 (1962). 4Taub, A. H., "General relativistic variational principle for perfect fluids," Phys. Rev., 94, 1468 (1954). GENE PRODUCTS OF CRM- MUTANTS AT THE TD LOCUS* BY S. KAPLAN, S. ENSIGN, D. M. BONNER, AND S. E. MILLS SCHOOL OF SCIENCE AND ENGINEERING, UNIVERSITY OF CALIFORNIA (LA JOLLA) Communicated January 20, 1964 The fungus Neurospora crassa normally possesses an enzyme, tryptophan synthetase (t'sase), which catalyzes the terminal reaction in the biosynthesis of the amino acid tryptophan. The enzyme is recognizable in vitro by virtue of its catalytic activity for three reactions: (1) indole-3-glycerol phosphate (InGP) + L-serine -*0 L-tryptophan, (2) indole + L-serine -- L-tryptophan, (3) InGP -- indole, and by the fact that injection into rabbits of partially purified preparations elicits the formation of neutralizing antiserums. Many mutations leading to a growth requirement for tryptophan are known at the genetic locus (td) which determines the primary structure of the enzyme. Of these mutant strains some, designated CRM+, synthesize easily recognizable gene products, in that extracts of such strains react with neutralizing antibodies to t'sase, and of these, some still possess enzymatic activity for either reaction 2 or 3. The other class of mutant strains, CRM-, yields extracts with no demonstrable enzymatic or serologic activity, in terms of neutralizing antibodies. CRM+ mutants have been intensively studied both enzymatically and genetically.1-5 Analogous studies of CRM- mutants have, until recently, been lacking.6'7 With the large number of CRM- isolates available at the td locus and the generalized occurrence of this mutant phenotype in many gene-enzyme systems, an understanding of the nature of the mutational event leading to a CRM- phenotype is essential to an understanding of the gene-protein relationship. Genetic analyses by means of intragenic three-point crosses and reversion tests performed on a number of CRM- mutants at the td locus reveal these strains as products of point mutations.6' 6 Speculation on the nature of the point mutation resulting in a CRM- phenotype has been published previously.6 Detailed phenotypic