THE UNIVERSITY OF MICHIGAN COLLEGE OF LITERATURE, SCIENCE, AND TEE ARTS Department of Mathematics Final Report THE GEOMETRY OF FLUID FLOWS IN RELATIVITY E. Ramnrath Suryanarayan No Coburn Project Supervisor CPEA Project 2767 under contract with: DEPARTMENT OF THE ARMY ORDNANCE CORPS DETROIT ORDNANCE DISTRICT CONTRACT NO. DA-20-018-ORD-17213 DETROIT, MICHIGAN.a.m.i ni ct.rd thrrll^. 4. OFFICE OF RESEARCH ADMINISTRATION April 1961 ANN ARBOR

ERRATA 1) P. 12, Eq. (3.23), replace "V" by "Vj" 2) P. 13, line 2, replace "Sij" by "si" 3) P. 13, line 6, reference 12 off 4) P. 15, in the second equation of (3.31), replace "- PM" by "- pM" 5) P. 18, Eq. (3.39), replace "dp/du" by "dp/ds" 6) P. 18, Eq. (3.40), replace "dp/du" by "dp/ds" 7) P. 19, Eq. (3.41), replace "dp/du" by "dp/ds" 8) P. 22, line 4, after "field equations2" add "(which determine gij)" 9) P. 22, line 9, after "special relativity" add "as well as general relativity" 10) P. 23, line 3, replace "(3.26)" by "(3.41)" 11) P. 25, on the right-hand side of Eq. (4.4) replace "xJ" by "xi" 12) P. 29, Eq. (4.19), replace "a" by "2" 13) P. 34, line 4, replace "(4.33)" by "(4.38)" 14) P. 37, Eq. (4.49), replace Z by l in the second sum a=l 15) P. 41, add the following paragraph after the statement c. "Thus we see that the hypersurface, p = constant, in the case of geodesic flow satisfies one of the conditions a, b, c. However in Newtonian case we know that when the stream lines are straight, the surfaces, p = constant, are one of the following classes of surfaces: parallel plane, concentric circular cylinders, concentric spheres. This result in the Newtonian case was proved by Wasserman (Formulations and Solutions of the Equations of Fluid Flow, Doctoral Thesis, University of Michigan, 1958). In proving this result Wasserman made use of the fact that orthogonal coordinate systems can be introduced on a V2. In our case since p = constant is a V3, we cannot in general introduce an orthogonal coordinate system on V3." 16) P. 42, line 5, replace "the space V3" by "our coordinates" 17) P. 44, after the last sentence in the page add "The flow is uniform." (Over)

18) 19) 20) 21) 22) 23) ERRATA (Concluded) P. 45, line 3, after "is given by" add "(at P - see Figure 1, page 28)" P. 47, Eq. (5.2), replace "=" by "-" P. 47, line 7, after "[see (4.64)]" add "at P" -12 1 Pc 56, Eq. (7.4), replace "t|g 2" by "I|g2" P. 56, line 4, replace "Eijk2" by,Eijk2f? P. 56, line 7, after "satisfy" add "(Ejkpg are not the covariant components of EJkpg)" 24) P. 58, Eq. (7.13), replace "s" by "S", -,wu," by 1 - k wiuj,, 25) P. 58, Eq, (716), replace " E jkwUi" by "- Eijkw 26) P. 59, Eq. (7.17), replace "- 1 k1" by'" k1" 2 2 27) P. 60, Eq. (7.19), replace "- klaijl by "kiai" 28) P. 60, Eq. (7.20), replace " "by" - 6x1 ~x1 29) P. 60, Eq. (7,21), replace " -1 " by" - 1 i oc2 ox ac 2 50) P. 61, in the second equation of (7 22), replace " uiJ; - iu i " by " ^u;i - iu;i 31) Po 62, line 1, replace "(7.12)" by "(7,13)" 32) P. 62, line 2, replace "vj" by "wj" 33) P. 62, line 3, replace "(7.12)" by "(7.13)" 34) PO 62, in the two equations u =. Q. j. =.. replace "s" by "S" 35) PO 62, Eqe (7.26), replace "s" by "S" 36) P, 62, after the last sentence add "The last two equations are valid in the Newtonian mechanicso"

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1961.

TABLE OF CONTENTS Page ABSTRACT iv CHAPTER I. INTRODUCTION 1 II. THERMODYNAMICS 5 III. INTRINSIC FORMULATIONS OF BASIC RELATIONS 8 A. Intrinsic Form of the Equations of Motion and the Equation of Conservation of Mass 8 B. Properties of Fluid Flow When the Orthogonal Ennuple Lies Along the World-Line Vectors and Along the First, Second and Third Normals of the World-Line 17 C. Integration of the Equal of Motion in the Case of a Degenerate - fect Fluid for Isentropic Flow 20 IV. GEODESIC FLOW IN TEE SPACE-TIME OF SPECIAL RELATIVITY 23 A. The Equations of Motion 24 B. The Hypersurface, p = Constant 25 C. The Codazzi Equations 57 D. Two Special Hypersurfaces, p = Constant 41 V. REDUCTION OF THE GEODESIC FLOW TO NEWTONIAN MECHANICS, IN THE CASE WHEN THE HYPERSURFACES, p = CONSTANT, ARE HYPERSPHERES 46 A. Transition to Newtonian Mechanics 46 VI. INTRINSIC FORMS OF EQUATIONS OF MOTION AND CONSERVATION OF MATTER, WHEN THE WORLD-LINES LIE ON A HYPERSURFACE S3 50 A. Hypersurfaces Containing the World-Lines 50 B. Intrinsic Equations 51 ii

TABLE OF CONTENTS (Concluded) Page CHAPTER VII. THE VORTICITY TENSOR AND THE VORTICITY VECTOR 55 A. General Flows 55 B. Steady Flows 61 REFERENCES 65 iii

ABSTRACT The purpose of this study is to relate fluid flow in the space-time of relativity to the geometry of the world-lines. This is done by introducing an orthogonal ennuple, such that one of its directions is along the world-line and the other three are along any three orthogonal directions perpendicular to the world-line, at each point of the world-line. By expressing the equations of motion and the equation of conservation of matter in terms of the ennuple, the variations of pressure, density, and generalized density along the four directions are determined in terms of the divergence of the world-line vectors, and the projections of the curvature vector of the congruence of world-line vector along these four orthogonal directions, Then eliminating the divergence of the world-line vector between two of these equations, a generalization is provided of a result proved by Taub in special relativity for isentropic motion in connection with the local sound speed (Relativistic Rankine-Hugoniot Equations, Physical Review, Vol. 74, No. 3, 1948, pp, 328-334). In particular, if the orthogonal ennuple is chosen along the directions determined by the world-line vectors, the principal normal vector of the world-line, the second and the third normals of the world-line, it is found that the pressure does not vary along the second and the third normals of the world-line. In the case of geodesic flows, it is shown that the motion is irrotational and the hypersurfaces, pressure = constant, form a system of geodesic parallel hypersurfaces orthogonal to the world-lines. The hypersurfaces, entropy = constant, are orthogonal to the hypersurfaces, pressure = constant, if and only if either a) the world-lines are geodesics, b) the entropy does not vary along the principal normal vector of the world-lines. The hypersurface, pressure = constant, in the case of geodesic flows in special relativity, has all its principal normal curvatures constant. It is found that hypersphere also belong to this class of hypersurfaces. The flow in the case when the hypersurface, pressure = constant, is a hypersphere, is reduced to the corresponding case in nonsteady Newtonian mechanics. The intrinsic forms of the equations of motion and the equation of conservation of matter are derived with reference to a hypersurface containing the world-lines. It is found that the flow properties depend on the normal curvature of the hypersurface in the direction of the worldlines and the geodesic curvature of the world-lines. Lastly, the geometric properties of the vorticity tensor and vorticity vector are studied. The flow is found to be irrotational if and only if the world-lines are geodesics. In the case of Beltrami flows, that is, if the vorticity vector vanishes, the vorticity tensor is found to be in the plane formed by the world-line vector and the principal iv

ABSTRACT (Concluded) normal vector. In the case of steady flows, it is observed that the Bernoulli hypersurfaces contain the world-lines; they contain also the vorticity vector if the entropy does not vary along the vorticity vector. v

CHAPTER I. INTRODUCTION The problem of determining the motion of a fluid subjected to its own gravitational and internal forces, is a problem in relativity. The geometric treatment of flow is simpler in relativity than in Newtonian mechanics. This is because the world-line vector, which corresponds to the velocity vector in relativity has its magnitude unity.1 In this work the fluid flow is related to the geometry. In Chapter II, basic concepts of thermodynamics in relativity needed for our study are considered. In Chapter III, a world-line is considered as a curve in spacetime V4 of general relativity, and the properties of the fluid flow are related to the intrinsic properties of the curve. This is done by introducing at each point of the world-line an orthogonal ennuple such that one of its directions is along the world-line and the other three along arbitrary directions. By expressing the equations of motion and the equation of conservation of matter2 in terms of the ennuple, the variations of p, the pressure, p, the density, and a, the generalized density, along the four directions are determined in terms of the divergence of the world-line vector and the projections of the curvature vector of the congruence of world-line vector along these four orthogonal directions. These are the intrinsic forms of the equations or motion ana the equation or conservation of matter. Then, elim1

2 inating the divergence of the world-line vector between two of these equations, a generalization is provided to the result proved by Taub3 in special relativity for isentropic motion in connection with the local sound speed. Lichnerowicz4 and Coburn5 have also provided the generalization by studying the discontinuity manifolds. Our next step in Chapter III is to choose the orthogonal ennuple along the directions determined by the world-vector, the principal normal vector of the world-line and the second and the third normals of the world-line. By this choice, we find that the pressure does not vary along the second and the third normals of the world-line. This property is a generalization of the known result in Newtonian mechanics that the pressure does not vary along the binormal of the stream lines.6 Then, the properties of the fluid flow are studied in the case the world-lines are geodesics. We find that the motion is irrotational5 and the hypersurfaces, p = constant, form geodesic parallel hypersurfaces normal to the world-lines. This result is also an extension of a result known in Newtonian mechanics, that the surfaces, p = constant, are parallel surfaces orthogonal to the stream lines when the stream 6 lines are straight lines. It is observed also that the hypersurfaces, S (entropy) = constant, are orthogonal to the hypersurfaces, p = constant, if and only if either (a) the world-lines are geodesics or (b) the entropy does not vary along the principal normal. We conclude Chapter III by expressing E, the internal energy and the generalized density explicitly in terms p, for a degenerate and

3 a classically perfect gas for isentropic motions (Chapter II). In Chapter IV, geodesic flows in special relativity are studied in detail. It is observed that the hypersurfaces, p = constant, have constant principal normal curvatures ka, kb, kn. It is found that hyperspheres and hyperplanes belong to the above class of hypersurfaces. The world-line vectors and the thermodynamic quantities are determined in terms of coordinates when the hypersurfaces, p = constant, are hyperplanes and hyperspheres. In Chapter V, the flows in the case when the hypersurfaces, p = constant, are hyperspheres in special relativity, are reduced to nonsteady flows in Newtonian mechanics. It is found that each component of velocity varies directly with the corresponding coordinate and inversely with time. It is found also that p and p are functions of time only. In Chapter VI, the intrinsic forms of the equations of motion and the equation of conservation of matter are derived, in the case when the world-lines lie on a hypersurface S3 of the space of general relativity V4. It is observed that the flow properties depend on the normal curvature7 of the hypersurface in the direction of the world-lines and the geodesic curvature7 of the world-lines. From the equations of motion (6.7), it is seen that the worldlines are geodesic on S3, if and only if the pressure does not vary along the relative curvature vector cX tht n1 *lId-lJrACO wi 1. u t o3. i Lb sCee aiLo Lihat a h Wurldlines are asymptotic on S3 if, and only if the pressure does not vary

along the normal to the hypersurface S3. Finally, in Chapter VII, the geometric properties of the vorticity tensor and the vorticity vector are studied. It is found that the fluid flow is irrotational if and only if the world-lines are geodesics. In the case of Beltrami flows, it is found that the vorticity tensor lies in the two-plane formed by the world-line vector and the principal normal vector of the world-lines. If the motion is isentropic, in addition to being Beltrami, the hypersurfaces, p = constant, are found to be orthogonal to the principal normal vector of the world-lines; and in this case all the thermodynamic quantities are found to be constant along the world-lines. In the case of steady flows, it is observed that the Bernoulli hypersurfaces contain the world-lines; they contain also the vorticity vectors, if the entropy does not vary along the vorticity vector.

CHAPTER II. THERMODYNAMICS In this chapter we consider notations and concepts used by Taub.2 A fluid is characterized by its caloric equation of state. This implies that e, the internal energy, is expressed as a function of p and p, where e, p, and p are measured by an observer at rest with respect to the fluid. Following Taub the equation of state may be written in the form e = e(p, p). (2.1) We introduce another function a, which we call the generalized density, given by a- p (1+ e- +P'" (2.2) \ c' pc2 where c is the speed of light, assumed to be a constant. In the thermodynamic plane only two of the six quantities p, p, a, c, T, the temperature and S the rest specific entropy are independent. Hence two of these six thermodynamic functions determine the other four. A differential equation connection the thermodynamic quantities is given by the first law of thermodynamics; namely Ts = b ^e up (2.3) axJw axJ a a i where x-: j O-, 1,,.3j denote a curvilinear coordinate system in V4. 5

6 A fluid is said to be perfect2 if p = pRT, (2.4) where R is the gas constant, and is said to be incompressible if = 0. (2.5) A fluid is termed as degenerate if2 + e = 3P (2.6) c2 pc~ A fluid motion in which S is identically constant throughout the medium of the fluid is said to be isentropic. For a degenerate and classical2 ly perfect gas, isentropic motion implies the following relation: p = p o( ) = Ap7 (2.7) Po where po, po, Y, A are constants. The local sound speed, a, or the speed with which sound waves are propagated, was obtained by Taub3 in special relativity for isentropic motion. It is given by a2 aPd -2 (2.8) 2 _ 2 c2 a dp p 2 From this relation and other formulas of Taub's paper, we find a = a2 a (2.9) dp p flimrninatin A hs+wpev n (P.A) anrl ( P-Q we obtain -.,., X.. * * -

7 ida 1 = 1 (210) a dp p c2a dp The result (2.10) will be shown to be valid in Chapter III in Equation (3.35) for non-isentropic compressible relativistic fluids in general relativity assuming that p varies along the world-lines. In the case of isentropic flow of a degenerate and classically perfect gas, the speed of sound can be written in the form a2 = 7 (2.11) a The above result is obtained when we substitute for p from (2.7) in (2.9).

CHAPTER III. INTRINSIC FORMULATIONS OF BASIC RELATIONS A. INTRINSIC FORM OF THE EQUATIONS OF MOTION AND THE EQUATION OF CONSERVATION OF MASS Let x3(j = 0, 1, 2, 3) denote a curvilinear coordinate system in the four dimensional space-time V4 of general relativity. Let gij be the covariant components of the metric tensor of V4. The indefinite form (ds)2 = gijdxdx (3.1) has the signature (+, -, -, -), ds being the element of arc. The scalar e is +1 if dxi determines a direction which is time-like; and | is -1 if a space-like direction is determined. The sign of e is so chosen to keep (ds)2 always positive. The unit time-like vector along-the world-line u is given byl u =dxi (3.2) ds From (3.1) and (3.2) we see that the unit four vector ui of a worldline satisfies the relation gijuiu = 1. (33) Now, we introduce the symmetric energy tensor Tij 2= ocUJ - P T = c u3- pg (.4) 8

9 where a is given by the Equation (2.2). The equations determining the fluid motion are2 Tij.j = 0 (3.5) where the semi-colon denotes covariant differentiation with respect to the space-time V4 with metric gij given by (3.1). For non-isentropic flows the conservation of mass relation (pui).i = 0 (3.6) must be added to the system. Let us now introduce arbitrary unit vectors a1, bi, ni at every point of a world-line such that the four vectors u, a, bi, and ni form an orthogonal ennuple in V4. Since ui is time-like the other three are space-like; that is, they satisfy gijaiaj = ib gii = gnin = -1. (5.7) Since the four vectors are mutually orthogonal, they satisfy uia = uib = ui n = aib = ain = bin =0. (3.8) At every point of the world-line the metric tensor can be written in the form (cf. Ref. 7, p. 96) gij = uiuj - aiaj -bib - ninj (3.9) We shall now express the variation of p, p, and a along the four

10 directions ui, ai, bi, and ni with the aid of the Equations of motion (3.5) and the Equation of conservation of mass (3.6). Substituting for Tij in the Equations (3.5) from (3.4) and expanding, we obtain c2a.;uiuj + c2o(uiu j + ujui j) - gip;j = 0. (.10) Expanding (356) we obtain p;ju j + puJ;j = 0. (51) Taking the scalar product of (3.10) with ui and using the Equation (3.3) and its consequence u; ui = 0, (3 12) i;j (3.10) reduces to c2;juj + c2Oruj up. = 0. (313) Ccnbining the first two terms of (3o13), we get (oui);j = U-p;j. (3.14) Substituting now for a from (2.2) in (3,14) and applying the product rule of covariant differentiation to the two functions pud and (1 + EC-2 + pp' —2) we get u;j (1+ + ) + 1+ + );, = p u'p.1. (5.15) + )+ Pu (1 + cp /315 ou~;j + c- + --.9 ~ ~ ~ ~ ~, ~+o —;j c u ~:J' 5o The first term of the left-hand side of the above equation vanishes by

11 virtue of the Equation (3.6). The second term, after performing the covariant differentiation becomes [puJ;j +p() up +up. c12 r puj;j tPJ;j Cancelling uJp;j/c2 from both sides of the Equation (3.15), Equation (3.15) can now be written in the form uJ;j + p () uj = 0. (316) For a scalar function f, f;ju denotes the directional derivatives of f along the world-line. Denoting u3f. df (5317) ds Equation (3.16) can now be written in the form d P E(d ) = o (3.18) ds ds p. Since the covariant derivative of a scalar is its partial derivative, the Equation (2,3), after taking the scalar product with ui, becomes, using the notation (3.17), T dS dc. d P 1 ds ds ds p Comparing Equations (3518) and (3o19), we see that dS %A 020 that is, the entropy is constant along the world-line, as observed by

12 Taub. Using the notation (3.17), Equation (3.11) can be written in the form d + puj;i = 0 ds (3.21) Now we shall consider the divergence term uj;j appearing in (3.13) and (3.21). For this we make use of the well-known fact that for a congruence of curves determined by a unit vector ui, the tensor Ui;j = Lii + uVi (3.22) where Vi is the curvature vector of the congruence ui and Lij is a covariant tensor lying locally in the subspace normal to ui.9 The following relations are satisfied by Vi: ji uuj;i = V uiVi = 0. (3.23) We write Lij = sij + rij where sij is symmetric and rij Lij from the above equation in is skew symmetric. Substituting for (3,22), we get i;j = Sij + rij + UjVi (3.24) Multiplying both sides by gij and using the second equation of (3.23) and the fact that rij is skew symmetric (that is, g-rij = 0), Equt-tiou (3.24) can be written in the form

15 u;j = giJui;j = Jsij (3.25) The invariant giJsij is the sum of the principal values of the symmetric tensor Sij. If the principal values are -pl, -P2, -p3, then (3.25) can be written in the form u;j = - (P1 + P2 + P3) = M. (3.26) In the case when there exist oo' hypersurfaces orthogonal to the worldlines, then M is the mean-curvature of these hypersurfaces.2 Substituting for uJ;j from (3.26) into (3.13) and writing the two Equations (3.13) and (3.21) in the notation (3.17), we get 1 dp = dy + aM (3.27) J^ = ^ aM (3.27) c2 ds ds dp + pM =. (.28) ds Equation (3.28) implies that the variation of log p along the worldlines is equal to the negative divergence of the world vector. In the case when hypersurfaces exist orthogonal to the world-lines, then M is the mean curvature of these hypersurfaces (cfo Ref. 7, p. 168), and the density does not vary along the world-lines if and only if M = 0. Hence we may conclude that: In the case when surfaces exist normal to the world-lines the variation of log p along the world-lines is equal to the negative mean curvature of these hypersurfaces and further the density is constant along the world-line if and only if these hyper

14 surfaces are minimal. To find the remaining intrinsic formulations of the equations of motion, we take the scalar product of the equations (3.10) with ai, bi, and ni and use the normal conditions given by (357) and (3.8) to obtain c2a(uJui;jai) - gaip;j = 0 c2 (uJui;jbi) - gibip;j = 0 (3.29) c2o(uJui;jni) - giJnip;j = 0 The factor uJui; j in the first terms of these equations can be replaced by Vi, by virtue of the first relation of (3.23). The second term of the first equation of (3.29) is aJp;j. Following the notation (3.17), let us write ajp.; = dp/da, etc.; now the Equations (3~29) assume the 9J form o(Viai) 1 dp c-2 da c db o(Vini) = 1 dp c dn Viai, V'bi, Vni are the projections of the curvature vector v of the world vector along the directions am, bi, ni. Equations (3.27), (3.28), and (3.30) are the intrinsic forms of the equations of motion and the equation of conservation of mass for fluid flow in general relativity. The Equations (3.27) and (3.28) are not independent. It can easily

15 be seen by differentiating (2.2) along the world-line and using (3.18) and (3.28) we get (3.27). Therefore the system of equations consisting of the first law of thermodynamics and ~a = p/i) P ( c2 PC2 = - PM, d = 0 ds ds ( Viai) = dp (3.31) c2 da a(V ) =A 1 dp (vi) c2 db a(Vni) = dp c2 dn determine the motion of fluid in general relativity. Comparing the Equations (3.27) and (3.28) and (3.30) with their analogues in Newtonian mechanics, we notice that the variations of p, p, and c along the directions ui, ai, b, and ni depend only on the geometry of the world-line and the congruences ai, bi, and ni The equations do not depend on the magnitude of the "velocity." Now let us provide a generalization of Taub's result in connection with the local sound speed (Chapter II). Assuming that p varies along the world-line, we prove that the Equation (2.10) holds good in the case of non-isentropic fluid motion in general relativity. To prove this let us eliminate M between the two Equations (3.27) and (3.28) to get 1 da 1 dp 1 d (3.32) a ds p ds c2a ds

16 Expressing a and p as functions of p and S, we have da aa dp + 3a dS ds ap ds aS ds -j;p dp +. dS ds ap ds aS ds By virtue of the Equation (3.20), the two above equations reduce to da a p ds ap ds - p dp (3.33) dp a 6p dp ds ap ds Eliminating da/ds and dp/ds among the three Equations (3.32) and (3.33), we get i T. dps 1 dp 1 _2 dp (3.34) a a p ds p ds ca p ds Since we have assumed that p varies along the stream line, that is, ^a 0 ds we cancel dp/ds throughout the Equation (3.34) and get 1 Ca 1 - - 1= (.55) a ap p c2a ap This above equation agrees with the Equation (2.10). The Equation (2.10) is obtained by Taub in special relativity for isentropic motion.

17 B. PROPERTIES OF FLUID FLOW WHEN THE ORTHOGONAL ENNUPLE LIES ALONG THE WORLD-LINE VECTORS AND ALONG THE FIRST, SECOND, AND THIRD NORMALS OF THE WORLD-LINE Equations (3.27), (3.28), and (3.30) are valid for arbitrary choice of the orthogonal ennuple, at each point of the world-lines. Now we choose an ennuple along the world vector, along the principal normal to the world-line, and along the second and the third normals of the world-line. Let ai, bi, n denote the directions along the principal, first, and second normals of the world-line. These vectors are mutually orthogonal and they satisfy the Frenet formulas (cf. Ref. 7, p. 106) ui; ju = - kai a uj = - klui - k2bi b u1;J (3.36) bi juj = k2ai + k3ni ni uj = k3bi;J where kl, k2, k3 are the first, second, and the third curvatures of the world-lines. Substituting for ui.uj from the first equation of (3.36) J in the Equations (3.30) and 1using the normality conditions of the vectors u1, ai bi, and ni, Equations (3.30) reduce to 1 p= ak c2 da (3.37) - = o = - ( db dn The above equations indicate that the variation of pressure along the first normal depends on the curvature and that the pressure does not

18 vary along the second and the third normals. This result agrees with the corresponding in the Newtonian mechanics, where the pressure does not vary along the binormal of the stream lines and that the variation of pressure along the principal normal depends on the curvature of the stream lines. Further, if the world-lines are geodesics, then kl = 0, which implies from the Equation (3.37) that d 0. (3.38) da And conversely if (3038) is true, then from (3.37) we have kl = 0; that is, the world-lines are geodesics. Hence we have: A necessary and sufficient condition that the world-lines are goedesics is that the pressure does not vary along its first normal. The gradient of pressure can be written as the sum of the gradients along ui ai bi and ni in the form dpui d - ai dp b - ni i du 1 da i db i dn From (3.37) and (3.39), the relation (3.39) can be written in the form P;i = dp ui - c2aklai. (40) du The above result shows that the normals to the hypersurfaces, p = constant, lie in the biplane parallel to the world vectors and the first normal. In the case of geodesic flow, k1 = 0. Therefore (3.40) reduces to

19 P;i du P;i = du i (3.41) The relation (3.41) shows that the "velocity" vector is normal to the hypersurfaces, p = constant; that is, ui forms a normal congruence. Therefore the vector ui forms a geodesic and normal congruence. These two properties imply that the world vector satisfies the conditionl;j j;i 0 (.2) Since the vorticity tensor wij is defined by5 2wij = i;j - ji (3.43) Equations (3.42) and (3.43) imply that Wi = 0; that is, the fluid motion is irrotational.5 Therefore, if the worldlines are geodesics then the fluid motion is irrotational. Also, since the world vector is orthogonal to the hypersurfaces, p = constant, the hypersurfaces, p = constant, form a system of geodesic parallel hypersurfaces. Now we wish to know under what conditions the two hypersurfaces, p = constant and S = constant, intersect orthogonally. To determine the conditions let us first express the gradient of S as the sum of the arji.rri n+. 1 f ulong.. a.. b.. and n...-' -.' 1'

20 as dS dS dS n S = - u - a- bi - ni (3.44) S;i = ds ui - da a4i i- db b dn ni The first term on the right-hand side of the above equation vanishes by virtue of the Equation (3.20). Therefore we may write (3.44) in the l0= daadS d i dSr* S. dS dSb dS (3.45) S.i = - -a. -- b. --- n:. (545) da db 1 dn L Let us now form the scalar product of the gradient vectors p;i and S;j. In the scalar product gijp.iS;j we substitute for p;i from (3.40) and for S;j from (3.45). Then we notice ijpiS - j gi ui -k2a ( das dSb dS g p j Si duydu - da1 db J dn tJh Simplifying the right-hand side of the above equation by using the normality conditions of the vectors ui, ai, bi, and ni, we get ii 2 dS gip;iS = - klc2a - (3.46) da gijp iS is zero if and only if either (a) k1 = 0 or (b) dS/da = 0.;1;J Therefore we have the following result. The hypersurfaces, p = constant, intersect S = constant orthogonally if and only if either of the followin conditions hold: (a) the world lines are geodesics; (b) the entropy does not vary along the first normal. C. INTEGRATION OF THE EQUATIONS OF MOTION IN THE CASE OF A DEGENERATE I.LT iILUI. r u r L V \ LR 1 -,l._wrLA.fi 1LA)WS In the case of isentropic fluid motion of a degenerate, classically

21 perfect gas, the thermodynamic variables p, E, and a, and the local sound speed a, can explicitly be expressed in terms of p. For such fluids in isentropic motion p and p satisfy the relation (2.7). Let us assume that p does vary along the world-line. Differentiating e = e(p, S) along the world-line, we get de e dp e d BdS ds op ds aS ds The above equation with the aid of (3.20) can be written in the form de be d ds pp ds Substituting for de/ds from the above equation, in the relation (3.18), Equation (3.18) becomes after cancelling dp/ds e p. = 0. (3.47) )P P2 ~p p2 Substituting for p from (2.7) in the above equation, and integrating the resulting equation with respect to p, we get A pT-1 + constant, (348) 7-1 a can also be expressed in terms of p. We will substitute for p from (2.7) and for e from (3.48) in the Equation (2.2). Then we get [1 +- c ( -) consta(3.49) To obtain the local sound speed, we will substitute for p from (2.7) and for a from (3.49) in the relation (2.11). Then we get

22 a2.A.-.-.. + constant. (3.50) c2 + pThe properties of fluid flow we have discussed in this chapter hold in the space of general relativity. In addition to the equations of motion and conservation of matter (3.5) and (3.6), the flow must satisfy also the field equations2 1 Kc2T (.51) Rij - Rgij (3 51) where R is the Ricci tensor of the space V4, R its scalar curvature and K a universal constant. However, in the case of flows in special relativity, Equations (3.51) are not valid but (3.5) and (3.6) do hold good. Therefore all the properties of flow we have obtained in this chapter are valid for the motion of fluid in special relativity.

CHAPTER IV. GEODESIC FLOW IN THE SPACE-TIME OF SPECIAL RELATIVITY In Chapter III we have seen that in the case where the world-lines are geodesics in V4, the oo' surfaces, p = constant, form geodesic parallel hypersurfaces orthogonal to the world-lines (3.26). In this chapter, we study these surfaces in detail in the space-time of special relativity E4. By use of geometry of the parallel hypersurfaces in the hyperbolic space E4, and the equations of motion (3531), we shall show that in the case where the fluid motion is isentropic, any hypersurface, p = constant, is such that all three of its principal normal curvatures are constants. From these classes of hypersurfaces we choose the following two hypersurfaces with the principal normal curvatures ka, kb, kn satisfying (1) ka = kb = kn = 0 (2) ka = kb = kn = constant 0. We shall show that the hypersurface satisfying condition (1) is a hyperplane and the hypersurface satisfying the condition (2) is a hypersphere. Then, we shall express the world-line vector and the thermodynamic quantities p, c, p, a, a in terms of the coordinates when the hypersurfaces, p = constant, are hyperplanes and hyperspheres. 23

24 A. THE EQUATIONS OF MOTION Let xi = (i = 0, 1, 2, 3) denote an orthogonal pseudo-cartesian coordinate system in E4. The components of the metric tensor gij of E4 are goo = 1 (4.1) gij = - Since E4 is a "flat" space, all the Christoffel symbols vanish, and the covariant derivatives are just the partial derivatives. We have assumed that the world-lines are geodesics; that is, the curvature vector Vi of the world-lines satisfy the equations Vi = ui; j = 0 (4.2) We substitute for Vi in the last three equations of (3.31) from the Equation (4.2). The equations of motion (3.31) now become a = p 1 + e- + c2 pc2:= - pM (4.5) ds dp = dp = dp = 0 da db dn where ai, bi, and ni are three mutually orthogonal directions in the hypersurfaces, p = constant, and M is the mean curvature of this hypersurface. Let us denote some particular hypersurface, p = constant, by V3.

25 B. THE HYPERSURFACE, p = CONSTANT We shall introduce now, a coordinate system on the hypersurface V3; and discuss the first and second fundamental tensors of V3. Let ya: (a -= 1,, 3) be a curvilinear coordinate system on V3. The components of the metric tensor g., of V3 is given by (cf. Ref. 7, p. 146) = g xI ci i = ) (4.4) g = g ijx~a = xix (Q) i=O where, Xi a ac' Ya' is the projection tensor. A unit vector di of E4 and lying in V3 has components da in V3 given by = dixi (4.5) Since the vectors ui, ai, bi, and ni form a mutually orthogonal system in E4, the metric tensor gij of E4, by virtue of (3.9), can be written in the form gij = uiuj aa - bibj - ninj (4.6) Since ui is orthogonal to V3 and since ai, bi, and ni lie on V3, we have from (4.5) that i Uix a = ~ aix a ba,bX = bi b n x n. (4.7) ia a ia a ia a

26 Multiplying both sides of the Equation (4.6) by xiaxiJ and using (4.7) and (4.4), the Equation (4.6) becomes ga = - a'a - bab b - nCBB, (4.8) Also, multiplying both sides of (4.4) by aaaO and using the second equation of (4.7), we obtain from (4.6) g9aC a a gijaa xi xJ 3 giaia = -1 l (4.9) The result (4.9) shows that the vector aa is a space-like unit vector in V3. Similarly bs and na are also unit space-like vectors in V3. The second fundamental tensor Pp is given by (cf. Ref. 7, p. 148) ui =- _ gX'7 (4.10) where the comma followed by a denotes covariant differentiation with respect to the sub-space V3. Multiplying both sides of (4.10) by gijXJ and using (4.4), the Equation (4.10) becomes - gijui xi =. (4.11) Now, let us choose the three directions aa, b, na along the principal directions of V3. If points are umbilical, we choose any three mutually orthogonal directions on V3. Let ka,, kb, kn be the principal normal curvatures along aa, bc, na, respectively. Since aa, ba, na are along the principal directions of V3, we have (cf. Ref. 7, p. 153)

27 ( Ka - kaga)aa = 0 ( ap- kng )nb = 0 Multiplying the above equations by at, b7, n7 respectively and adding, we obtain Qnc(aaa7 + bc7 + nanT) = g o(kaaCaa + kbbbY + knan7). Substituting for aaa7 + bb7 + nacn7 on the left-hand side of the above equation by -gC', the equivalent of the result (4.8), the above equation becomes - Qapcga = ga(kaaCa` + kbb%7 + knncn7). (4.12) Let us now multiply the above result by g58 and use the results g7g yp = y gGaG = ap, etc.. (4.13) Then the Equation (4.12) takes the form 5 = ka,-aa + kbbsb5 + knnn. (4.14) The mean curvature, M of V3, from (4.9) and (4.14), is M = g po = ka + kb + kn. (4.15) We shall now consider a hypersurface V3 which is parallel to V3, and express the first and second fundamental tensors g., and 2a., Of

28 V3 in terms of the first and second fundamental tensors of V3 and a parameter r. Let us introduce the pseudo-cartesian coordinates 3i (i = 0, 1, 2, 3), such that, on the original V3 xi = xi(yoG) (4.16a) and on any parallel V3 = xi(ya) + ruii(y") (4.16b) where r is a parameter. P(ya, r) on V3. In the figure, P(yi, O) on V3 corresponds to Figure 1 In the case when the metric is positive definite, r is the distance between the two parallel hypersurfaces along their common normal;here it is geodesic distance. Differentiating the Equations (4.16b) along the hypersurface V3, we have

29 x = + rua. (4.17) Our procedure in obtaining gaB and 20c of V3 is similar to that of the Euclidean case. The metric tensor gal of V3 is, by definition, gapB = gi x cp (4.18) Substituting for xi from (4.17) in the above equation, the Equation (4.18) becomes g.(xi + )(xj + gg = gij(xa + ru ) r u Simplifying the above result, we get g3 = gijxlx + cOrgijxiCu + r2giju' u,. (4.19) The first term on the right-hand side of (4.19) is gc by (h.4). In the second term on the right-hand side, we substitute for uj from the Equation (4.10). Then this term reduces to -2rflX. In the third term of the right-hand side of (4.19), we substitute for ui, from (4.10) and use (4.4) and (4.13). The Equation (4.19) then reduces to gaB = gaOr - 2rOp + r2 y2g6. (4.20) We will now substitute for g.g and Qoa in the above result from (4.8) and (4.14) and use (4.9). Then the Equation (4.20) becomes pg = - (1 - rka)a - (1 - rkb) ab - (1 -rkn)2 n (4.21) gap a a p a p ~~n a

30 Now, we define the vectors a, b, n on V3 at P corresponding to the vectors a, ba, nc on V3 at P, in the following manner: a = i a, b, n- na (4.22) 1- rkaa 1 -rkb 1 rkn We shall show that the vectors a, b, n are mutually orthogonal, spacelike unit vectors on V3. From (4.21) and (4.22), we have gQclad = [- (1 - rka)2aaa - (1 - rkb)% 2bab aa B (1 - rkn)2nan] (- rka) From the orthogonal properties of the vectors a(X, bC, na the above equation reduces to s aQ; = -1 which shows that a' is a unit, space-like vector. Similarly we conclude bP and na are also unit, space-like vectors. To show that aa, ba, na are mutually orthogonal on V3, we have from (4.21) and (4.22) gCaCbB2 = [- (1 - rka) aa - (1 - rkb)%b - (1 - rkn)2n ] ( - (1 a)(- rkb From the orthogonality conditions of a, b, na it is clear that the above result reduces to ", - - The above result shows that aa and ba are orthogonal. Similarly we can

31 prove that a, b, are mutually orthogonal. The covariant components ap are given by, from (4.21) and (4.22) and the condition that aa is space-like, a = 9e = (1- rka) a (4.23a) Similarly we find b = (1 - rkb)b, n = (1 - rkn)np. (4.23b) The metric tensor gFC of V3 can now be written in terms of a., ba, nC. From (4.21), (4.23a), and (h.23b) we have gap = - aa - bolb - nnp. (4.24) We shall now express the second fundamental tensor "ou of V3 in terms of the principal normal curvatures of V3. From (4.11), SO can be written in the form iap = - gijxiu, P (.25) Substituting for xia from (4.17) and for uJ, from (4.10) in the above equation, the Equation (4.25) becomes - Cl = - + r(foaf6ig) (4.26) Substituting again for f2 from (4.14) and - abaE - b~be - n nE for g56 in the above result, Equation (4.26) becomes

32 k = ka(l - rka)acaa + kb(l - rkb)bab + kn(l - rkn)nrang (4.27) or, in terms of a=, ba, nC', Q, can be written by use of (4.27) and (4.22) in the form k kb _ _ = -a a>a0 + b bab b+. (4.28) -: 1- rk u' a1-rkb + p 1- rkn..8p Let us now define ka - kb ka = rka' b 1 - rkb' kn = - r * (4*29) Comparing (4.28) with (4.14), we can write (4.28) by use of (4.29) in the form -S kA = kaa + kbbp + knnC (4-30) a.- n... The above result shows that ka, kb, kn are the principal normal curvatures along aa, ba, ns respectively. The mean curvature M of V3 at P is given by (4.30) and (4.29). The mean curvature: M gJ ~ - ka + kb + kn - + +. k (4.31) k kb+akn = (1 - rka - rkb 1 kn Since the world-lines are geodesics orthogonal o' hypersurfaces, p = constant, we parametrise any world-line by the variable r and write the metric in E4 in the form (cf. Ref. 7, p. 57) 5(ds)2 = dr2 + gdPdydy (4.32) where E is given by the Equation (3.1).

33 We now use differential geometry to determine the properties of fluid flow, in the case when the world-lines are geodesics. We assume that the metric E4 is given by the Equation (4.32) where any curve, r = variable, is a world-line. We shall now show that for isentropic fluid motion, the hypersurface V3 (p = constant) is such that all its three principal normal curvatures ka, kb, kn are constant. We prove this by the aid of the equations of motion (4.3). The last equation of (43,) indicates that the pressure varies only along the world-line. Therefore we write p = p(r). (4.33) Thus, any r = constant is a hypersurface and can be chosen as the initial p = constant. Since the motion is isentropic p and p are connected by the relation (2o7), Equations (4.33) and (2,7) imply that p = p(r). (4.34) The second equation of (453) can now be written with reference to the hypersurface V3 in the form a = -M (4.55) p or where M is the mean curvature of V3a In the above equation, we notice that the left-hand side is a function of r by virtue of (4.34), and tnerefore M is a function of r only. Hence we have

34 M = 0 a6 (4.36) Substituting for M from (4.31) in the above equation, we find that (4,36) becomes j _il- k kb + 1 - rkb kn- J 1 - rkn = 0 (4.37) or, the above equation may be written as 1 aka (1 - rka) Y 1 akb +.- -a-l (1 - rkb ay 1 akn + - 2'- - = (1 - rkn) ay 0 (4.38) Differentiating (4.33) with respect to r and using the fact that ka, kb, kn of initial V3 are not functions of r, we get ka (1 - rka)3 aka kb akb ay + (1 - rkb)3 ^ay kn akn (1 -rkn)3 ay + (- k)3~ =^ 0. (4.39) Differentiating (4.39) again with respect to r, we get k2 ~k k ak a a (1 - rka) 4 ~ya __ _ akb + - -rkb) 4 - (1 - rkb)4 8ya k2 ak + n 0 (1 - rkn)4 aya (4.40) Now we shall show, by use of the three Equations (4h38), (4.39), and (4.40), that the normal curvatures ka, kb, kn are all constants. We write Ca = (1 - rka), p = (1 - rkb), v = (1 - rkn). (4.41) The determinant of the system (4.38), (4.39), and (4,40) is

v55 1 3. 1 C2 2 2 as 72 V2 v k kb k, 1 D = a3 k k kn 2 2 2 kakb n -k ka2 2 k2 a4 P4 V4 Expanding the determinant, the above result can be written in the form a4 4v4D = (kb - ka) (Pkn - vkb) (Ckn - vka) (4.42) = (kb- ka)(kn - b)(k - ka) We consider the following two cases in solving the Equations (4L38), (4.39), and (4.40) for aka/JYc, akb//yac, and akn/Ycx. Case (a) ka, kb, kn are distinct. (4.43) Case (b) At least two of ka, kb, kn are equal. If ka, kb, kn are distinct, then from (4.42) it is clear that D A 0, which implies from the Equations (4o38), (4o39), and (4.40) that aka kb n 4). =... - = 0 (4.44) which imply that ka, kb, kn are constants. Let us now consider the case (b). Let us assume that ka = kb. This implies from (4,42) that D = Oo Therefore, we put kb = ka in the two Equations (4.38) and (4o39) and solve for aka/ ya and n kn/d y. Substituting ka in place of kb in (4.38) and (4.39), we obtain

36 2 aka 1 akn 2 ^ ka+ ~ (4.45) (2- rka2 k (- krn)2 y a a+ 2k kn kn 0 o.(4.46) (i1 rka,)3 lY' (1 - rk,)3'y ( The determinant D of the above system is 2 1 e 2 v2 D = 2ka kn a3 v3 Arguing as in the two cases (4.43), we find that if ka and kn are distinct then both ka and kn are constant. If ka = kn, then D = 0. Substituting ka = kn in the Equation (4.45), we find akn = aya or kn is a constant and therefore ka is also equal to the constant. Therefore case (b) implies either ka = kb = constant, kn = constant, or ka = kb = kn = constant. In the case when ka = kb, we have seen that ka = kb = constant; kn = constant, by using only the two Equations (4.38) and (4~39). The above equations satisfy also the other equation of the system, namely, the Equation (4.4h)n Ws.mlmmTr7. 7 olr ri-lt1+.t, E. follosw. The hypersurface V3 (p = constant), with principal normal curvatures ka, kb, kn, in the case of isentropic geodesic motion,

37 is such that i. if ka, kb, kn are distinct, then k's are constants ii. if ka = kb, kn A ka, then k's are constants iii. if ka = kb = kn, then k's are constants. Thus we see that k's are always constants. C. THE CODAZZI EQUATIONS Now we shall consider the Codazzi relations (cf, Refo 7, p. 150) and get all the possible cases of the hypersurface V3. Let us write 1 2 3 - 8, n = ba, n = n, so that k = k, k = k, k = k. 1 2 3 The Codazzi equations (cf. Refo 7, po 150) are - 0 QcX,7'Y y, = Substituting for pg in (4.48) from (4o14) using the notation (4.14), the Equation (4o48) becomes a a a a niii6 te WiB a p knl) k - ^UU af w4,4 (4.49)e c a=l a a FoJmi~ the............... vf (4'~.......b8 r.r ll.eS IL.L c=L6 a11CbC" t" pl- UC I \ Lic~ Lfl W -L W.LU' WC-6cu~C

n+ aa bp a ak = a (4.50) a~ a a a b where Z stands for summation with respect to the index a, which runs from 1 through 3. Since all n. are space-like, we shall write aaa b b I -1 if a =b nan = 5 = (4.51) 0 if a b. We also have the equation nP"Y nE3 _ O. (4 52) By use of (4.51) and (4.52) the Equation (4~50) becomes - kn: 7n - knhbn - k a + akni a 0 a Y7 b,7Y a ca Yn = Forming the scalar product of the above equation with OnC, we get, by use of (4.51) and (4.52) cbpr - kC,7n - a k a nbnna. + kC b = 0 Again forming the scalar product with n7, the above equation becomes bpf b f bfca c f be - ~,7nCn7 - abkna 7 + fa n + kn7nn (453) Let us now consider all the possible cases of the Equation (4.53). CZ — 1: f - b - In this case, the Equation (4.53) reduces to an identity in virtue of the Equation (4.52).

39 Case 2: b = c A f. Let us take b = c 2 2, f = 1. Then the Equation (4.53) becomes Since kSn,n + kn, np = 0. 2 1 2a nB7 = - a,, =yay. (4.54) (4.55) substituting for n. na in the first term of (4.54) from (4.55), the Equation (4.54) becomes (k - k) n7, 7 = O 1 2 p (4.56a) Similarly we can show (k - k) 7 = 0 2 5 7 (4.56b) Case 3: f, b, c are distinct. Let us take b = 1, c = 2, f = 3. With these values the Equation (4.53) becomes -k2 f r - n ^,a7 + ka n 1fp~a + 0. (4.57) In the above equation we put 1 2a n, 7n 2 37 n7 gfn 2 a = - n3, 3 27 = - n. 9 pn

40 by virtue of (4.55). With these substitutions the Equation (4.57) becomes (k - n 7n3r + (k - k) n = 0. (4.58) 1 2 33 2 Similar equations can be obtained for the values b = 2, c = 3, f = 1 and b = 3, c = 1, f = 2, etc. If k, k, k, are distinct constants, then from (4.56a) we find that n. ~ _ = o. And from (4.56b) we find that 2 2 n7 7 = 0. The above equations and the Equations (4.52) indicate that the curvature 2 1 2 3 vector of n7 is orthogonal to n7, n, n7. Therefore we conclude that 2 2p npn = 0 Similarly we have = o0, n 0n = 0. 7, The above three equations imply that the principal directions of V3 are along the geodesics of the hypersurface V3. Therefore we have the following result: If k, kb, kn are distinct then the lines of curvature of V are geodesics.

41 Again if kl = k2, then from (4.59) we see that either k3 = k2; or 1n 0. a, P These conditions show that either all the curvatures are equal, or 3 12 3n cn = 0. a,P From the above results, we see that the Codazzi relations are satisfied if: a. All the principal curvatures are equal constants. b. The principal curvatures are distinct constants. This implies that the lines of curvatures of V3 are geodesics. c. Only two of the principal curvatures are equal constants. D. TWO SPECIAL HYPERSURFACES, p = CONSTANT From the above classes of hypersurfaces, we examine the following two: (A) ka = kb = k, = O (B) ka = kb = kn = constant 0, and get sufficient conditions to satisfy the Codazzi relations. We shall show that, if the condition (A) holds, then the hypersurface is a hyperplane. If the condition (B) holds, then we shall show that the hypersurface is a hypersphere. If the condition (A) holds, we find then from (4o14) that Qa = ~. (4.59)

42 Substituting for Sg from the above equation in the Gauss equation (cf. Ref. 7, P. 197) %a76 = c7ayf 8 %s 17r a we find that %576 = ~ O The above condition implies that all the Christoffel symbols vanish for the space V3. Therefore, we have (cf. Ref. 7, p. 147) xi 0 = oPui. (4.60) The Equation (4.60) becomes, by virtue of (4.59), x i = 0 The above equations become on integration xi = i + pi where ia and pi are constants. The above equations show that the coordinates are linear in yA. Therefore V3 is a hyperplane. If the condition (B) is satisfied, then all the principal curvatures are equal. Therefore, the Equation (4.14) can be written in the form ca = - +b(aba + bb + na nj

The above equation, by virtue of (4.8), becomes ap = kga which shows that all the points of V3 are umbilics. Substituting the above value of Pa into (4o10), the Equation (4.10) becomes ui =- kxi (4.61) Since uei X ui xi axi we can integrate the Equation (4.61) to get we can integrate the Equation (4.61) to get ui = - kxi + kei (4,62) where ei is a constant vector. Taking the scalar product of the above equation with itself, we obtain gij(kxi - kei)(kxj - kej) = 1 Writing xi ei = xi, the above equation can be written in the form gijxtix' = k-2 or, removing the primes and making use of the Equation (4.1), we find (x)2 _ (x1)2 _ (x2)2 _ (x3)2 = k-2 (4.63) which is a hypersphere in E4.

44 Now, we shall find the thermodynamic quantities and the components - of the world-line vector in both the cases, when the hypersurface V3, that is, the hypersurface, p = constant, is (i) a hyperplane, (ii) a hypersphere. In the case when the hypersurface is a hyperplane, since the principal normal curvatures are zero, from the Equation (4031) we find that M = 0. Therefore, substituting this value of M into the Equation (4~35), we find that -P = 0 dr Since p and p are related by the Equation (2~7), from the above equation and the Equation (2~7), it follows that d = o 0 dr which shows that the pressure does not vary along the world-line. Since the pressure is constant on the hyperplane and also along the normal, it follows that the pressure is constant throughout he medium of flow, Since p and S are both constants, it follows that all the thermodynamic quantities are constants throughout the medium of the fluid ui is normal to the hyperplane and is a constant vector. Therefore, we find that the components of the world-line vector are constants also,

45 Let us now consider the case the hypersurfaces, p = constant, are hyperspheres. We first note that the unit normal vector of the hypersphere (4.63) is given by ui = [kx~, - kx, - kx2 kx3] (4.64) Since all the principal normal curvatures of the hypersphere are equal to k, the mean curvature M from (4.31) becomes 3k 1 - rk Substituting for M from the above equation in the Equation (4035), we find that 1 p 3k (4.65) p or 1 - rk Since p and p are related by (2.7) and p is constant on the hypersphere, p is a function of r only. Therefore, integrating the above equation with respect to r, we find that p = B(1 - rk)3 (4.66) where B is an arbitrary constant. Substituting the above value of p into the Equations (2.7), (3.48), (3.49), and (3.50), we obtain p, e, a, a, respectively. The unit normal vector of the parallel hypersphere gives the world-line. Therefore the result (4.64) gives the worldline vector.

CHAPTER V. REDUCTION OF THE GEODESIC FLOW TO NEWTONIAN MECHANICS, IN THE CASE WHEN THE HYPERSURFACES, p = CONSTANT, ARE - HYPERSPHERES In Chapter IV, we studied the properties of the geodesic flows in the space of special relativity. In this chapter, we shall reduce the geodesic flows to Newtonian mechanics in the case when the hypersurfaces, p = constant, are hyperspheres. We employ the technique used by Levi-Civita (cfo Refo l)o We find that the flow in Newtonian mechanics is three dimensional. We find also that each component of the velocity vector is the ratio of the corresponding coordinate and time; and that the density is directly proportional to the cube of time. Thus for a degenerate gas (for the isentropic case). all the thermodynamic quantities are functions of time only. A. TRANSITION TO NEWTONIAN MECHANICS In order to make transition to Newtonian mechanics, we shall need a result of the text of Levi-Civita (cf. Ref 1) o To state the result in the present notation, we note that x = ct, =, = y, x = z (5.1) are the pseudo-cartesian orthogonal coordinates of a system of special relativity with metric tensor gij satisfying the Equations (4ol)o The metric tensor of the corresponding system of Newtonian mechanics is 46

(the prime indicates a tensor in Newtonian mechanics)'gll ='g22 ='g33 = 1'gCP = 0 The Greek indices run from one through three (a, P = 1, 2, 3). Thus if corresponding relativistic and Newtonian quantities are related by an arrow, then 9ij ~'gcg i = ac, = B. The result of the paper8 in the coordinate system (5.1) is u~ 1, j = - (5.2) c where va are the components of the velocity in Newtonian case. The world-line vector ui is also given by [see (.4.64)] u = (x, - kx, - kx, - 3) (5.) where k is given by (4.29). Substituting for x~ from (5.1), k from (4.29), the component u~ becomes u = kct 1 - rk The first result of (5.2) implies that, in the above equation we need have k k -- ct (5.4) 1 - rk or (5.4) can be written in this form:

48 r - ct k From the above result we have -- a - Jr c at Substituting the value of )/3r from (5.5) into the left side of (4.65), and from (5.4) into the right side of (4.65), the Equation (4,65), in Newtonian mechanics, becomes p - 3 = 0 (5~5) at t Integrating the above equation, we get p = Ct (5o6) where C is an arbitrary constant. Since p is a function of time only, we have p is also a function of time orly; that is, p P p = _ -p = _P = = = = ax by az ax by 3z To obtain the velocity vector v%, from the second relation of (5..2), the relation (5,4) and (4.6), we have b X1 X2 X3 [ t t which can be written from (5.1) in the form [ t'] ~ (a&)

49 Now we shall show that v, p, p satisfy the equation of motion and continuity in Newtonian mechanics. The equations of motion are aV- + vP. V. 1 5P (5 9) and the equation of continuity is bL + v- = O ( o10) Substituting in (5.9) and (5.10) for v~ from (58), and' for o/)x4. and ap/ox from (5.7), we see that (5.9) is identically satisfied and (5o10) becomes -p 3 = 0, at t which is the Equation (5T5)o Therefore v, p, and p identically satisfy the equation of motion and continuity. The stream lines of flow are given by the velocity components (5o3)o They are given by the differential equations dx _ a = dz -x y z which imply that the stream lines are straight lines, From the velocity components (5o8), it is clear that the motion is irrotationalo

CHAPTER VI. INTRINSIC FORMS OF EQUATIONS OF MOTION AND CONSERVATION OF MAITER WHEN THE WORLD-LINES LIE ON A HYPERSURFACE S3 In this chapter we shall derive the intrinsic forms of the equations of motion and the equation of conservation of matter in the case when the world-lines lie on a hypersurface S3 of the space of general relativity V4. The intrinsic forms of the equations of motion (6.7) show that the flow properties depend on the normal curvature of the hypersurface in the direction of the world-lines and the geodesic curvature of the world-lines. From the equations of motion we find that the worldlines are geodesics on S3 if, and only if, the pressure does not vary a'ong the relative, curvature vector (cf. Ref. 7, p. 151) of the worldlines with respect to S3. We find also that the world-lines are asymptotic lines on S3 if, and only if, the pressure does not vary along the normal to the surface. A. HYPERSURFACES CONTAINING THE WORLD-LINES Let xJ (j = 0, 1, 2, 3) denote a curvilinear coordinate system in the four dimensional space V4 of general relativity, with metric tensor gij. The world vector ui along a world-line is given by (3.2); that is, dxi i,. = i-u ds where ds is the element of arc along the world-line. The congruence of curves determined by the vector field ui is such that the value of ui 50

51 at any point is tangent to the curve of the congruence through that point; that is, if dxi are the components of a displacement in the direction of ui, we have dxo dxl dx2 dx3 ~ 1 2 3 U U U U This system of differential equations admits three independent solutions ~i(x0, X1, x2, x3) = ci (6.1) where ci are arbitrary constants. Each of the equations in (6,1) is a hypersurface S3 in V4. The intersection of the three hypersurfaces is a curve of the congruence; or a world-lineo Also, given a worldline, there exist three surfaces (6.1) such that the world-line is the intersection of these three hypersurfaces. We shall study the properties of the world-line in relation to one of these hypersurfaces. B. INTRINSIC EQUATIONS At each point on a world-line, we introduce the four unit vectors, ui along the world-line, ai along the normal to the surface, bi along the relative curvature vector (cf. Refo 7, p. 151) of the world-line with respect to S3, and ni orthogonal to ui, ai, and bi. The following relations are satisfied between the curvature vector ui ju of the world-line and the vectors ai and bi: ui u = - k ai + k bi (6.2);J u g'

52 where k. is the normal curvature of S3 of the world-line and kg is the relative curvature (cf. Ref. 7, P. 151) of the world-line with respect to S3. The above equation is the generalization of Meusnier's theorem (cf. Ref. 7, p. 152). The vectors ai and bi are orthogonal. We shall now express the variation of p, p, and a along the four directions ui, a, bi, and ni with the aid of the equations of motion (3.10) and the equation of conservation of mass (3o11). Using the notation (3.17), the Equation (3.11) can be written in the form (3.21). The divergence of ui appearing in the Equation (3.21) is the sum of the principal values of the symmetric tensor Sij (3.24), which we denoted by M. Therefore the equation of conservation of mass reduces to the Equation (3.28). Let us now substitute in (3.10) for uJ;j from (3.26), for ui;ju from (6,2). Then the Equation (3.10) becomes;juJ c;juJui + c2a(uiM - kuai + kgb) gjp = 0. (63) Forming the scalar product of (6.3) with ui, ai, bi, and ni and using the orthogonal properties of these vectors, we obtain c a;ju + C2^ - p;ju = 0 c2ok - p;ja = 0 (6.4) -c2okg - p;jb = 0 p.jn = 0 form

553 d a+ M = 1 gl ds c2 ds _ 1 d, ku = u c2 da (6.5) -k 1c dpb - kg cya db 0 = -kP dn To this system we have to add the equation of conservation of mass dp + pM = 0.(6.6) ds From the first three equations of (3531), Equations 6.5) and (6.6), we have the equations = p ( - + - += 0, = - pM c( 2 pc2 )' ds ds 1 dp c cT da (6.7) -kg = C2cdb 0 = a - dn These equations and the first law of thermodynamics (2.3) constitute the basic equations of the system. From the equations of motion, we shall deduce the properties of fluid flow. The last equation of (6.7) shows that the pressure does not ~va~ry alotUhe dit Uliteet A.iOl il'wo thile il'Lih tquauiOul, we ~iiLu Li-aL, iL the world-line is a geodesic on V3, then the pressure does not vary along

54 the relative curvature vector of the world-line, and conversely it follows from the same equation, that if the pressure does not vary along the relative curvature vector of the world-line, the world-line is a geodesic on S3. Hence we have that the world-lines are geodesics on S3 if, and only if, the pressure does not vary along the relative curvature vector of the world-line with respect to S3. From the third equation of (6.7), it follows that if the normal curvator ku of the world-line is zero. then the pressure does not vary along the normal to the hypersurface S3; and conversely. But the normal curvature of the world-lines is zero provided that the world-line is an asymptotic line on S3 (cf. Ref. 7, p. 156). Therefore, we have the following result. The world-lines are asymptotic lines on S3 if, and only if, the pressure does not vary along the normal to the hypersurface S3.

CHAPTER VII. THE VORTICITY TENSOR AND THE VORTICITY VECTOR In this chapter we study the geometric properties of the vorticity tensor and the vorticity vector, and we shall prove the following results: (a) The fluid flow is irrotational if and only if the world-lines are geodesics. (b) In the case of the Beltrami flows, the vorticity tensor lies in the two-plane formed by the world-line vector and the principal normal vector of the world-line. (c) In the case of isentropic Beltrami flows, the hypersurfaces, p = constant, are orthogonal to the principal normal vector of the world-lines; in this case, all the thermodynamic quantities are constant along the world-lines. (d) For steady flows, the Bernoulli hypersurface contains the world-lines. It contains the vorticity vector also, if the entropy does not vary along the vorticity vector. A. GENERAL FLOWS The vorticity tensor is defined by wij I (ui;j uj;i) Let us define the vorticity vector wi by 55

56 wi = EiJklwklUJ (7.2) Note that if wk vanishes then wi vanishes, and not conversely. The tensors EiJkL and Eijk are related to the tensor density Eijkl and tensor capacity eijkl byll 1 Eijkl = lg2 Ei k (7.3) 1 Eijkl = Ig I eijk (7.4) Note that Eijkl and eijk~ have the values 1, when i, j, k, I is an even permutation; -1, when i, j, k, I is an odd permutation; 0, otherwise. And g is the determinant of the matrix gij. The tensors Eijkl and Eipst satisfy, EiikEipst = - 53'ipbks5 t] (7.5) where bJ 5k 56 is the generalized Kronecker tensor; the brackets de[p s t] noting the alternate sum. From the definitions of the tensor Eijkl and the vorticity vector wi, and from the relation (7.2), it follows that wiui =, (7.6) that is, the velocity vector and the vorticity vector are orthogonal. The fluid motion with Wij = 0 (7.7)

57 is said to be irrotational. We define the fluid motion with wi = 0 (7.8) as Beltrami flow. Note, Taub12 defines the flow to be irrotational if wi = 0. From (7.2), (7.7), and (7.8), it follows that irrotational motion implies Beltrami flow. Following the notation of Taub (cf. Ref. 12), we introduce the vector vi = e- i (7.9) where the function A is given by /- =:I. (7.10) Po oc Po in the above equation is independent of the coordinates of the points of space-time. The integrand is considered a function of p and S, and the integration is carried out with S constant. Defining the skewsymmetric tensor ij = (i;j - j; i) (7.11) Taub has shown that wi vanishes and the motion is isentropic, if and only if (cf. Ref. 12) 4^ = 0,(7.12) and also that

58 ij = wViJl (s;jUi - 8;i ) (7.13) where T is the proper temperature. From (7.1), (7.9), and (7.11), it follows that "ij = e[wj - j (jui - U;i)]. (7.14) Now, using these above equations, we shall find the intrinsic properties of flow when the fluid flow is (a) irrotational, (b) Beltrami flow. Multiplying both sides if the Equation (7.2) by EFmnp, and using the Equation (7.5), the Equation (7.2) becomes wiE = -35 5 w u imnp [m n p]skluj Expanding the Kronecker tensor, the right-hand side of the above equation can be written in the form wEimn -p 3 + unwm + UpWmn) which by virtue of (7.1) becomes 3 ijk! = uJ(uk; uI;k) + Uk(ul;j - uj;I) + U(Uj;k - Uk;j) (7.15) Multiplying the above equation by uj and making use of the Equations (3.3), (3.12), and (7.1), the above equation becomes k2W Eijk w'uJ + U kUk;J - U l;J (7.16)

59 If wkl is zero, then we know from (7.2) that wi is zero. Therefore, if wk, is zero, then forming the scalar product of (7.16) with uI, the curvature vector uk;juJ of the world-line vector is zero. Hence we conclude that, if the fluid motion is irrotational, then the world-lines are geodesics. In Chapter III, we have seen that the fluid motion is irrotational when the world-lines are geodesics. Thus, from the result of Chapter III and the present result, we find that the fluid motion is irrotational if, and only if, the world-lines are geodesics. This result is not valid in the Newtonian case. There, if the motion is irrotational, the stream lines are not necessarily straight lines. Let us now consider the properties of flow when the vorticity vector vanishes; that is, the flow is Beltrami. When wi is zero, from (7.16) it follows that wki = (uuk;juJ - uku;jud) Substituting for the curvature vector ui; juJ from the first equation of (3.36), the above equation becomes kl = - 1 kl(ua - u uak) (7.17) where ai is the principal normal vector of the world-lines. The above equation shows that if the flow is Beltrami, that is, if wi vanishes, then the vorticity tensor lies in the two-plane formed by the worldline vector and the principal normal vector of the world-lines. Further, in addition to being Beltrami flow, if the motion is

60 isentropic, that is, if Qij is zero, then from (7.14) and Taub's result [see (7.12), it follows that "w 1= I (8;jUi Uj;i) X (7-18) Comparing (7.17) and (7.18), we find that O;i = - klai, (7.19) and from the Equation (7.10), since the integrand is a function of p and S, we have x1= aC2 = x Is 5 & (7.20) The last term of the right-hand side of the above equation is zero, since the motion is isentropic. Therefore the Equations (7.19) and (7.20) imply that 2 = klai. (7.21) ac The above result shows that, in the case of isentropic Beltrami flows, the hypersurfaces, p = constant, are orthogonal to the principal vector of the world-lines. Again, comparing (7.21) with (3.40), we find that the pressure does not vary along the world-line. Since both p and S do not vary along the world-line, and since only two of the six thermodynamic quantities c, p, p, T, a, S, are independent (Chapter II), it follows that all the thermodynamic quantities are constant along a world-line. Therefore, we have, in the case of isentropic, Beltrami

61 flows all the thermodynamic variables are constant along a world-line. These constants may vary from one world-line to another. B. STEADY FLOWS Now we shall study the properties of the vorticity vector, in the case of steady flows. In describing the study flows of fluids in their own gravitational fields, it is assumed that the space-time and the velocity field are invariant under a time-like one parameter group of motions. That is, there exists a Killing vector (cf. Ref. 12), satisfying the following equations: i;j + j;i = ui;i - iuJ = 0 (7.22) pi - lUpi S;iti = 0 = P; Taub has shown (cf. Ref. 12), that the invariant H = vii = e-u (7.23) is the Bernoulli function for the fluid flow in relativity; and also, that H;i = Qjitj. (7.24) Now we shall show that the Bernoulli hypersurface, that is, the hypersurface, H = constant, contains the world-lines; and that it contains the vorticity vector also, if the entropy does not vary along the vor

62 ticity vector. To show this, let us form the scalar product of (7.12) with uj and vj and use the fact that the entropy does not vary along the world-line, and the Equation (7.6). The the Equation (7.12) becomes ijuj= 2 s8;i ijw = T s;jwui Multiplying both sides of the above equations by i and using (7.22), (7.23), and (7.24), the above equations become H..u = o (7.25) H; jw = (THe)s- jw. (7.26) The Equation (7.25) implies that the world-line vector is on the hypersurface, H = constant. Equation (7.26) shows that, if the entropy does not vary along the vorticity vector, then the hypersurface, H = constant, contains the vorticity vector also. In particular, we have that in the case of isentropic flow the Bernoulli hypersurfaces contain the worldline vector and the vorticity vector.

REFERENCES 1. Levi-Civita, T., The Absolute Differential Calculus, Blackie and Son Ltd., London (1929), p. 358. 2. Taub, A. H., Isentropic Hydrodynamics in Plane-Symmetric Space-Time, Tlhe Physical Review, Vol. 103 (1956), pp. 454-467. 3. Taub, A. H., Relativistic Rankine-Hugoniot Equations, The Physical Review, Vol. 74, No. 3 (1948), pp. 328-334. 4. Lichnerowicz, A., Theores relativistes de la gravitation et 1'Electromagnetisme, Masson et Cie., Paris (1955). 5. Coburn, N., The Method of Characterisitics for a Perfect Compressible Fluid in General Relativity and Non-Steady Newtonian Mechanics, Journal of Mathematics, Vol. 7 (1958), pp. 44.9-482. 6. Coburn, N., Intrinsic Relations Satisfied by the Vorticity and Velocity Vectors in Fluid Flow Theory, Michigan Mathematical Journal, Vol. I, No. 2, pp. 113-130. 7. Eisenhart, L. P., Riemannian Geometry, Princeton University Press, Princeton (1949), p. 151. 8. Coburn, N., Intrinsic Form of the Characteristic Relations for a Perfect Compressible Fluid in General Relativity and Non-Steady Newtonian Mechanics, Journal of Mathematics and Mechanics, Vol. 9, No. 3 (1960), pp. 421-458. 9. Schouten, J. A., and Struik, D. J., Einfuhrung in die Neuren Methoden der Differentialgeometrie, P. Noordhoff, Groningen, Batavia (1958), p. 28. 10. Weatherburn, C. E., Introduction to Riemannian Geometry and Tensor Calculus, Cambridge University Press (1957), p. 105. 11. Coburn, N., Relativity Theory, Lecture Notes, University of Michigan (1960), p. 27. 12. Taub, A. H., On Circulation in Relativistic Hydrodynamics, Archive fnor RPfl-nnnl M oIhmnm o ai n Ana!ry-ni *,A Vnl % Non L (10OC;'Q -.Al n 12-24 63