Quantum Physics or Physics course

Please Answer Problem 2, (a, b, and c)

Problem 2 Relativistic Stress-Energy Tensor for Perfect Fluids and their Dynamics We studied the stress-tensor, T, in 3-dimensional Euclidean space. As one might expect, it can be generalized to a rank-2 tensor in 4-dimensional Minkowski spactime, T(_ _), called the stress-energy tensor. Like the stress-tensor is a 3-momentum flux density, so too is the stress-energy tensor a 4-momentum flux density. That is, T(_. E) = total 4-momentum P that flows through surface S. (Our notation here matches that of midterm 2, where a 3-dimensional vector is denoted A, a 3-dimensional tensor by T. 4-dimensional vector by A and a 4-dimensional tensor by T.) Although a bit more subtle to work through in the present case, a component-by- component analysis of the stress-energy tensor leads to the understanding that physically, its contravariant components are 70% = energy density, 71 = momentum density, 70) = energy flux, Tik = stress. Although not obvious at first, the stress-energy tensor is symmetric, To = Toa (yes, the momentum density is equal to the energy flux). As expected, it is conserved: the law of conservation of 4-momentum is ToB d58 = 0. ne where OV is a closed 3-surface bounding a 4-volume V. Applying Gauss's theorem converts this global integral conservation law into a local differential one: VT's - 0. V . T= 0. In a specific, but arbitrary, Lorentz frame, the temporal part is OTO aTok at Ork = 0. which states that the time derivative of the energy density plus the 3-divergence of the energy flux vanishes; and the spatial part is OTj0 OTik at + ark = 0.which states that the time derivative of the momentum density plus the 3-divergence of the stress (i.e., of momentum flux) vanishes. As anticipated, the conservation of stress-energy encapsulates both conservation of energy and conservation of momentum. As an important example of a stress-energy tensor, we consider a relativistic perfect fluid. This is an ideal fluid (i.e. it supports only pressures and no shears) whose speed is comparable to the speed of light. so that we must replace the Newtonian description of the fluid by a relativistic one. Throughout this problem, we work in units where c = 1. For the perfect fluid, it's stress-energy tensor evaluated in its local rest frame (a frame where 770 = 0, i.e. it is not moving), has the form To = p, Tik = pgik where p is short-hand notation for the energy density (density of total mass-energy, including rest mass), and the stress tensor T" is an isotropic pressure, P. This is the form of the stress-energy tensor in the local rest frame, from which it follows that the geometric, frame- independent expression in terms of the 4-velocity u of the local rest frame (i.e. of the fluid itself), the metric tensor of spacetime g, rest-frame energy density p and pressure P is ToB = (p + Pu'u + Pgas; T = (p+ PjuQu+Pg. (a) Derive/explain the frame-independent expression for the perfect fluid stress-energy tensor starting from its rest-frame components. Hint: start from the fact that the stress-energy tensor is a symmetric tensor made from u. p, P, and g and so it must be of the form To = Audu + By 5. Use the local rest-frame to determine A and B. It is convenient to write the fluid's total density of mass-energy as p = po(1 + w). where Po is the density of rest mass and w is the specific internal energy. The dynamics of our relativistic, ideal fluid are governed by five equations. The first equation is the law of rest- mass conservation: Va(pou") = 0. (b) Show that the law of rest-mass conservation can be writen as dpo dr = -poV . 1, or d( po V ) =0, dr where d/ dr = u . V is the derivative with respect to proper time moving with the fluid, V . u= (1/V)(dV/dr) is the divergence of the fluid's 4-velocity, and V is the volume of a fluid element. The second equation is energy conservation, in the form of the vanishing divergence of the stress-energy tensor projected onto the fluid 4-velocity: U.(VsTo8) = 0. (c) Show that the law of energy conservation, when combined with the law of rest-mass conservation, reduces to ap = -(p+ P)V . u, d(PV) dv or dT dT