what happens to the energy momentum tensor when a total divergence is added to the lagrangian

Energy-Momentum Tensor

The EMT Tinvμν given by (A.11) is actually traceless off-beat out for north=4.

From: Nuclear Physics B , 2021

Field Variational Principles for Irreversible Energy and Mass Transfer

Stanislaw Sieniutycz , in Variational and Extremum Principles in Macroscopic Systems, 2005

xi Free energy-momentum tensor and conservation laws in the estrus theory

The energy-momentum tensor is defined as

(47) $G^{jk}\equiv {\displaystyle \sum _l\frac{\partial 5_l}{\partial {\chi }^j}\left[\frac{\partial \Lambda }{\partial (\partial v_{\mathrm{fifty}}\text{/}\partial {\chi }^k)}\right]-{\delta }^{jk}\Lambda ,}$

where δ^jk is the Kronecker delta and χ = (x, t) comprises the spatial coordinates and fourth dimension. The conservation laws are valid in the absenteeism of external fields; they describe and then the vanishing four-divergences (∇, ∂/∂τ) of K^jk. Our arroyo here follows those of Stephens [13] and Seliger and Whitham [14], where the components of G^jk are calculated for Λ gauged past utilize of the difference theorem along with differentiation by parts. The link of the components of tensor G^jk with the partial derivatives of four principal functions S_j that are solutions of Hamilton–Jacobi equations is known [31].

Any physical tensor G = G^jk has the following general structure

(48) $\mathrm{Chiliad}=\left[\begin{array}{cc}T& -\Gamma \\ Q& E\end{array}\right],$

where T is the stress tensor, Γ is the momentum density, Q is the energy flux density, and East is the total free energy density.

When external fields are nowadays, the kinetic potential Λ contains explicitly some of coordinates χ^j. And then the balance equations are satisfied rather than conservation laws

(49) ${\displaystyle \sum _{\mathrm{grand}}\left(\frac{\partial {\mathrm{One\; thousand}}^{jk}}{\partial {\chi }^k}\right)}+\frac{\partial \Lambda }{\partial {\chi }^j}=0,$

for j, m = ane, 2, 4. Eq. (49) is the formulation of rest equations for momentum (j = 1, 2, iii) and energy (j = four).

We shall focus first on the heat model considered in Sections one–8. We think the assumption of the small deviation from equilibrium at which that model is physically consistent. With this assumption and for the kinetic potential of Eq. (7) gauged every bit described to a higher place, the approximate activeness assures that the components of the free energy-momentum tensor are multiplier independent. These components are given past Eqs. (fifty)–(53) beneath. Respectively, they draw: momentum density Γ^α , stress tensor T^αβ , total energy density E, and density of the full energy flux, Q^β , which approximately equals q^β.

The momentum density for the mass menses of the medium at rest is, of grade, J = 0, where J is the mass flux density. The momentum density of heat flow follows as

(50) ${\Gamma }^{\alpha }=-G^{\alpha 4}=c_0^{-2}\frac{{\rho }_{\text{e}}}{\epsilon }{q}^{\alpha }\cong c_0^{-\mathrm{ii}}{q}^{\alpha },$

or, in the vector form, $\Gamma =q{c}_0^{-\mathrm{ii}}$ , whereas the stress tensor T^αβ has the form

(51) ${\mathrm{1000}}^{\alpha \beta }=T^{\alpha \beta }={\epsilon }^{-\mathrm{one}}\{-c_0^{-2}{q}^{\alpha }{q}^{\beta }+{\delta }^{\alpha \beta }(\frac{1}{\mathrm{ii}}{q}^{\mathrm{ii}}{c}_0^{-2}-\frac{\mathrm{one}}{2}{\rho }_{\text{e}}^2+\frac{1}{2}{\epsilon }^2)\}.$

This quantity represents stresses acquired by the pure heat menses; it vanishes at equilibrium. The full energy density is

(52) ${\mathrm{Grand}}^{44}=E^{\text{tot}}=\frac{1}{2}{\epsilon }^{-1}{c}_0^{-1}{q}^2+\frac{\mathrm{one}}{\mathrm{two}}{\epsilon }^{-\mathrm{ane}}{\rho }_{\text{e}}^2+\frac{1}{\mathrm{ii}}\epsilon \cong \frac{\mathrm{ane}}{2}{\epsilon }^{-1}{c}_0^{-2}{q}^{\mathrm{ii}}+{\rho }_{\text{e}}.$

Finally, nosotros find for the energy flux

(53) $G^{4\beta }=Q^{\beta }={\epsilon }^{-1}{\rho }_{\mathrm{due\; east}}{q}^{\beta }\cong q^{\beta }.$

In the quasiequilibrium situation ρ _e is very close to ε, then the formal density of the free energy flux Thou ^4β coincides with the heat-flux density, q.

As the estrus flux, q, is both the process variable and the entity resulting from the variational procedure, the fact that it is recovered here may be regarded as a positive test for the self-consistency of the procedure.

The associated conservation laws for the energy and momentum have the class

(54) $\frac{\partial (\frac{\mathrm{i}}{2}{\epsilon }^{-1}{c}_0^{-\mathrm{ii}}{q}^2+{\rho }_{\text{east}})}{\partial t}=-\nabla \cdot ({\epsilon }^{-1}{\rho }_{\text{eastward}}q)$

(55) $\frac{\partial (c_0^{-2}{\epsilon }^{-1}+{\rho }_{\text{e}}{q}^{\alpha })}{\partial t}=\nabla \cdot \{{\epsilon }^{-1}(-c_0^{-\mathrm{ii}}{q}^{\alpha }{q}^{\beta }+{\delta }^{\alpha \beta }(\frac{1}{\mathrm{ii}}{q}^2{c}_0^{-2}-\frac{1}{2}{\rho }_{\text{eastward}}^{\mathrm{ii}}+\frac{1}{2}{\epsilon }^2))\}.$

The energy-conservation law (54), which stems from Eqs. (49), (52), and (53), refers to nonequilibrium total energy Due east that differs from the nonequilibrium internal free energy ρ _e by the presence of the "kinetic energy of oestrus" (explicit in 50 of Eq. (seven) or in Eq. (52)). The necessity of stardom between E and ρ _{due east} is caused by the property of finite thermal momentum (50) in the frame-work of a stationary skeleton of a rigid solid, in which we work. The physical content of results stemming from the quadratic kinetic potential L thus seems adequate when the system is close to equilibrium.

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THE ENERGY-MOMENTUM TENSOR IN MACROSCOPIC ELECTRODYNAMICS

V.L. GINZBURG , in Theoretical Physics and Astrophysics, 1979

Publisher Summary

This chapter discusses the energy–momentum tensor in macroscopic electrodynamics. It discusses the applications of the energy and momentum conservation laws to the radiation of electromagnetic moving ridge (photons) in a medium. The tensor is the energy–momentum tensor for a uniform medium at balance. Abraham force is genetically continued with the strength due to the magnetic field (Lorentz strength) acting upon the deportation current. In the field of applications of the energy and momentum conservation police force, usually only 2 points are important: (ane) what is the energy and momentum lost (or gained in absorption) in the emitting particle or system and (two) what is the field energy emitted in a given direction. The chapter further discusses the issues of the free energy–momentum tensor and the forces in a medium.

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Particle Physics and Cosmology: The Fabric of Spacetime

R.U.H. Ansari , ... F.R. Urban , in Les Houches, 2007

3 Gravitons [2]

In this section nosotros will hash out how gravitons couple to affair fields on the brane. Since we piece of work in an infinite extra–dimensional model, we have a continuous spectrum of KK graviton states, including the conventional 4d graviton as the zero mode. Nosotros will work in conformally flat infinite, keeping in listen that at high curvatures (due east.grand. in the inflationary epoch) this approximation must be corrected.

Matter couples to gravitational fluctuations h_AB via the energy–momentum tensor T_AB ,

(3.1) $\begin{array}{cc}{S}_{\mathrm{int}}\sim {\displaystyle \int d^{4}xdy{h}_{AB}{T}^{AB},}& A,B=0\dots \mathrm{four},\end{array}$

where the 5d energy–momentum tensor of affair confined to the brane becomes (in the weak gravitational field limit) $T_{AB}={\eta }_A^{\mu }{\eta }_B^5{T}_{\mu v}\left(\mathrm{ten}\right)\delta \left(y_c\right)$ , and $h_{AB}\left(x,y\right)$ are the perturbations specified beneath in (3.two). This means, that after integrating out the fifth dimension y in (3.1), the constructive 4d coupling will be determined by the aamplitude of $h_{AB}\left(\mathrm{ten},y\right)$ , at the (Planck or probe) brane position y _c. We parametrise the 4d perturbations ³ every bit

(3.two) $\begin{array}{cc}d{s}^2=\left({\mathrm{east}}^{-\mathrm{ii}\mathrm{one\; thousand}\left|y\right|}{\eta }_{\mu 5}+h_{\mu \mathrm{five}}\left(x,y\right)\right)d{\mathrm{ten}}^{\mu }d{x}^v-d{y}^{\mathrm{ii}},& \mu ,v=0\dots 3,\end{array}$

where we will introduce the factorisation $h_{\mu v}\left(x,y\right)={\widehat{h}}_{\mu 5}\left(x\right)\psi \left(y\right)$ . Plugging this ansatz into the 5d Einstein equations, 1 finds that $\psi$ obeys a Schrödinger–like equation. Its solutions show a strong localisation at the Planck brane for the zero mode, whereas the massive graviton modes are suppressed there and asymptotically approach a plane moving ridge as nosotros movement far away from the brane. This means that the aught mode coupling at the Planck brane, given by $\psi \left(0\right)$ , volition be much larger than that of private massive KK states (their contribution might be meaning nevertheless, as we sum over a large number of them). On a TeV probe brane, on the other manus, we discover the opposite coupling hierarchy.

Nosotros would like to study the bear upon of gravitational corrections on stage transitions. We will therefore compute the effective potential for a scalar field that undergoes spontaneous symmetry breaking, like the Higgs. For the Electroweak phase transition (which would be interesting e.thou. for baryogenesis models) we would have to couple all standard model fields to the KK gravitons. Equally a slightly less ambitious stride, let us consider the 1–loop effective potential of a single scalar field coupled to gravity, which could be relevant for the inflationary era.

The one–loop effective potential is given as a sum over all 1PI diagrams with a single scalar/graviton loop. This can be re–summed into a i–loop vacuum diagram (a loop with all external legs removed). Our task is therefore to calculate this diagram for the massless graviton also every bit the continuum of massive KK states. If we are located on the Planck brane, nosotros should integrate over the full KK mass spectrum (upwardly to a cutoff of the society of Planck mass); if we alive instead on a probe brane somewhere in the infinite 5th dimension nosotros only need to take masses 10⁻⁴ eV ≤ m ≤ 1 TeV into account (assuming a hierarchy such that the fundamental scale on the probe brane is precisely ane TeV). In both cases we have to impose a UV cutoff on the loop integral. This will non yield a normalisable result: fifty-fifty at one loop nosotros will meet an explicit cutoff dependence. This is due to the well–known fact that gravity cannot be renormalised as a QFT.

The KK decomposition, graviton propagators and their coupling to matter and gauge fields have been considered for toroidal extra dimensions; we will have to repeat the analysis for the infinite AdS majority. We promise to come across some not–negligible contribution to the symmetry–breaking potential that could result in interesting consequences for the (order of the) phase transition, inflationary dynamics and baryogenesis.

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Judge THEORY OF THE CONFORMAL Group

J.P. Harnad , R.B. Pettitt , in Group Theoretical Methods in Physics, 1977

3.ane Identities and Conservation Laws

Now, by a standard method of minimal replacement ^{(half dozen)} , ane can go from a Lagrangian density L(ψ,∂_μ ψ) (henceforth the subscript on ψ_A volition be suppressed, a particular choice of local section σ_A being implicit) to £(ψ,ψ;i) where the second Lagrangian is an invariant for arbitrary (non-rigid) group transformations provided L is invariant nether rigid ones. The consequences of such invariance may be expressed by the following set up of identities and conservation laws.

(i)

Human relationship betwixt dynamically and kinematically defined energy-momentum tensors:

(3.i) $T_{\upsilon }^{\mu }={\text{t}}_{\text{υ}}^{\text{μ}}-{\text{South}}_{\text{a}}^{\text{μ}}{\text{ω}}_{\text{υ}}^{\text{a}}$

where

(iii.two) ${\text{t}}_{\text{υ}}^{\text{μ}}\equiv {\text{h}}_{\text{i}}^{\text{μ}}\frac{\partial \text{h}}{\partial {\text{h}}_{\text{i}}^{\text{μ}}}$

is a generalization of the dynamical energy-momentum density.

(three.3) $T_{\upsilon }^{\mu }\equiv -{\delta }_{\upsilon }^{\mu }\mathfrak{Fifty}+\frac{\partial \mathfrak{L}}{\partial \left({\partial }_{\mu }\text{ψ}\right)}{\partial }_{\upsilon }\text{ψ}$

is the canonical (kinematically defined) energy-momentum tensor density and

(three.four) ${\text{Due south}}_{\text{a}}^{\text{μ}}=\frac{\partial \mathfrak{L}}{\partial {\text{ω}}_{\text{μ}}^{\text{a}}}$

is the intrinsic current density coupling to the judge field ω_ω ^a.

(ii)

Relationship between dynamically and kinematically defined current densities:

(three.5) ${\text{S}}_{\text{a}}^{\text{μ}}=\frac{\partial \mathfrak{L}}{\partial {\partial }_{\text{μ}}\text{ψ}}{\text{T}}_{\text{a}}\text{ψ}$

(intrinsic current associated with the one-parameter subgroup generated past t_a).

(iii)

Covariant conservation laws for currents:

(3.six) ${\nabla }_{\mu }{\text{S}}_{\text{a}}^{\text{μ}}=0(\text{non-linear\hspace{0.17em}role\hspace{0.17em}of\hspace{0.17em}G)}$

(3.vii) ${\nabla }_{\mu }{\text{Southward}}_{\text{j}}^{\text{i}\text{μ}}={\text{t}}_{\text{υ}}^{\text{μ}}{\text{b}}_{\text{μ}}^{\text{i}}{\text{h}}_{\text{j}}^{\upsilon }\equiv {\text{t}}_{\text{j}}^{\text{i}}\text{(linear\hspace{0.17em}part\hspace{0.17em}of\hspace{0.17em}G)}$

(where = covariant difference of S_a ^μ).

(3.8) $\begin{array}{l}\text{(where}{\nabla }_{\mu }{\text{Southward}}_{\text{a}}^{\text{μ}}={\partial }_{\text{μ}}{\text{S}}_{\text{a}}^{\text{μ}}-{\text{f}}_{\text{ab}}^{\text{c}}{\text{ω}}_{\text{μ}}^{\text{b}}{\text{S}}_{\text{c}}^{\text{μ}}\\ \text{=\hspace{0.17em}covariant\hspace{0.17em}departure\hspace{0.17em}of}{\text{S}}_{\text{a}}^{\text{μ}})\end{array}$

(iv)

Covariant Divergence of Energy-Momentum Tensor:

(three.9) ${\partial }_{\text{μ}}{\text{t}}_{\text{υ}}^{\text{μ}}+{\Gamma }_{\text{μυ}}^{\alpha }{\text{t}}_{\alpha }^{\text{μ}}={\text{R}}_{\text{μυ}}^{\text{a}}{\text{S}}_{\text{a}}^{\text{μ}}$

where

(3.10) $\begin{array}{l}{\Gamma }_{\text{μυ}}^{\alpha }=\left({\text{b}}_{\text{μ,}\upsilon }^{\text{i}}+{\text{ω}}_{\text{jυ}}^{\text{i}}{\text{b}}_{\text{μ}}^{\text{j}}\right){\text{h}}_{\text{i}}^{\text{α}}\\ \begin{array}{ll}\hfill & =-\left({\text{h}}_{\text{i,}\upsilon }^{\text{α}}-{\text{ω}}_{\text{jυ}}^{\text{i}}{\text{h}}_{\text{i}}^{\text{α}}\right)\hfill \end{array}{\text{b}}_{\text{μ}}^{\text{i}}.\end{array}$

Equation (3.9) has the interpretation that a coupling between the curvature and the intrinsic currents gives ascent to a force density causing deviations from geodesic motion.

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Properties of Higgs bosons

50.B. OKUN , in Leptons and Quarks, 1984

24.half-dozen Digression on the trace of the energy-momentum operator

The result obtained above stems from a spectacular property of the energy-momentum tensor. We shall at present discuss this property. The trace of the total energy-momentum operator of gluons and quarks is written in the course

${\theta }_{\mu }^{\mu }={\displaystyle \sum _l{m}_{q_{\mathrm{50}}}{\overline{q}}_{\mathrm{50}}{q}_l}+{\displaystyle \sum _h{m}_{q_h}{\overline{q}}_h{q}_h}-\frac{\text{b}{\alpha }_s}{8\pi }{\mathrm{1000}}_{\mu \upsilon }^a{\mathrm{Thousand}}_a^{\mu \upsilon },$

where $b=\mathrm{xi}-\frac{\mathrm{ii}}{3}{N}_l-\frac{2}{\mathrm{iii}}{N}_h\equiv \tilde{\text{b}}-\frac{\mathrm{two}}{3}{N}_h$ , and indices 50 and h refer to light and heavy quarks, respectively. In contrast to the preceding sections of the present affiliate, we define hither the u-, d-, and s-quarks whose masses are beneath μ_c (1/μ_c is the confinement radius), as light quarks, and all other quarks as heavy. (In the general example, therefore, N_h ≥ N_H ; the equality N_H = Due north_h and then holds but if one thousand_H ≃μ_c).

The last term in the expression for ${\theta }_{\mu }^{\mu }$ is the contribution of the so-called triangle anomaly. A pertinent image is that of the contribution due to the "shadow cabinet" of gluons and quarks with infinitely high masses, that is the so-called Pauli-Willars particles which regularize (cut off) divergent graphs. These particles have entered the free energy-momentum tensor from the lagrangian where they are introduced in order to regularize the theory. The triangle anomalies in question are represented in the graphs of fig. 24.9 (recall that gravitons interact with the energy-momentum tensor). The graphs (a) and (c) of this figure contain concrete particles, while the graphs (b) and (d) contain regularizing particles in loops. The graphs (c) and (d) cancel each other out, and heavy quarks drop out if the virtualities of external gluons are small compared to quark masses in the triangle.

Fig. 24.9.

Graph 24.9a gives nix contribution to the trace of the energy-momentum tensor since the gluon mass is zero. For light quarks, the graph (c) contains a small factor 1000 _q/μ_c, where 1/μ_c is the confinement radius. Therefore, the regularizing contribution to ${\theta }_{\mu }^{\mu }$ is non cancelled out for lite quarks and gluons.

In the instance of the nucleon the matrix elements

$〈N\left|m_{qh}{\overline{q}}_h{q}_h\right|\mathrm{Northward}〉\text{and}〈N\left|\frac{2}{\mathrm{three}}{N}_h\frac{{\alpha }_{\mathrm{due\; south}}}{\mathrm{viii}\pi }{G}_{\mu \upsilon }^a{G}_a^{\mu \upsilon }\right|\mathrm{Northward}〉$

cancel to an accuracy of terms of the club of (μ_c/m_qh )². If we as well utilize the fact that in the chiral limit m_q50 → 0, we come up to the decision that in this limit the nucleon mass is adamant by the contribution of ${\tilde{\theta }}_{\mu }^{\mu }$ .

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Particle Physics and Cosmology: The Fabric of Spacetime

Due north. Bartolo , ... A. Riotto , in Les Houches, 2007

Appendix B.ii Radiation-dominated era

We consider a universe dominated by photons and massless neutrinos. The energy-momentum tensor for massless neutrinos has the same grade as that for photons. During the phase in which radiation is dominating the energy density of the Universe, a ∼ η and we may combine Eqs. (A.iii) and (A.vii) to obtain an equation for the gravitational potential Ψ^(one) at first order in perturbation theory

(B.5) $\begin{array}{ll}{\Psi }^{\left(1\right)\text{'}\text{'}}+\mathrm{iv}\mathscr{H}{\Psi }^{\left(1\right)\text{'}}-\frac{1}{\mathrm{iii}}\hfill & {\nabla }^{\mathrm{ii}}{\Psi }^{\left(1\right)\text{'}}=\mathscr{H}{Q}^{\left(\mathrm{i}\right)\text{'}}+\frac{\mathrm{i}}{\mathrm{iii}}{\nabla }^2{Q}^{\left(1\right)},\hfill \\ \hfill & {\nabla }^{\mathrm{two}}{Q}^{\left(\mathrm{i}\right)}=\frac{\mathrm{nine}}{\mathrm{ii}}{\mathscr{H}}^2\frac{{\partial }_i{\partial }^j}{{\nabla }^2}{\displaystyle {\prod }_T^{\left(\mathrm{i}\right)i}j,}\hfill \end{array}$

where the total anisotropic stress tensor is

(B.half-dozen) ${\displaystyle {\prod }_{T\text{undefined}j}^i=}\frac{{\overline{\rho }}_{\gamma }}{{\overline{\rho }}_T}{\displaystyle {\prod }_{\gamma \text{undefined}j}^i+}\frac{{\overline{\rho }}_v}{{\overline{\rho }}_T}{\displaystyle {\prod }_{\mathrm{five}\text{undefined}j}^i.}$

Nosotros may safely neglect the quadrupole and solve Eq. (B.5) setting $u\pm ={\Phi }_{\pm }^{\left(\mathrm{ane}\right)}\eta$ . Then Eq. (B.five), in Fourier space, becomes

(B.7) $u^{\text{'}\text{'}}+\frac{2}{\eta }{u}^{\text{'}}+\left(\frac{k^{\mathrm{ii}}}{3}-\frac{2}{{\eta }^2}\right)u=0.$

This equation has as independent solutions $u_{+}=j_{\mathrm{i}}\left(k\eta /\sqrt{\mathrm{three}}\right)$ the spherical Bessel function of lodge 1, and $u_{-}=n_1\left(k\eta /\sqrt{3}\right)$ , the spherical Neumann role of lodge 1. The latter blows upwardly as η gets small and nosotros discard it on the basis of initial weather condition. The final solution is therefore

(B.8) ${\Phi }_{\mathrm{thousand}}^{\left(1\right)}=3{\Phi }^{\left(1\right)}\left(0\right)\frac{\sin \left(k\eta /\sqrt{3}\right)-\left(k\eta /\sqrt{3}\right)\cos \left(\mathrm{thousand}\eta /\sqrt{3}\right)}{{\left(k\eta /\sqrt{3}\right)}^3}$

where Φ⁽¹⁾(0) represents the initial condition deep in the radiations era.

At the same lodge in perturbation theory, the radiation velocity can exist read off from Eq. (A.4)

(B.9) ${\upsilon }_{\gamma }^{\left(1\right)i}=-\frac{1}{\mathrm{two}{\mathscr{H}}^2}\frac{{\left(a{\partial }^i{\Phi }^{\left(1\right)}\right)}^{\text{'}}}{a}.$

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Hydrodynamics

Ramona Vogt , in Ultrarelativistic Heavy-Ion Collisions, 2007

5.2 Free energy-momentum tensor

To obtain the equations of movement of the systems, we begin with the free energy-momentum tensor, T ^μν. We start in the fluid balance frame with T_R ^μν where the subscript R denotes the remainder frame. Nosotros will and so generalize to the moving frame past Lorentz transformation. The free energy-momentum tensor is the 4-momentum component in the μ direction per three-dimensional surface area perpendicular to the ν direction. Thus T ^μν has dimensions of free energy per volume.

If nosotros take differential four-momentum Δp = (ΔE, Δp_x , Δp_y , Δp_z ) with differential space-time four-vector Δx = (Δt, Δx, Δy, Δz), then, for μ= ν = 0,

(v.14) $T_R^{00}=\frac{\Delta E}{\Delta x\Delta y\Delta z}=\frac{\Delta E}{\Delta V}=\varepsilon ,$

the free energy density. We take μ= ν= 1 (the x coordinate) equally an example of the spatial directions,

(5.fifteen) $T_R^{00}=\frac{\Delta p_x}{\Delta t\Delta y\Delta z},$

where Δp₁₀ /Δ t is the forcefulness in the ten direction acting on a surface of area ΔyΔ z perpendicular to the force. The force exerted per area is the force per unit area. For a perfect fluid, the pressure is the aforementioned in all directions or isotropic. Then

(5.sixteen) $T_R^{ij}=P{\delta }^{ij}\cdot$

Thus, in the fluid residual frame,

(5.17) $T_R^{\mu \nu }=\left(\begin{array}{cccc}\varepsilon & 0& 0& 0\\ 0& P& 0& 0\\ 0& 0& P& 0\\ 0& 0& 0& P\end{array}\right)\cdot$

We at present boost into the moving organization. Nosotros Lorentz heave the velocities using the form,

(5.18) ${\text{u}}^{\mu }={\Lambda }_{\nu }^{\mu }{\text{u}}_R^{\nu }$

where u ^v _R is the velocity unit vector in the residuum frame of the system,

(5.19) ${\text{u}}_R^{\nu }=\left(\mathrm{ane},0,0,0\right),$

and ${\Lambda }_v^{\mu }$ is a Lorentz transformation. Thus the velocity in the additional frame defines the ν = 0 component of the transformation,

(five.20) ${\text{u}}^{\mu }={\Lambda }_0^{\mu }{\text{u}}_R^0={\Lambda }_0^{\mu }\cdot$

To find the components of the transformation in the spatial dimensions, nosotros use

(5.21) ${\mathrm{1000}}^{\rho \sigma }={\mathrm{yard}}^{\mu \nu }{\Lambda }_{\mu }^{\rho }{\Lambda }_{\nu }^{\sigma }\cdot$

Since the off-diagonal components of g ^{μ v} are zero, we can expand Eq. (5.21) to obtain

(5.22) ${\mathrm{one\; thousand}}^{\rho \sigma }=g^{00}{\Lambda }_0^{\rho }{\Lambda }_0^{\sigma }+g^{ii}{\Lambda }_i^{\rho }{\Lambda }_i^{\sigma }\cdot$

Using g ⁰⁰ = one and thousand² = −1 and rearranging terms, we have

(v.23) $\begin{array}{l}{\Lambda }_i^{\rho }{\Lambda }_i^{\sigma }={\Lambda }_0^{\rho }{\Lambda }_0^{\sigma }-g^{\rho \sigma }\\ ={\text{u}}^{\rho }{\text{u}}^{\sigma }-k^{\rho \sigma }\cdot \end{array}$

In the concluding step we have used Eq. (5.xx) to substitute Λ^ρ ₀ = u ^ρ and Λ^σ ₀ = u ^σ. We tin now utilize the Lorentz transformation to the rest-frame value of the free energy-momentum tensor to obtain the general grade

(5.24) $\begin{array}{l}{T}^{\rho \sigma }={\Lambda }_{\mu }^{\rho }{\Lambda }_{\nu }^{\sigma }{T}_R^{\mu \nu }\\ ={\Lambda }_0^{\rho }{\Lambda }_0^{\sigma }\varepsilon +{\Lambda }_i^{\rho }{\Lambda }_i^{\sigma }P\\ ={\text{u}}^{\rho }{\text{u}}^{\sigma }\varepsilon +\left({\text{u}}^{\rho }{\text{u}}^{\sigma }-k^{\rho \sigma }\right)P\\ =\left(\varepsilon +P\right){\text{u}}^{\rho }{\text{u}}^{\sigma }-{\mathrm{thou}}^{\rho \sigma }P\cdot \end{array}$

The energy-momentum tensor tin also exist derived from kinetic theory. For a detailed discussion of fluid mechanics and kinetic theory, run across the book by Csernai [69]. Only a brief sketch is given here. The scalar function N (x, q) gives the distribution of fluid elements with four-momentum q ^μ at space-time points x ^μ. B ecause N is a scalar, information technology can only depend on relativistic invariants, i.e. pro ducts of four-vectors such as x ², x · q or q ⁱⁱ. (Note that q ^μ is the momentum of a fluid element and not that of a unmarried particle.) Starting from North (10, q), the energy-momentum tensor of a fluid is defined equally

(5.25) $T^{\mu \nu }\left(\text{x}\right)=\int d^{4}q{\text{q}}^{\mu }{\text{q}}^{\nu }N\left(\text{x},\text{q}\right)\cdot$

In the rest frame of a perfect fluid with uniform temperature and no temperature gradients, N (x, q) is independent of 10 · q. Then, when μ = 5 = 0,

(five.26) $T^{00}\left(\text{x}\right)=\int d^{\mathrm{iv}}q{q}_0^{\mathrm{two}}N\left(\text{x},\text{q}\right).$

To meliorate see that Eq. (v.26) is actually an energy density, we recall from Chapter 2 that

$d^{4}q=d{q}^{\mathrm{two}}\delta \left({\text{q}}^{\mathrm{ii}}-M^2\right)\frac{d^{\mathrm{iii}}q}{q_0}\cdot$

On mass shell, the integral over dq ^two vanishes, leaving

(5.27) $T^{00}\left(\text{x}\right)=\int d^{3}q{q}_0\mathrm{North}\left(\text{ten},\text{q}\right)=\varepsilon$

where d ⁱⁱⁱ q has units of GeV³ or fm⁻³ in ħ = c = 1 units. The pressure is divers from

(5.28) $\begin{array}{l}{T}^{ij}=\int d^{4}q{q}^i{q}^j\mathrm{Due\; north}\left(\text{x},\text{q}\right)\\ =\frac{1}{3}{\delta }^{ij}\int d^{4}q{\left|\overrightarrow{q}\right|}^2\mathrm{North}\left(\text{x},\text{q}\right)\end{array}$

then that

(five.29) $P=T^{ii}\cdot$

Thus the rest frame form, $T_R^{\mu v}$ , is seen to only be applicable to rotationally symmetric systems, independent of x · q. To describe systems without rotational symmetry, three additional quantities are needed: the shear and bulk viscosities, η and ζ respectively, and the thermal conductivity, χ. Run across Ref. [70] for some more general calculations including the viscosities.

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On the problem of the singularities in the general cosmological solution of the Einstein equations

V.A. BELINSKII , ... Eastward.M. LIFSHITZ , in Perspectives in Theoretical Physics, 1992

(1)

All our piece of work is based on the Einstein equations with the energy–momentum tensor for macroscopic, electrically neutral matter described by the pressure level and energy density just, related to each other by a sure equation of state. That the state of such cosmological models must be determined by viii 'physically capricious' independent functions of the spatial coordinates (ref. [2], §one) is physically completely evident. Expressing doubts about this obvious fact, BT of course practice not signal which physical properties ought to be described by a larger number of independent functions. Their reference to the 'Reissner–Nordstrom initial information' has no begetting to the problem (the Cauchy horizon is unstable [7]). Completely irrelevant is also the question of time-like singularities – our work is concerned with singularities non of this type. However, we note in passing, that for the fourth dimension-like fictitious singularity (caustic in the synchronous system of reference) information technology is indeed necessary to have one contained function more to describe metrics in its neighbourhood-as a result of an additional arbitrariness due to the choice of the initial hypersurface (equally explained in ref. [two], appendix B).

(2)

We ever emphasised that a concrete solution with the necessary number of capricious functions is general just in the sense that information technology covers some finite region of the functional space of initial (prescribed at a certain moment t =t ₀) data; not in the least must information technology encompass all (or near all) this space. Such a solution is ipso facto stable in the same region-it does not modify its character for any (non merely modest) modify of the initial data in this region. BT desire to utilize the term 'full general' for a solution which would cover well-nigh the entire (open up dumbo) space of initial data; such a definition would just confuse the problem: it is so broad that probably no physically meaning dynamical property of the solutions of the Einstein equations could exist general in this sense of the give-and-take.

The problem of elucidation of the region of existence of our general solution with the oscillatory way of arroyo to the singularity and of its possible connections with the global properties of the spatial geometry is non solved; we could only assert that at that place is no direct connexion with the model existence open or closed. This is a difficult problem which cannot be solved past idle talk.

(3)

All solutions we have plant are not exact but asymptotic in the sense that they refer (in time) to the region of sufficiently small-scale t in the neighbourhood of the singularity at t = 0. Information technology is understood that the 'initial' hypersurface S (t = t ₀) is chosen in the same region. It is seen from these solutions that in this region the determinant of the metric, |g |, increases monotonically starting from the value |g | = 0 at t = 0. This means that no intersection of the coordinate lines of time of the synchronous reference system occurs here. The possible beingness of a caustic far from the singularity and the properties of the metric in its neighbourhood (which interested the states in our earlier piece of work) have really no begetting to the problem of truthful singularities. As has already been mentioned, we shall not comment on BT's remarks on this matter.

(4)

BT assert that a small change of initial data on S can lead to a global change of the synchronous reference organisation with a simultaneous singularity. Merely in the oscillatory regime the region of influence of such a change is modest if S is sufficiently about to the singularity, hence also the alter of the reference system will be minor.

(v)

Contrary to BT's assertions, our solutions do not imply whatsoever closeness to homogeneous models. The homogeneous Friedmann and Kasner solutions are indeed contained in our quasi-isotropic and generalised Kasner solutions, but in the derivation of the latter it is not assumed that the arbitrary functions (contained in these solutions) are close to those which would correspond to the homogeneous case. The same refers to the oscillatory regime. Its properties in the general example are indeed similar to those of the homogeneous models of the Bianchi types Eight and IX, but its actual construction [6] starts from the generalised Kasner solution and does not imply whatever closeness to the homogeneous models. We can likewise add that the stability of the homogeneous Kasner solution with respect to the relevant small perturbations (which correspond to its generalisation in the 'generalised' Kasner solution) was proved in ref. [ii], Appendix F]. Equally to the homogeneous oscillatory regime, the verbal definition of the notion of its stability (if one is interested in this question) is non at all so unproblematic as BT seem to imagine. Stable is only the mere existence of the oscillatory regime. But the actual behaviour of specific solutions in their development towards the singularity can differ profoundly in the values of certain parameters. The oscillatory authorities corresponds to the motion of the dynamical system (the cosmological model) forth trajectories in the neighbourhood of a sure strange attractor.

This motion is accompanied by stochastisation, inherent in strange attractors and past 'forgetting' the initial data [iv].

(half-dozen)

Opposite to BT'south exclamation, expressions(1),(ii) in their paper do non represent our general solution. Really these expressions represent only the generalised Kasner solution†. As to the general solution (the oscillatory regime), it cannot be represented in a airtight analytical form, although information technology admits a very detailed clarification [6] (the same refers even to the simpler case of the oscillatory regime in homogeneous models). The construction of this solution does non imply any a priori assumptions most its character and is based but on a thorough analytic investigation and on the interpretation of the terms in the equations which are omitted in the asymptotic limit. It is but these estimates (and non an a priori assumption of a 'passive inhomogeneity' as asserted past BT) which bear witness the possibility to neglect the particular terms with the spatial derivatives in the equations. All the details of the calculations have been published and a conscientious criticism of the results ought to contain an analysis of these calculations. As far as 1 can gauge by their newspaper, BT did not even attempt such an analysis (and their misinterpretation of expressions(i),(2) or their obvious misunderstanding of the basic difference in the origin of the oscillatory regime in type Eight, 9 models and in particular cases of type VI, Vii models, even casts doubts on whether they have studied and understood our derivations at all).

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The Catholic Dynamics

Thou. Dalarsson , N. Dalarsson , in Tensors, Relativity, and Cosmology (2nd Edition), 2015

24.ii Friedmann Equations

In order to construct the gravitational field equations (24.36 ) we now need the free energy-momentum tensor $T_n^k$ of the cosmic affair. The supposition is that the matter in the universe is on the large-calibration evenly distributed in the form of the cosmic fluid with no shear-viscous, majority-viscous and oestrus-conductive features. Thus, nosotros may utilize the mixed energy-momentum tensor of an ideal fluid, which is in the covariant form given by (19.72). We may and so write

(24.xl) $\begin{array}{l}\hfill T_{\mathrm{north}}^k=\left(p+\rho c^2\right)u_n{u}^k-{\delta }_{\mathrm{due\; north}}^{k}p,\end{array}$

where p is the force per unit area and ρ is the matter (or rest free energy) density of the cosmic fluid. In the co-moving frame the cosmic fluid is at rest and we accept

(24.41) $\begin{array}{l}\hfill u^0=1,u^{\alpha }=0(\alpha =1,\mathrm{two},3).\end{array}$

Using the results (24.41) and observing that p ≪ ρc ², we obtain

(24.42) $\begin{array}{l}\hfill T_0^0=\rho c^{\mathrm{ii}},T_1^1=T_{\mathrm{ii}}^{\mathrm{two}}=T_3^{\mathrm{three}}=-p.\end{array}$

Thus, the energy-momentum tensor of the catholic fluid in the co-moving frame tin can be structured in the following matrix:

(24.43) $\begin{array}{l}\hfill \left[T_n^k\right]=\left[\begin{array}{llll}\rho c^2& 0& 0& 0\\ 0& -p& 0& 0\\ 0& 0& -p& 0\\ 0& 0& 0& -p\\ \end{array}\right].\end{array}$

Substituting the results (24.39) and (24.42) into (24.36), we obtain the gravitational field equations in the form

(24.44) $\begin{array}{l}\hfill \frac{{\dot{R}}^2+\mathrm{thou}{c}^2}{R^{\mathrm{ii}}}=\frac{8\pi \mathrm{Thousand}}{\mathrm{iii}}\rho ,\end{array}$

(24.45) $\begin{array}{l}\hfill 2\frac{\stackrel{\dot{}\dot{}}{R}}{R}+\frac{{\dot{R}}^2+\mathrm{grand}{c}^2}{R^2}=-\frac{\mathrm{viii}\pi \mathrm{Thousand}}{c^2}p.\end{array}$

Equations (24.44) and (24.45) are called the Friedmann equations and their solution describes the cosmic dynamics. Combining (24.44) and (24.45) we may write

(24.46) $\begin{array}{l}\hfill 2\frac{\stackrel{\dot{}\dot{}}{R}}{R}+\frac{8\pi \mathrm{Thou}}{3}\rho =-\frac{8\pi \mathrm{Thou}}{c^2}p.\end{array}$

Furthermore, from (24.44) we have

(24.47) $\begin{array}{l}\hfill {\dot{R}}^2+k{c}^2=\frac{8\pi G}{3}\rho R^2.\end{array}$

Differentiating (24.47) with respect to the cosmic time t, nosotros obtain

(24.48) $\begin{array}{l}\hfill 2\dot{R}\stackrel{\dot{}\dot{}}{R}=\frac{\mathrm{viii}\pi \mathrm{One\; thousand}}{3}\dot{\rho }{R}^2+\frac{8\pi G}{3}\rho \mathrm{ii}R\dot{R},\end{array}$

or

(24.49) $\begin{array}{l}\hfill 2\frac{\stackrel{\dot{}\dot{}}{R}}{R}=\frac{8\pi \mathrm{Thou}}{\mathrm{three}}\dot{\rho }\frac{R}{\dot{R}}+\frac{\mathrm{viii}\pi K}{3}2\rho .\end{array}$

Substituting (24.49) into (24.46) we obtain

(24.fifty) $\begin{array}{l}\hfill \frac{8\pi \mathrm{Chiliad}}{3}\dot{\rho }\frac{R}{\dot{R}}+\frac{8\pi G}{3}3\rho =-\frac{\mathrm{viii}\pi G}{3}3\frac{p}{c^2},\end{array}$

or

(24.51) $\begin{array}{l}\hfill \dot{\rho }\frac{R}{\dot{R}}+\mathrm{three}\left(\rho +\frac{p}{c^2}\right)=0.\end{array}$

Observing that the universe as a system of galaxies in the shine fluid approximation behaves as an incoherent dust, we can again apply the approximation p ≪ ρc ^two to obtain

(24.52) $\begin{array}{l}\hfill \dot{\rho }+\mathrm{three}\rho \frac{\dot{R}}{R}=\frac{1}{R^{\mathrm{three}}}\frac{\mathrm{d}}{\mathrm{d}t}\left(\rho R^3\right)=0.\end{array}$

Integrating (24.52) nosotros obtain

(24.53) $\begin{array}{l}\hfill \rho R^{\mathrm{three}}={\rho }_0{R}_0^3=\mathrm{Constant},\end{array}$

where ρ ₀ and R ₀ are the affair density and scale radius at the present epoch (t = t ₀). From the result (24.53) we see that the affair density varies in time as R ⁻³ and that the quantity of matter in a co-moving volume chemical element is abiding during the expansion of the universe. This conclusion is relevant to the present matter-dominated epoch in the history of the universe.

In the early on phases of its history when the universe was radiations-dominated this decision was not valid because at that stage the supposition p ≪ ρc ² was not valid either. In the radiations-dominated universe equation (24.51) becomes

(24.54) $\begin{array}{l}\hfill \dot{\rho }+3\left(\rho +\frac{p}{c^2}\right)\frac{\dot{R}}{R}=0,\end{array}$

and the pressure component cannot be neglected. Using (24.53) and neglecting the pressure, we obtain the Friedmann equations for the matter-dominated epoch of the universe as follows:

(24.55) $\begin{array}{l}\hfill 2\frac{\stackrel{\dot{}\dot{}}{R}}{R}+\frac{{\dot{R}}^2+k{c}^{\mathrm{ii}}}{R^2}=0,\end{array}$

(24.56) $\begin{array}{l}\hfill \frac{{\dot{R}}^2+k{c}^2}{R^2}=\frac{\mathrm{eight}\pi G{\rho }_0}{3}\frac{R_0^{\mathrm{three}}}{R^3}.\end{array}$

The solutions of the Friedmann equations for the matter-dominated epoch of the universe given past (24.55) and (24.56) volition be the bailiwick of the following chapter.

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Relativity, General

James L. Anderson , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

5.East Fields with Matter—Gravitational Plummet

In addition to the empty-space solutions discussed here there are numerous solutions of the Einstein equations with nonvanishing energy–momentum tensors. One can, for example, construct a spherically symmetric, nonsingular interior solution that joins on smoothly to an exterior Schwarzschild solution. The combined solution would and so correspond to the field of a normal star, a white dwarf, or a neutron star. What distinguishes these objects is the equation of state for the matter comprising them. Information technology is surprising that no interior solutions that could be joined to a Kerr–Newman field accept been found.

Normal stars, of form, are not eternal objects. They are supported against gravitational collapse by pressure forces whose source is the thermonuclear called-for that takes place at the center of the star. Once such burning ceases due to the depletion of its nuclear fuel, a star will brainstorm to contract. If its total mass is less than approximately 2 solar masses it will ultimately become a stable white dwarf or a neutron star, supported by either electron or neutron degeneracy pressure. If, however, its total mass exceeds this limit, the star will continue to contract under its ain gravity down to a signal, a issue first demonstrated past Oppenheimer and Snyder in 1939. Although they treated just cold matter, that is, affair without pressure level, their issue would still agree in one case the star has passed a certain disquisitional stage even if the repulsion of the nuclei comprising the star were infinite. This critical stage is reached in one case the radius of the star becomes less than its Schwarzschild radius. When this happens, the surface r  =   twoYard becomes an event horizon and the matter is trapped inside, forming a blackness hole. Fifty-fifty an infinite pressure level would then exist unable to halt the continued wrinkle to a indicate because such a pressure level would contribute an space corporeality to the free energy–momentum tensor of the affair, which in turn would event in an even stronger gravitational allure.

Considering of the disquieting features of black hole formation (neither Eddington nor Einstein was prepared to accept their being), theorists looked for ways to avoid their germination. However, theorems of Penrose and Hawking show that collapse to a singularity is inevitable once the gravitational field becomes stiff enough to drag back whatever low-cal emitted by the star, that is, when the escape velocity at the surface of the star exceeds the velocity of low-cal.

What has not been proved to engagement is what Penrose calls the hypothesis of "cosmic censorship." This hypothesis asserts that affair volition never collapse to a naked singularity, just rather that the singularity will e'er be surrounded by an event horizon and hence not be visible to an external observer. While the hypothesis is suported past both numerical and perturbation calculations, it has so far not been shown to be a rigorous upshot of the laws of motion of the general theory.

The detection of black holes is complicated by the fact that they are blackness—past themselves they can emit no radiations. If they be at all and so, they can exist detected simply through the effects of their gravitational field on nearby matter. If a black hole were a fellow member of a double star organisation, it would get the source of intense X rays when its companion expanded during the later stages of its own evolution. Equally matter from the companion fell onto the black hole it would become compressed and thus heated to temperatures high enough for it to emit such high-energy radiations. Of course, it is not enough to find an 10-ray-emitting binary system in order to evidence the existence of a black pigsty. It is besides necessary that the mass of the x-ray-emitting component be larger than the upper limit on the mass of a stable neutron star, that is, larger than 3 M _⊙ (M _⊙ denotes a solar mass of i.99   ×   10³⁰  kg).

The offset object to be definitely identified every bit a black hole in 1971   was a fellow member of a binary system Cygnus X-1 in the constellation Cygnus which was an intense emitter of x-rays. Measurements established that the mass of the compact component of the system was about eight M _⊙. Since and then eight more black holes have been found in binary systems. In improver to stellar mass black holes at that place is at present compelling evidence that massive black holes exist in the centers of galaxies. By measurements of the orbital speed of the gaseous deejay around the nucleus of the screw galaxy NGC 4258, Makoto Miyoshi and his collaborators were able to plant in 1995 that the mass of the key object is 4   ×   10^vii M _⊙ and its diameter is a one-half a lite year, thereby establishing its identity as a blackness hole. Our own milky way has been shown to comprise a black hole at its center of mass 2.6   ×   10⁶ M _⊙. To date, there are about 15   masses that have been determined for blackness holes in the centers of nearby galaxies. The nigh massive such candidate for a blackness hole is in the giant elliptic galaxy M87   with an estimated mass of 3   ×   10⁹ M _⊙ although information technology is not certain that this object is indeed a blackness hole. Finally it is now believed that the free energy source of quasars is the inflow of vast amounts of gas into massive blackness holes in the centers of very young galaxies.

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