Relativistic Viscous Fluids

George E. Hrabovsky, MAST
george@madscitech.org

Presented to the 2020 Midwest Relativity Meeting, University of Notre Dame, IN

Introduction

This talk will proceed in the following way, all of this work being done in Mathematica using xAct:

We will present the stress-energy tensor for the perfect fluid.

We will then show how we can get the fluid equations of motion for a perfect fluid using the principle of source conservation applied to the stress-energy tensor.

We will present the stress-energy tensor for a viscous fluid.

I will show the xAct implementation of this tensor.

I will then show how we can get the fluid equations of motion for a viscous fluid using the principle of source conservation applied to the stress-energy tensor.

The Stress-Energy Tensor for a Perfect Fluid

Given the pressure p, the density ρ, the metric tensor (Subscript[g, αβ]), and the 4-velocity (u^α), we can write the stress energy tensor for a perfect fluid

(T^(αβ))=p (Subscript[g, αβ])+(ρ + p)(u^α)(u^β).

How do we tell Mathematica to write this? First we have to load xAct.

<<xAct`xTensor`;
<<xAct`xPert`;
<<xAct`xCoba`; 
<<xAct`SymManipulator`;
$DefInfoQ=False;
$CVVerbose=False;

The Stress-Energy Tensor for a Perfect Fluid II

Then we define our spacetime manifold.

DefManifold[ℳ,4,{α,β,γ,ι,λ,μ,ν}];

Then we define the abstract metric.

DefMetric[-1,g[-μ,-ν],CD,{";","∇"}, PrintAs->"g"];

We now establish our chart. We will label it cb for coordinate basis

DefChart[cb,ℳ,{0,1,2,3},{t[],r[],θ[],ϕ[]}]

Then we can specify how we want these written in output,

cb/:CIndexForm[0,cb]:="t";

cb/:CIndexForm[1,cb]:="r";

cb/:CIndexForm[2,cb]:="θ";

cb/:CIndexForm[3,cb]:="ϕ";

The Stress-Energy Tensor for a Perfect Fluid III

We are using geometrized units in this work where G=1 and c=1. We need to define the constant for the central mass, M.

DefConstantSymbol[M]

We will adopt the Schwarzschild metric.

MatrixForm[met=DiagonalMatrix[{1-(2 M)/r[],(((1-(2 M)/r[]))^(-1)),-((r[])^2),2 r[] ((Sin[θ[]])^2)}]]

The Stress-Energy Tensor for a Perfect Fluid IV

We then enter this into our coordinate basis.

MetricInBasis[g,-cb,met]//TableForm

We can show the symmetries.

TensorValues@g

The Stress-Energy Tensor for a Perfect Fluid V

We can also enter it directly using CTensor.

gm=CTensor[met,{-cb,-cb}];

SetCMetric[gm,-cb];

The t t component of the tensor can be displayed.

gm[{0,-cb},{0,-cb}]

We then define the density field.

DefTensor[ρ[],ℳ];

Here we define the 4-velocity.

u=CTensor[{1,0,0,0},{-cb}];

The Stress-Energy Tensor for a Perfect Fluid VI

For the perfect fluid we need to define a pressure field.

DefTensor[p[],ℳ];

The stress-energy tensor for the perfect fluid is,

Tf[α_,β_]:=(ρ[] u[-α] u[-β])-(p[] u[-α] u[-β])+p[] gm[-α,-β]

Tf[α,β]

We can specify a particular set of components.

Tf[{-0,cb},{-0,cb}]

The Stress-Energy Tensor for a Perfect Fluid VII

We can make a command to do this,

SET[α_,β_]:=Tf[{-α,cb},{-β,cb}]

SET[0,0]

SET[0,1]

SETTable=Table[Row[{(Subscript["T", Grid[{{α,β}}]]),"->",SET[α,β]}],{α,0,3},{β,0,3}]//TableForm

Using the Source Conservation Law to Derive the Equations of a Perfect Fluid

The source conservation law is often called the conservation of 4-momentum. Why the distinction? There is no general conservation of 4-momentum in general relativity, only in the source of the gravitation field.

We can write this conservation law

(Subscript[(T^(αβ)), ;γ])=0.

In our case, by (1) we can write this, in terms of the velocity of the fluid v,

(Subscript[(T^00), ,0])+(Subscript[(T^00), ,j])=(∂_t)ρ+ρ (Subscript[(v^i), ,j])=0.

Looking at this long enough, we will realize that it is the continuity equation.

We also have the spatial components

(Subscript[(T^(i j)), ,0])+(Subscript[(T^(i j)), ,j])=(∂_t)ρ (v^i)+(Subscript[(ρ (v^i)(v^j)), ,j])+(Subscript[p, ,i])=0.

This is none other than the Euler equation for a fluid.

Using the Source Conservation Law to Derive the Equations of a Perfect Fluid II

In general we can write the Christoffel symbols

Christoffel[CD,PDcb][α,-β,-γ]

We can calculate the covariant derivative of the metric.

cdg=CovDOfMetric[gm];

We can calculate the Christoffel symbols.

chr=Christoffel[cdg, PDcb];

We can determine a specific Christoffel symbol

chr[{0,cb},{0,-cb},{1,-cb}]

We can make a command to do this,

Γ[α_,β_,γ_]:=chr[{α,cb},{β,-cb},{γ,-cb}]

Γ[0,0,1]

We can get all the components.

ChrisTable=Table[Row[{(Subscript["Γ", Grid[{{"",β,γ}}]]^(Grid[{{α}}])),"->",Γ[α,β,γ]}],{α,0,3},{β,0,3},{γ,0,3}]//TableForm

Using the Source Conservation Law to Derive the Equations of a Perfect Fluid III

We can write the covariant derivative of the stress-energy tensor.

cdstpf[α_,β_,γ_]:=cdg[-α][Tf[β,γ]]

We can write this,

cdstpf[α,β,γ]//ToCanonical//Simplify

We can specify the specific components of this. Here we have the covariant derivative of the time-time component.

cdstpf[{0,cb},{0,cb},{0,cb}]//ToCanonical//Simplify

We can write a program for this,

cdsetpf[α_,β_,γ_]:=cdstpf[{α,cb},{β,cb},{γ,cb}]

cdsetpf[0,0,0]//ToCanonical//Simplify

cdsetpf[1,0,0]//ToCanonical//Simplify

cdsetpf[2,0,0]//ToCanonical//Simplify

cdsetpf[3,0,0]//ToCanonical//Simplify

Using the Source Conservation Law to Derive the Equations of a Perfect Fluid IV

Putting together all of the derivatives of the time components

tcdsetpft=Total[Table[cdsetpf[α,0,0],{α,0,3}]]//ToCanonical//Simplify

Setting this to 0 will give us the continuity equation for a perfect fluid in Schwarzschild spacetime. This is how (3) is adapted to our specific geometry.

Using the Source Conservation Law to Derive the Equations of a Perfect Fluid V

In order to get the equations of momentum we need to apply a projection tensor.

We begin by defining the projection tensor.

P[α_,β_]:=gm[-α,-β]+u[-α]u[-β]

P[α,β]

We can specify a particular set of components.

P[{-0,cb},{-0,cb}]//Simplify

We can make a command to do this,

PT[α_,β_]:=P[{-α,cb},{-β,cb}]//Simplify

PT[0,0]

PTTable=Table[Row[{(Subscript["P", Grid[{{α,β}}]]),"->",PT[α,β]}],{α,0,3},{β,0,3}]//TableForm

Using the Source Conservation Law to Derive the Equations of a Perfect Fluid VI

So we then write,

mom[α_,β_,γ_,μ_,ν_]:=P[μ,ν]cdstpf[α,β,γ]//ToCanonical//Simplify

mom[α,β,γ,μ,ν]

The Stress-Energy Tensor for a Viscous Fluid I

It is important to realize that for a Non-Perfect Fluid the stress-energy tensor will have the form of the stress-energy tensor plus some additional terms,

(T^(αβ))=Stress-Energy Tensor: Perfect Fluid + Viscous Contributions + The Generation of Energy Fluxes.

We will adopt the Eckert frame for our presentation as this is most generally applicable (though not for some many-body collision cases).

The stress-energy tensor for the Eckert frame of a viscous fluid exhibiting heat flow, having an anisotropic stress tensor (π^(αβ)), the viscous bulk pressure Π, a projection tensor (P^(αβ)), the heat-flux 4-vector (q^α),

(T^(αβ))=ρ (u^α)(u^β)+(π^(αβ))+(p+Π )(P^(αβ)) + (q^α)(u^β)+(q^β)(u^α).

(P^(αβ))=(g^(αβ))+(u^α)(u^β).

Π=p-(Subscript[p, eq]).

We also have the heat flux 4-vector,

(q^α)=(0,(q^(Overscript[i, ⋀])))

where the hat represents spatial components in the comoving frame of the fluid.

The Stress-Energy Tensor for a Viscous Fluid II

We then establish our equilibrium pressure.

DefConstantSymbol[peq]

Then we define the viscous bulk pressure.

Π[]:=p[]-peq

The heat flux 4 - vector is.

q=CTensor[{0,q1,q2,q3},{-cb}];

c=CTensor[{(Sqrt[(1/(1-((2 M)/r)))]),0,0,0},{-cb}];

DefTensor[S[],ℳ];

σ[α_,β_]:=(Sqrt[3])S[](c[-α]c[-β]-(1/3)(u[-α] u[-β]-gm[-α,-β]))

σ[α,β]

The Stress-Energy Tensor for a Viscous Fluid III

Tvf[α_,β_]:=ρ[]u[-α]u[-β]+σ[α,β]+(p[]+Π[])P[α,β]+q[-α]u[-β]+q[-β]u[-α]

Tvf[α,β]//ToCanonical

We can specify a particular set of components.

Tvf[{-0,cb},{-0,cb}]

We can make a command to do this,

SETV[α_,β_]:=Tvf[{-α,cb},{-β,cb}]

SETV[0,0]

The Stress-Energy Tensor for a Viscous Fluid IV

SETVTable=Table[Row[{(Subscript["T", Grid[{{α,β}}]]),"->",SETV[α,β]}],{α,0,3},{β,0,3}]//TableForm

Using the Source Conservation Law to Derive the Equations of a Viscous Fluid I

We can write the covariant derivative of the stress-energy tensor.

cdstpfv[α_,β_,γ_]:=cdg[-α][Tvf[β,γ]]

We can write this,

cdstpfv[α,β,γ]//ToCanonical//Simplify

We can specify the specific components of this. Here we have the covariant derivative of the time-time component.

cdstpfv[{0,cb},{0,cb},{0,cb}]//ToCanonical//Simplify

We can write a program for this,

cdsetpfv[α_,β_,γ_]:=cdstpfv[{α,cb},{β,cb},{γ,cb}]

cdsetpfv[0,0,0]//ToCanonical//Simplify

cdsetpfv[1,0,0]//ToCanonical//Simplify

cdsetpfv[2,0,0]//ToCanonical//Simplify

cdsetpfv[3,0,0]//ToCanonical//Simplify

Using the Source Conservation Law to Derive the Equations of a Viscous Fluid II

Putting together all of the derivatives of the time components

tcdsetpftv=Total[Table[cdsetpfv[α,0,0],{α,0,3}]]//ToCanonical//Simplify

Setting this to 0 will give us the continuity equation for a viscous fluid in Schwarzschild spacetime.

Using the Source Conservation Law to Derive the Equations of a Viscous Fluid III

In order to get the equations of momentum we need to apply a projection tensor.

momv[α_,β_,γ_,μ_,ν_]:=P[μ,ν]cdstpfv[α,β,γ]//ToCanonical//Simplify

mom[α,β,γ,μ,ν]

Once again, you can read the components off the lists to get the specific components.

More work on this needs to be done to extract features like the vorticity tensor.

References

Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, (1973), Gravitation, W. H. Freeman and Company.

George E. Hrabovsky, (2020), Twenty Lessons to General Relativity, In preparation.

George E. Hrabovsky, (2015), GR Calculations in Specific Bases Using Mathematica, Midwest Relativity Meeting, 2015, CIERA, Northwestern University

Luciano Rezzolla, Olindo Zanotti, (2013), Relativistic Hydrodynamics. Oxford University Press

J.R.Wilson and G.J.Mathews, (2003), Relativistic Numerical Hydrodynamics. Cambridge University Press, digitally printed in 2007.

T. Tsumura, T. Kunihiro, K. Ohnishi, (2007), “Derivation of covariant dissipative fluid dynamics in the renormalization-group method”. Physics Letters B 646 (2007) 134–140

D.D.Pawar, (2012), “Spherically Symmetric Fluid Cosmological Model with Anisotropic Stress Tensor in General Relativity”, International Scholarly Research Notices, vol. 2012

Thank You!