Energy conservation in a FLRW spacetime

In summary, energy conservation in a FLRW (Friedmann-Lemaitre-Robertson-Walker) spacetime is the principle that the total energy in a closed system remains constant over time. This is derived from the symmetries of the spacetime and has important implications for the behavior and evolution of matter and energy in the universe. It holds true in all FLRW spacetimes and plays a crucial role in understanding phenomena such as the expansion of the universe and the formation of structures within it.
  • #1
andrewkirk
Science Advisor
Homework Helper
Insights Author
Gold Member
4,116
1,712
Consequences of Energy conservation in a FLRW spacetime

This entry uses the local energy conservation law to derive an equation that can be used, together with the Einstein field equation, to derive Friedman's equations for the dynamics of a homogeneous, isotropic universe.

The energy conservation rule is ##{T^{0b}}_{;b}=0##.

We want to use this to get information about the equation of state

We use FLRW coordinates for this, in which the metric tensor components are all zero except for:
\begin{align*}
g_{00}&=-1, g_{11}=\frac{R^2}{1-kr^2}, g_{22}=R^2r^2, g_{33}=R^2r^2sin^2\theta\\
\end{align*}
Here ##R## is the scale factor, which depends only on ##t##.

The stress-energy tensor is T##=(\rho+p)\vec{v}\otimes\vec{v}+p##g-1 where ##\vec{v}## is the average four-velocity of the particles of the fluid which, in the FLRW coordinates, is ##(1,0,0,0)##, because of the isotropy assumption.
The representation of T in FLRW coordinates is a diagonal matrix with elements:
##T^{00}=\rho##, ##T^{11}=p \frac{1-kr^2}{R^2}##, ##T^{22}=\frac{p}{R^2r^2}##, ##T^{33}= \frac{p}{R^2r^2sin^2\theta}##

Now we write ##0={T^{0b}}_{;b}={T^{0b}}_{,b}+T^{ab}\Gamma^0_{ab}+T^{a0}\Gamma^b_{ab}##
We calculate the three terms in turn.

First, we have ##{T^{0b}}_{,b}={T^{00}}_{,0}=\frac{\partial\rho}{\partial t} = \dot{\rho}## because T inherits a diagonal nature in the FLRW coordinates from ##\vec{v}\otimes\vec{v}## and g.

For the next step we use
\begin{align*}
\Gamma^0{ab}&=\frac{1}{2}g^{0s}(g_{as,b}+g_{bs,a}-g_{ab,s})\\
&=\frac{1}{2}g^{00}(g_{a0,b}+g_{b0,a}-g_{ab,0})\\
&=\frac{1}{2}g^{00}(g_{00,b}+g_{00,a}-\delta^a_b g_{aa,0})\\
&=\frac{1}{2}\delta^a_b g_{aa,0}\\
\end{align*}
since ##g^{00}## is constant at ##-1##. Hence the second term is:
\begin{align*}
T^{ab}\Gamma^0_{ab}&=T^{ab}\frac{1}{2}\delta^a_b g_{aa,0}\\
&=\frac{1}{2}T^{aa} g_{aa,0}\\
&=\frac{1}{2}\Big(\rho(g_{00,0})+p \frac{1-kr^2}{R^2}g_{11,0}+\frac{p}{R^2r^2}g_{22,0}+\frac{p}{R^2r^2sin^2\theta}g_{33,0}\Big)\\
&=\frac{1}{2}\Big(0+p \frac{1-kr^2}{R^2}\frac{\partial}{\partial t}(\frac{R^2}{1-kr^2}) +\frac{p}{R^2r^2}\frac{\partial}{\partial t}(R^2r^2)+\frac{p}{R^2r^2sin^2\theta}\frac{\partial}{\partial t}(R^2r^2sin^2\theta)\Big)\\
&=3p\frac{\dot{R}}{R}
\end{align*}The third term is
\begin{align*}
T^{a0}\Gamma^b_{ab}&=T^{00}\Gamma^b_{0b}\\
&=\frac{\rho}{2}g^{bs}(g_{0s,b}+g_{bs,0}-g_{0b,s})\\
&=\frac{\rho}{2}g^{bb}(g_{0b,b}+g_{bb,0}-g_{0b,b})\\
&=\frac{\rho}{2}g^{bb}g_{bb,0}\\
&=\frac{\rho}{2}\Big(0+\frac{1-kr^2}{R^2}\frac{\partial}{\partial t}(\frac{R^2}{1-kr^2})+\frac{1}{R^2r^2}\frac{\partial}{\partial t}(R^2r^2)+\frac{1}{R^2r^2sin^2\theta}\frac{\partial}{\partial t}(R^2r^2sin^2\theta)\Big)\\
&=3\rho\frac{\dot{R}}{R}
\end{align*}

Putting the three terms together, we have:
\begin{align*}
{\rho}+3(\rho+p)\frac{\dot{R}}{R}=0\\
\end{align*}

We can express this in a manner that better reflects the relationship of work and energy by considering the rate of change of energy in a co-moving volume:
\begin{align*}
\frac{d}{dt}(\rho R^3)&= 3\rho R^2\dot{R}+\dot{\rho}R^3 =R^3\Big(\dot{\rho}+3\rho\frac{\dot{R}}{R}\Big) = -3p R^3\frac{\dot{R}}{R}
=-3p\dot{R}R^2=-p\frac{d}{dt}(R^3)\\
&\textrm{In summary:}\\
\frac{d}{dt}(\rho R^3)&=-p\frac{d}{dt}(R^3)\\
\end{align*}

The left-hand side is the increase in energy in the co-moving volume as it expands and the rightmost formula is the negative of the work done in the expansion.

In the current universe, in which most mass-energy is matter, and it is very sparsely distributed on average, average pressure ##p## is low compared to ##\rho## and can be ignored, so we get the approximation ##\frac{d}{dt}(\rho R^3)=0## in the matter-dominated phase of the universe.

In the early universe, there was not much matter and such particles as there were would be traveling very fast. Hence the equation of state for a photon gas or a highly relativistic gas applies, which is ##p=\frac{1}{3}\rho##. Inserting that in the above equation we get:
\begin{align*}
\frac{d}{dt}(\rho R^3)= -\frac{1}{3}\rho\frac{d}{dt}(R^3)\\
\end{align*}
Multiply by ##R## and collect on one side to get:
\begin{align*}
0&=R\frac{d}{dt}(\rho R^3)+R\frac{1}{3}\rho\frac{d}{dt}(R^3)\\
&=R\frac{d}{dt}(\rho R^3)+R\frac{1}{3}\rho(3R^2)\dot{R}\\
&=R\frac{d}{dt}(\rho R^3)+(\rho R^3)\dot{R}=\frac{d}{dt}(\rho R^4)\\
\end{align*}
So, in the early, energy-dominated era, we have:
\begin{align*}
\frac{d}{dt}(\rho R^4)&=0\\
\end{align*}

V.361.
 
Last edited:
  • Like
Likes fresh_42
Physics news on Phys.org
  • #2
0

In summary, the consequences of energy conservation in a FLRW spacetime are:

1. The energy conservation law, together with the Einstein field equation, can be used to derive Friedman's equations for the dynamics of a homogeneous, isotropic universe.

2. The equation of state for the fluid in the universe can be determined using the local energy conservation law.

3. In the matter-dominated phase of the universe, the energy density is constant as the universe expands.

4. In the early, energy-dominated era, the energy density and scale factor are related by ##\frac{d}{dt}(\rho R^4)=0##.
 

What is energy conservation in a FLRW spacetime?

Energy conservation in a FLRW (Friedmann-Lemaitre-Robertson-Walker) spacetime refers to the principle that the total energy in a closed system remains constant over time. This means that energy cannot be created or destroyed, but can only be transformed from one form to another.

How does energy conservation apply to FLRW spacetimes?

In FLRW spacetimes, energy conservation is a fundamental principle that is derived from the laws of physics. It is a consequence of the symmetries of the spacetime, which dictate that the laws of physics remain constant throughout the universe.

What are the implications of energy conservation in FLRW spacetimes?

The principle of energy conservation has important implications for the behavior of matter and energy in FLRW spacetimes. It means that the total energy in the universe remains constant, and any changes in the distribution or form of energy must be balanced by corresponding changes elsewhere in the system.

Does energy conservation hold true in all FLRW spacetimes?

Yes, energy conservation is a universal principle that applies to all FLRW spacetimes. It is a fundamental aspect of the laws of physics, and has been tested and verified in numerous experiments and observations.

How does energy conservation impact the evolution of the universe in FLRW spacetimes?

The principle of energy conservation plays a crucial role in the evolution of the universe in FLRW spacetimes. It helps to determine the distribution and dynamics of matter and energy, and is a key factor in understanding phenomena such as the expansion of the universe and the formation of structures within it.

Similar threads

  • Special and General Relativity
Replies
6
Views
1K
  • Special and General Relativity
Replies
11
Views
1K
  • Special and General Relativity
Replies
2
Views
520
  • Special and General Relativity
Replies
5
Views
1K
  • Special and General Relativity
Replies
1
Views
789
  • Special and General Relativity
Replies
5
Views
223
  • Special and General Relativity
Replies
2
Views
1K
  • Special and General Relativity
Replies
4
Views
3K
  • Special and General Relativity
Replies
8
Views
1K
  • Special and General Relativity
Replies
7
Views
1K
Back
Top