- #1
Das apashanka
while deriving the friedmann equation using Newtonian Mechanics the 2nd term of the r.h.s is coming to be 2^U/(r^2*a^2) where U is a constt,but it is replaced by -kc^2/(r^2*a^2)?
Das apashanka said:while deriving the friedmann equation using Newtonian Mechanics
Das apashanka said:the 2nd term of the r.h.s is coming to be 2^U/(r^2*a^2) where U is a constt,but it is replaced by -kc^2/(r^2*a^2)?
Das apashanka said:from the concept of kinetic energy and potential energy being constt for a test particle situated over a gravitational sphere having radius at time t to be a(t)r[o]
Yes, because in the last step you make a substitution for the constants in the last term, and call it curvature parameter k.Das apashanka said:my question is that it is coming 2U/(..) but everywhere it is written -kc^2/(...)
will you please explain why is term of 2U is written in terms of kc^2Bandersnatch said:Yes, because in the last step you make a substitution for the constants in the last term, and call it curvature parameter k.
When one arrives at the first Friedmann equation:Das apashanka said:will you please explain why is term of 2U is written in terms of kc^2
Sorry, love. I was using 'you' to mean 'one', or 'we', or 'it's how it's done'. I did not mean you in particular. Just a figure of speechDas apashanka said:No I didnt make a substitution my question is why it is being substituted
no no nothing happened like that ,thanks for replying I mean for last paragraphBandersnatch said:When one arrives at the first Friedmann equation:
$$H^2(t)=\frac{8πG}{3}\rho(t)+\frac{2U}{m R_0^2 a^2(t) }$$
in the first term on the r.h.s. we have some time-variable ##\rho(t)##, and some universal constants. In the second term, we have a time-variable ##a^2(t)##, and a bunch of parameters ##\frac{2U}{m R_0^2}## which are all time-independent. Whatever the total energy is, it is conserved throughout expansion. So is test particle mass (and it cancels out with itself in total energy anyway), and the comoving distance ##R_0## is likewise unchanging. So, for convenience, we gather all of these parameters into one constant, and call it ##k## (where ##c^2## is just a unit-conversion factor).
From the derivation one gets some intuitive understanding that the constant ##k## is related to the total energy - which as far as I understand is the main reason for using Newtonian derivation.
It's to give intuitive meaning to what pops up in the General Relativistic derivation.Sorry, love. I was using 'you' to mean 'one', or 'we', or 'it's how it's done'. I did not mean you in particular. Just a figure of speech
I believe it's because that's what you get from the General Relativistic derivation.Das apashanka said:Ok that's fine but why the constant k is taken to be the curvature?
Das apashanka said:why the constant k is taken to be the curvature?
The Friedmann Equation is a fundamental equation in cosmology that describes the expansion of the universe. Replacing U with -kc^2 allows us to account for the effects of curvature on the expansion of the universe. This term takes into consideration the geometry of space and can provide insight into the overall structure of the universe.
Replacing U with -kc^2 introduces a negative term into the Friedmann Equation. This negative term represents the effects of curvature on the expansion of the universe. It can change the overall behavior of the equation and provide a more accurate representation of the universe's expansion.
The constant k in the Friedmann Equation represents the curvature of space. It can have a value of +1, -1, or 0, which correspond to a closed, open, or flat universe, respectively. This constant is essential in understanding the overall structure and evolution of the universe.
The value of k directly affects the expansion rate of the universe. A positive value of k (closed universe) will result in a slower expansion rate, while a negative value of k (open universe) will lead to a faster expansion rate. A flat universe (k=0) will have a constant expansion rate over time.
Replacing U with -kc^2 allows us to consider the effects of curvature on the expansion of the universe. This can provide valuable insights into the overall structure and evolution of the universe. It also helps us to better understand the role of dark energy in the expansion of the universe and its implications for the future of our universe.