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mysearch
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Hi, I am trying to make some assessment and interpretation of Friedmann’s equation, which is often presented in the form:
[1] [tex]H^2 = \frac{8}{3}\pi G \rho - \frac {kc^2}{a2} + \frac{\Lambda}{3}[/tex]
While I understand that Friedmann’s equation is said to be a solution of Einstein’s field equation within general relativity, it seems that a basic derivation can be made by considering the conservation of energy of the kinetic and potential energy of a unit mass [m] at the surface of an expanding sphere of radius [R] to a central mass [M]. The following equations are only to provide a basic frame of reference:
[2] [tex] E_T = E_K + E_P = \frac{1}{2}mv^2 + \left(-\frac{GMm}{r}\right) [/tex]
[3] [tex] E_T = \frac{1}{2}mv^2 - G\left(\frac {4}{3}\pi R^3 \rho\right) \frac {m}{r} [/tex]
[4] [tex] \frac{E_T}{r^2} = \frac{1}{2}m\left(\frac{v}{r}\right)^2 - \frac {4}{3}\pi G \rho m [/tex]
[5] [tex] H^2 = \frac {8}{3}\pi G \rho + \frac{2 E_T}{mr^2} [/tex]
My first question is whether the last 2 terms in [1] can be considered comparable to the last term in [5]? However, if we drop [tex] + \frac{\Lambda}{3}[/tex] for the moment, we would have to equate :
[6] [tex] - \frac {kc^2}{a^2} = + \frac{2 E_T}{mr^2} [/tex]
To be compatible to equation [1], the units must resolve to [tex]1/sec^2[/tex]. Now on the assumption that (k) has no units and the units of [c] are known, then [a] must have units of distance as per [r]. In fact, if I equate [a=r] and describe this distance as a function of expansion of the universe over time, can we rationalise [k] as follows:
[7] [tex] k = \frac{2E_T}{mc^2}[/tex]
The units seem to cancel out correctly, but I am left wondering about the value of [k]. It seems that [tex]mc^2[/tex] would correspond to the rest energy of the unit mass [m], but what figure can be put on the total energy and would it give k->0?
Taking equations [1] & [5] as a whole, the value of [H] has been established from redshift measurements. If I extend the linearity of [H] to the edge of a visible universe, the value of [r=c/H=13.9 billion light-years]. As such, I appear to be left with 2 unknowns, i.e. density [tex][\rho][/tex] and total energy [Et]. While the mass-density is often substituted into the Friedmann equation, I would have thought this would be relatively meaningless given the speculation that matter only accounts for about 4% of the required energy density of a universe where r=13.9 billion light-years?
Therefore, would greatly appreciate any technical insights regarding any of the issues raised? Thanks
P.S. In part this is an extension of another thread about 'Distance & Hubble Constant'
https://www.physicsforums.com/showthread.php?t=245769
[1] [tex]H^2 = \frac{8}{3}\pi G \rho - \frac {kc^2}{a2} + \frac{\Lambda}{3}[/tex]
While I understand that Friedmann’s equation is said to be a solution of Einstein’s field equation within general relativity, it seems that a basic derivation can be made by considering the conservation of energy of the kinetic and potential energy of a unit mass [m] at the surface of an expanding sphere of radius [R] to a central mass [M]. The following equations are only to provide a basic frame of reference:
[2] [tex] E_T = E_K + E_P = \frac{1}{2}mv^2 + \left(-\frac{GMm}{r}\right) [/tex]
[3] [tex] E_T = \frac{1}{2}mv^2 - G\left(\frac {4}{3}\pi R^3 \rho\right) \frac {m}{r} [/tex]
[4] [tex] \frac{E_T}{r^2} = \frac{1}{2}m\left(\frac{v}{r}\right)^2 - \frac {4}{3}\pi G \rho m [/tex]
[5] [tex] H^2 = \frac {8}{3}\pi G \rho + \frac{2 E_T}{mr^2} [/tex]
My first question is whether the last 2 terms in [1] can be considered comparable to the last term in [5]? However, if we drop [tex] + \frac{\Lambda}{3}[/tex] for the moment, we would have to equate :
[6] [tex] - \frac {kc^2}{a^2} = + \frac{2 E_T}{mr^2} [/tex]
To be compatible to equation [1], the units must resolve to [tex]1/sec^2[/tex]. Now on the assumption that (k) has no units and the units of [c] are known, then [a] must have units of distance as per [r]. In fact, if I equate [a=r] and describe this distance as a function of expansion of the universe over time, can we rationalise [k] as follows:
[7] [tex] k = \frac{2E_T}{mc^2}[/tex]
The units seem to cancel out correctly, but I am left wondering about the value of [k]. It seems that [tex]mc^2[/tex] would correspond to the rest energy of the unit mass [m], but what figure can be put on the total energy and would it give k->0?
Taking equations [1] & [5] as a whole, the value of [H] has been established from redshift measurements. If I extend the linearity of [H] to the edge of a visible universe, the value of [r=c/H=13.9 billion light-years]. As such, I appear to be left with 2 unknowns, i.e. density [tex][\rho][/tex] and total energy [Et]. While the mass-density is often substituted into the Friedmann equation, I would have thought this would be relatively meaningless given the speculation that matter only accounts for about 4% of the required energy density of a universe where r=13.9 billion light-years?
Therefore, would greatly appreciate any technical insights regarding any of the issues raised? Thanks
P.S. In part this is an extension of another thread about 'Distance & Hubble Constant'
https://www.physicsforums.com/showthread.php?t=245769
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