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Wednesday, March 26, 2014

fundamentals of cosmology

can you give an intuitive explanation why one needs general relativity for describing the dynamics of the Universe?

CQW's answer would be that the Hubble constant $H_0$ provides together with Newton's constant $G$ a density scale, $\rho_\mathrm{crit}=3H_0^2/(8\pi G)$ with a numerical factor of order "unity". The typical matter density is comparable to this number. of course relativity provides a way of incorporating negative pressure fluids or curvature into the field equation.

the second reason is a description of everything that's related to curvature and other negative pressure "substances", which is of course not possible without general relativity, although a term analogous to curvature would appear as an integration constant in Newtonian gravity.

a third reason would be that without general relativity one would not be able to incorporate the Copernican principle as one would necessarily need to be in the centre of the Universe from which all galaxies fly away.

The nature of time has had extensive attention in part down through the ages, such as Plato, St. Augustine, Pascal, Leonardo, Newton etc. For example, Newton considered time to flow uniformly, as if it were a separate manifold (1-surface) from the 3-surface of his mechanics described universe.

‘Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external…’ Newton’s Principia

For a manifold, this would give a product space description M3 x M1 , the simplest fiber bundle description. Hence such description would be universal; that is the same common time for throughout the universe. Subsequently, the relativistic model refers to time as the interval between events for finite propagation, wherein clocks are associated with respective observers. However an event such as Big Bang, and concomitant Big Expansion of our 3-manifold (i.e. 3-surface), does not have such a General Relativity Theory description; nor is ‘initial’ 3-expansion (i.e. Hubble expansion) of our 3-volume limited by velocity of light, as in Special Relativity. Hence the possibility of further modeling in regards to how our 3-space and contents evolves.

Might there be another common time description as to how our 3-volume evolves? Just as Gauss described curvature of a 2-surface intrinsic to such surface, and Riemann described curvature of a 3-surface as intrinsic to such surface, might not one analogously describe time as intrinsic to our changing 3-surface? Could the nonlinear Hubble expansion be utilized as such common time description for our 3-surface and perhaps for a set of such 3-surfaces i.e. 3-volumes, 3-manifolds; that is for misnomer, ‘multiple universes’? 'Universe' denotes all inclusiveness, rather than multi-universe which implies a set of such all inclusiveness. Hence better to refer to a set of 3-volumes i.e. 3-manifolds.

Also non-linearity to Hubble expansion might even be of an always exponential nature, if it is just a specific example of the more general case: all explosiveness declines exponentially. So do all locations of our 3-volume, and for a possible set of 3-volumes, share the same common time i.e. common cosmic time? That is, perceiving the same Big Bang ~13.8 billion years ago; and thus the same ~2.7 degree kelvin temperature of cosmic background radiation for our now i.e. common positive definite modified global instant; a set of such instants mapping to the integers? Thus is there any necessity for inflation models?

Also for a set of 3-volumes, contained in an array of planes, spherically symmetrical about a modified central force, might one also have a concomitant common cosmic time description intrinsic to System S≡{UT', ...} wherein UT constitutes an array of planes comprised of 3-volumes i.e. UT≡{{VvR}p} ? That is, might one utilize a positive definite thick spherical shell intersecting respective centers of all 3-volumes, denoting such common cosmic time for all patches of a given manifold, and for all 3-manifolds? This might be referred to as Modified Global Simultaneity (MGS) construct. Such set of successive concentric MGSs would also match (i.e. function) to the integers; having positive definite moments, corresponding to {ΔSTm ->0+} set of modified time separations. Also such set of MGSs could be rendered as always exponentially changing i.e. part of an exponential curve. Likewise for changing rate of Hubble expansion, intrinsic to each and all changing 3-volumes.

Thus in such modeling, would one herein have two concomitant descriptions of an overall common cosmic time; the Hubble Expansion, intrinsic to a changing 3-volume, and MGS, intrinsic to an unending evolving dynamic System?

M^3 x M^1 refers to a product space. U_T={{V_R}_p} refers to a set of 3-volumes. ∆S_Tm->0_+ refers to a modified time-like separation going to positive definite. System S={U_T^', ...}, a divergent set.

CQW's answer would be that the Hubble constant $H_0$ provides together with Newton's constant $G$ a density scale, $\rho_\mathrm{crit}=3H_0^2/(8\pi G)$ with a numerical factor of order "unity". The typical matter density is comparable to this number. of course relativity provides a way of incorporating negative pressure fluids or curvature into the field equation.

ReplyDeletethe second reason is a description of everything that's related to curvature and other negative pressure "substances", which is of course not possible without general relativity, although a term analogous to curvature would appear as an integration constant in Newtonian gravity.

ReplyDeletea third reason would be that without general relativity one would not be able to incorporate the Copernican principle as one would necessarily need to be in the centre of the Universe from which all galaxies fly away.

ReplyDeletePhilosophy of Time

ReplyDeleteThe nature of time has had extensive attention in part down through the ages, such as Plato, St. Augustine, Pascal, Leonardo, Newton etc. For example, Newton considered time to flow uniformly, as if it were a separate manifold (1-surface) from the 3-surface of his mechanics described universe.

‘Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external…’ Newton’s Principia

For a manifold, this would give a product space description M3 x M1 , the simplest fiber bundle description. Hence such description would be universal; that is the same common time for throughout the universe. Subsequently, the relativistic model refers to time as the interval between events for finite propagation, wherein clocks are associated with respective observers. However an event such as Big Bang, and concomitant Big Expansion of our 3-manifold (i.e. 3-surface), does not have such a General Relativity Theory description; nor is ‘initial’ 3-expansion (i.e. Hubble expansion) of our 3-volume limited by velocity of light, as in Special Relativity. Hence the possibility of further modeling in regards to how our 3-space and contents evolves.

Might there be another common time description as to how our 3-volume evolves? Just as Gauss described curvature of a 2-surface intrinsic to such surface, and Riemann described curvature of a 3-surface as intrinsic to such surface, might not one analogously describe time as intrinsic to our changing 3-surface? Could the nonlinear Hubble expansion be utilized as such common time description for our 3-surface and perhaps for a set of such 3-surfaces i.e. 3-volumes, 3-manifolds; that is for misnomer, ‘multiple universes’? 'Universe' denotes all inclusiveness, rather than multi-universe which implies a set of such all inclusiveness. Hence better to refer to a set of 3-volumes i.e. 3-manifolds.

Also non-linearity to Hubble expansion might even be of an always exponential nature, if it is just a specific example of the more general case: all explosiveness declines exponentially. So do all locations of our 3-volume, and for a possible set of 3-volumes, share the same common time i.e. common cosmic time? That is, perceiving the same Big Bang ~13.8 billion years ago; and thus the same ~2.7 degree kelvin temperature of cosmic background radiation for our now i.e. common positive definite modified global instant; a set of such instants mapping to the integers? Thus is there any necessity for inflation models?

Also for a set of 3-volumes, contained in an array of planes, spherically symmetrical about a modified central force, might one also have a concomitant common cosmic time description intrinsic to System S≡{UT', ...} wherein UT constitutes an array of planes comprised of 3-volumes i.e. UT≡{{VvR}p} ? That is, might one utilize a positive definite thick spherical shell intersecting respective centers of all 3-volumes, denoting such common cosmic time for all patches of a given manifold, and for all 3-manifolds? This might be referred to as Modified Global Simultaneity (MGS) construct. Such set of successive concentric MGSs would also match (i.e. function) to the integers; having positive definite moments, corresponding to {ΔSTm ->0+} set of modified time separations. Also such set of MGSs could be rendered as always exponentially changing i.e. part of an exponential curve. Likewise for changing rate of Hubble expansion, intrinsic to each and all changing 3-volumes.

Thus in such modeling, would one herein have two concomitant descriptions of an overall common cosmic time; the Hubble Expansion, intrinsic to a changing 3-volume, and MGS, intrinsic to an unending evolving dynamic System?

M^3 x M^1 refers to a product space. U_T={{V_R}_p} refers to a set of 3-volumes.

ReplyDelete∆S_Tm->0_+ refers to a modified time-like separation going to positive definite.

System S={U_T^', ...}, a divergent set.