Is the Expanding Universe Affecting Our Solar System?

[Mark M]

Was not aware of such a distinction...but it makes no sense to me...'intrinsic' usually means we have no good ideas! In this case, assigning such a 'constant' factor of integration and ascribing a specific physical meaning amid the mess of GR is byond my paygrade!

Well, you could think of a cosmological constant as being a geometric feature of the particular space-time. For example, a space with a cosmological constant and no matter is a de Sitter space. A de Sitter space just comes with this constant force (the cosmological constant). Nothing present in the space causes this, it's 'built in' to it's equations.

Specifically, you get the cosmological constant by inserting it into the Einstein-Hilbert action, and then deriving the Einstein Field Equations as usual.

the exact mathematical relationship between vacuum energy and the cosmological constant is interesting...have not seen it...

It comes from the normal Einstein Field equations. Take a look: $$R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}$$ Now, say no matter is present, so we can ignore the the Ricci tensor and scalar. (Those are the two 'R's, the tensor has a subscript). Now the equation becomes $$g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}$$ Using a little algebra, you can solve for the stress-energy tensor (the 'T'), and get this $$T_{\mu \nu} = - \frac {\Lambda c^{4}} {8 \pi G} g_{ \mu \nu}$$ From which you could derive the expression I posted above.

Do you think the assumptions that go into the FLRW metric solution
to the EFE apply within gravitationally bound systems?

They most definitely don't apply for non-homogenous distributions of matter, such as galaxies. So, as John Baez explained on that page, we say that no expansion is occurring in the galaxies (dark energy is a different matter).

 

[Naty1]

I found my notes on the other discussion...Wallace is a practicing cosmologist...explaining that cosmological expansion assumptions do not carry to lumpy environments...

thread # 162727

If anything there is a vanishingly small FRW element to the metric of bound structures. If the FRW metric 'prevail(ed) on all scales and everywhere, even inside gravitationally bound structures or within atoms' then why do galaxies maintain a constant size as the distance between them expands? Commonly we are told that the local mass concentration 'overcomes' the expansion preventing this from occurring. This is one of the worst and most fallacious explanations you could possibly give someone! What really happens then?

The FRW metric is the inevitable result of the cosmological principle, CP. which is that the universe is homogeneous and isotropic. The metric is only valid if these principles hold. Consider now a galaxy, solar system or planet. Does the CP hold? No. Is it a remotely useful approximation? Not at all! Unsurprisingly then the dynamics of bodies in these systems and on these scales bears no resemblance to the dynamics of galaxies. So for instance, there is no redshift of light due to a(t) when we observe light from the other side of our galaxy, or from say Andromeda. The FRW metric simply is not valid on these scales.

….. The better way to look at it is that the presence of the mass in the galaxy gives the metric of space-time around this mass a form that would look much more like a Schwarzschild metric than FRW (though we cannot fully solve GR for a galaxy.). The point is though that there is not expansion to 'overcome' since the 'expansion' is merely the result of the metric [variable over time] formed by a homogeneous and isotropic mass distribution. If the mass doesn't obey these principles we shouldn't be surprised that we don't see any 'expansion'.

If you don't believe me hold an object in each hand with outstretched arms. When you let them go what happens? I think you will find that they both plummet towards the local centre of mass (the centre of the Earth) rather than drift off into the Hubble flow! The local mass concentration can hardly be described as a mere perturbation to the FRW metric!

Wallace: #63

….the 'expansion' (which we both definitely agree is a bad term for it!) is a result of the FRW metric, in particular a(t). The metric in the region of bound structure looks nothing like the FRW metric, in particular it has no global time dependence (though will of course evolve). For this reason I stand by the statement that the FRW metric is not valid on scales which are significantly inhomogeneous, since the metric has no component that reflects the global a(t), and hence the FRW picture does not relate to the dynamics of the system.

 

[Naty1]

MarkM..I posted #37 before I saw yours immediately preceding...we all agree on cosmological expansion.

Well, you could think of a cosmological constant as being a geometric feature of the particular space-time. For example, a space with a cosmological constant and no matter is a de Sitter space. A de Sitter space just comes with this constant force (the cosmological constant). Nothing present in the space causes this, it's 'built in' to it's equations.

I understand what you mean ..but being a former 'engineer' [supposedly] , I require a physical construct for my brain to function...