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I am aware about the balloon expanding analogue, where the galaxies are stuck on the balloon surface and here the balloon is our space time and as the balloon expand our galaxies also go apart.

But here is my doubt.

  1. when we say universe is expanding, we say spacetime is expanding, at the same time we say that due to this the galaxies can go apart even greater than the speed of light, since its not the galaxies are moving but the spacetime itself is moving.

  2. We know that the universe is expanding by looking at the doppler shift. but again the velocity of source we use in this formula is the actual relative velocity of source so if the source speed is higher than speed of light then how is the use of doppler formula possible?

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    $\begingroup$ "when we say universe is expanding, we say spacetime is expanding" - not spacetime, we say space is expanding. For details, take a look at Metric expansion of space $\endgroup$
    – Hal Hollis
    Commented Apr 2, 2018 at 13:22
  • $\begingroup$ I guess in many place its considered as spacetime (may be a misinterpret) BTW does the term spacetime expansion makes any sense? also if I am not wrong space and time are interlinked (just read it in some book i don't remember) $\endgroup$
    – shinigami
    Commented Apr 2, 2018 at 14:10
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    $\begingroup$ Possible duplicate of Is the universe actually expanding? $\endgroup$
    – Stéphane Rollandin
    Commented Apr 2, 2018 at 15:17

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Say some cosmic object spews forth a certain number of photons of a certain frequency, which make their way towards an observer.

The photon flux is a measure of distance: Assuming isotropy, it will decrease with $1/r^2$, constant across a growing spherical surface. The photon frequency is a measure of relative velocity, as evaluated by parallel transport through a curved spacetime.

In Friedmann universes specifically, spacetime can be sliced into spatial hypersurfaces of constant cosmological time, where the distance between 'stationary' points evolves according to a global scale factor. This distance can increase arbitrarily fast, and such 'recession velocities' can be greater than $c$. We can look beyond the Hubble sphere (the place where recession velocities hit the speed of light) without a problem. In contrast, relative velocities will always be smaller than $c$: When relative velocities approach the speed of light, redshift goes to infinity, and we cannot look beyond this cosmological event horizon.

Regarding your second point, note that you cannot just plug-in recession velocities into Doppler's formula and expect the correct answer: There's no distance parallelism in arbitrary spacetimes, and you'd have to integrate infinitesimal Doppler shifts along the photon trajectory. In Friedmann universes, this will yield a simple answer: Cosmological redshift between comoving emitters and observers ends up being anti-proportional to the scale factor at time of emission.

Some historic remarks, prompted by the comments:

Einstein was looking for a relativistic theory of gravity, inventing general relativity. Friedmann then derived a certain class of maximally symmetric solutions to the Einstein equations, described by the FLRW metric and evolving accordig to the Friedmann equations. Einstein wasn't a fan, though, because he believed in a static univere (which necessitated the introduction of the cosmological constant).

Lemaître re-discovered these solutions, and, among other things, derived cosmological redshift. However, this calculation was omitted from the English translation of his paper, which meant few people where aware of it when Hubble derived the law that now bears his name heuristically: He (incorrectly) interpreted cosmological redshift special-relativistically, when the general-relaivistic interpretation had in fact already been proposed...

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  • $\begingroup$ I have two more doubt now $\endgroup$
    – shinigami
    Commented Apr 2, 2018 at 13:58
  • $\begingroup$ (a) lets assume a hypothetical case if a galaxy is having a relative velocity and another galaxy is in rest but is moving due to space expansion -- how will we distinguish between that? (b) Did the idea of space expansion was invented for covering up for the fact that few galaxies might have velocities greater than 'c' - which is forbidden by theory of relativity? $\endgroup$
    – shinigami
    Commented Apr 2, 2018 at 14:04
  • $\begingroup$ re (a), you've got a couple of quantities you can use to derive distances (eg photon flux, angular diameter), and redshift as a measure of relative motion (including all types ot 'motion' such as spatial expansion or 'standing still' in a gravity well); they all have to be consistent, and must fit related experimental quantities (eg intensity) $\endgroup$
    – Christoph
    Commented Apr 2, 2018 at 15:05
  • $\begingroup$ re (b), see edit: the evolution of maximally symmetric universes had been calculated from general relativity before observational evidence was found; once the observations were in, we just went with the model that fit the data (though we 'recently' had to re-introduce the cosmological constant to account for accelerated expansion) $\endgroup$
    – Christoph
    Commented Apr 2, 2018 at 15:05
  • $\begingroup$ so if the readings of distances and velocities don't match then its a Space expantion? right $\endgroup$
    – shinigami
    Commented Apr 2, 2018 at 15:14

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