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It is said that the universe is expanding and the galaxies are moving apart. I understand that the space between every two galaxies is increasing. Doesn't this seem to imply that the galaxies will have relative motion and so have a moving velocity. According to relativity and the modern belief in physics, it is said that everything depends on relative motion and nothing else (I am talking about the concepts such as Time Dilation etc.). So in brief I would like to ask that why is it said that galaxies do not move even though the distance between them is increasing.

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It isn't said that galaxies don't move, or at least it isn't said by any physicists I know. Distant galaxies are moving away from us, and indeed their recession velocity can be (approximately) calculated using Hubble's Law.

The question is why the galaxies are moving away from us, or more precisely is there a theory we can use to explain not only why distant galaxies are moving away but also predict how they will move in the future. This is where General Relativity and the FLRW metric are used.

The galaxies are moving away from us because the space in between us and the galaxies is expanding. What does this mean? Well the way we measure distance in GR is using a geometrical object called the metric. The metric is a function of how matter is distributed, so for example around a black hole the metric is the Schwarzschild metric. If we take a system where matter is evenly distributed the metric that GR predicts is the FLRW metric.

The metric tells us how to calculate distances between objects. I won't go into details or this would turn into a book length answer. I'm afraid you'll have to accept that we can do this. So let's use the FLRW metric to calculate the distance to some distance galaxy, and suppose we get a billion light years. But the FLRW metric is a function of time, which means the distance we calculate depends on what time we do it. If we wait about 13.8 billion years and calculate the distance again we'll get the result two billion light years. Now we physicists would say this is because the space in between us and the galaxy has expanded, so there is more space to cross to reach the other galaxy. Yes, but the fact remains that the distance to the galaxy has increased by a billion light years in the 13.8 billion years between our two measurements so it must be moving away from us. We can even estimate its velocity as about 1/13.8 times $c$.

So the bottom line is that the galaxies are moving away from us because space is expanding. And General Relativity predicts how fast they are moving away from us.

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Thanks for answering but I had not asked anything about the FRW/FLRW metric because it is not a part of my question and I already know about it. And for that matter how can space even expand. It is just said that more space is just added and misleading analogies such as of the rubber balloon are given. But I just do not understand the whole concept of expansion because I personally feel that it might at some point be contradicting some of the physics laws definitely. –  rahulgarg12342 Dec 20 '13 at 17:26
    
If you can give an example of a potential contradition I'll have a look at it. At the moment I'm not sure what you're asking. –  John Rennie Dec 20 '13 at 17:36
    
Thanks for the help. Anyway I got my answer and the contradiction part was just a thought. –  rahulgarg12342 Dec 20 '13 at 18:12
    
Of course, the real part of it, though, is that from the perspective of the FLRW metric, aliens living in the other galaxies will say that they aren't moving, and that WE are receding from them. There is no local motion -- the motion we observe is due to a global effect of the spacetime. –  Jerry Schirmer Jan 20 at 1:12
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Relative velocities are a problematic concept in general relativity: Whereas you can directly compare any two velocity vectors in a flat Minkowski universe, you cannot do so in a curved one.

Of course, we can locally approximate curved spacetime with Minkowski spacetime the same way we can approximate the surface of the earth with flat Euclidean space. This is most obvious in normal coordinates and works as long as the metric tensor $g_{\mu\nu}$ differs from the Minkowski version $\eta_{\mu\nu}$ only negligibly due to small curvature or short distances: $$ g_{\mu\nu} = \eta_{\mu\nu} + \frac13R_{\mu\rho\sigma\nu}x^\rho x^\sigma + \mathcal O(x^3) $$

In general, that's not an option: The tangent spaces (which is where velocities live) rooted at different spacetime events are separate and you need to perform a path-dependent parallel transport to meaningfully compare velocity vectors.

If I'm not mistaken (hopefully, someone will correct me if I am), gravitational redshift in Schwarschild spacetime and cosmological redshift in FLRW spacetime actually reduce to special-relativistic doppler shift via parallel-transported velocities along the appropriate null geodesic (ie light path).

In a FLRW spacetime (which we use to model our expanding universe), in addition to the local approximate and the generic path-dependent concept of relative velocities, there is yet another variant:

A certain class of distinguished inertial reference frames can be used to define comsmological time as well as a comoving coordinate grid in space. The proper distance between two points at rest relative to that grid increases, which yields the recessional velocity that appears in Hubble's law. At small distances, it agrees with our other concepts of relative velocities, but at large distances, it does not and can in particular exceed the speed of light $c$ (which relative velocities really shouldn't).

The common physical interpretation of this velocity is as rate of expansion of space: It measures the stretching of the fabric of the universe between two galaxies analogous to the growing distance between dots on an expanding balloon. In addition to that, we also need the so-called peculiar velocity to fully describe relative motion, which is measured relative to the rest frame defined by our comoving grid and will contribute a doppler part to the total redshift.

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May be this is what was meant wherever you read it. Isotropic observers don't move, the distance between any two of them grows. The isotropy allows you to choose coordinates in a spacial way, so that isotropic observers have space coordinates which do not change with time(cosmological time), so they don't move with respect to each other, but the distance between any two is a function of time, so it can change over time. In the expanding balloon analogy if take meridians and parallels for coordinates, a galaxy at the intersection of the equator and the Greenwich meridian will always be there. Same for a galaxy at the equator and ten degree meridian. But the distance between them will change as the balloon expands.

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According to relativity and the modern belief in physics, it is said that everything depends on relative motion and nothing else

This is not true. In General Relativity there is such thing as pecular velocity, the velocity relative to the local comoving frame. The galaxies have limited pecular velocity but their distance to Earth can increase faster than speed of light.

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This is only true if the particular spacetime geometry has some special reference frame. –  Jerry Schirmer Jan 20 at 1:26
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protected by Qmechanic Mar 14 at 23:29

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