If space itself is expanding, then why would it have any effect on matter (separates distant galaxies)?
Space is "nothing", and if "nothing" becomes bigger "nothing" it's still a "nothing" that shouldn't interact with matter in any way (it doesn't have mass, energy, etc).
Gravity doesn't have a cutoff distance afaik, so even the most distant galaxies should be attracted to each other. Gravity force would be very very tiny, but it would still dominate "nothing" from space expansion.
Lets take inertia from Big Bang into account. Inertia would be the primary force that moves galaxies away comparing to their tiny gravity and even more tiny, if any, force of our "nothing" that is still expanding inbetween. Wouldn't expansion decelerate if driven mostly by inertia?
Answer
Space actually has energy, vacuum energy. It has been shown by various experiments and can be explained by Quantum Mechanics. So more space means more energy. Although it probably can't be used to do work, it does act to increase expansion. Check out the wikipedia page for more info.
Although galaxies are massive, they are far away and thus the resulting acceleration towards each other is weak. If the space between galaxies is expanding at a faster rate than their mutual atraction; galaxies will move apart.
Imagine a football player running from one endzone toward the other as fast as he could. However the distance between them endzones doubles every 4 seconds. The endzones are not moving; the space between them is growing. There is no way the player could hope to reach his goal. In a short time he would not be able to see the other end.
The scary thing is this not only goes for the field but everything. The player himself would be ripped apart as his own body parts moved farther and farther apart. If space was expanding fast enough the attractive forces keeping black holes or even atoms together would not be enough. This what is commonly referred to as the Big Rip.
Luckily the expansion is slow enough that only "distant" galaxies are moving away from each other. Gravitational forces in a solar system or a galaxy are more than enough to withstand expansion. The larger the scale of the system the more expansion has to play.
When we talk about expansion we are not talking about objects moving away because of their great velocities against a static backdrop.
What you're describing is like marbles on a grid. The marbles get farther away due to their velocities relative to the grid pointing in opposite directions.
Instead expansion is like the grid growing in scale. This effects the distances between the marbles independently of their velocities; which is why we observe all far away objects as moving away from us. If it wasn't for expansion we would expect a more random distribution. It is only at smaller scales where expansion is less of a factor that we can see objects moving toward us such as Andromeda's Galaxy.
We are not in any sense moving away from the center of the universe. The idea of the Big Bang and Inflation is that not only every thing, matter and energy; but everywhere was contained in the Big Bang.
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