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Things in the Universe which are thousands of light years away from Earth can be seen, or pictures can be taken by Hubble Telescope, because those stars, galaxies, etc. emitted light thousands, millions or billions of years ago. Does this mean the shape of the Milky Way galaxy that we see now was actually the shape millions of years back, and now it has a different shape that we cannot see yet?

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Our observations are of course limited by causality. The Milky Way galaxy (or at least the visible part of it) is a flattened disk of diameter $\sim 100,000$ light years, with a bulge in the middle. The Sun sits in the disk about 2/3 of the way from the centre. The maximum light travel time is therefore of order 100,000 years and is in fact much less for most of the stars that we can easily see.

To answer your question we then need to know something about the speeds that stars travel in the Milky Way and we also need to make the (reasonable) assumption that no deus ex machina suddenly changes the gravitational potential in which they move. If not, then we can extrapolate these speeds up to $\sim 100,000$ years into the future and ask how much this changes the stellar positions compared to the size of the galaxy.

The answer is not much. Relative to the Sun, the fastest stellar velocities are in hundreds of km/s and more normally in the region of tens of km/s. This is only 0.0001-0.001 of the speed of light and so this is by what fraction the stars move (10-100 light years with respect to the Sun) compared with their distance from the Sun.

Another way of looking at this is how fast the stars rotate about the Galactic centre. Again, this is of order 200 km/s, so stars will move at most by around 100 light years, compared with the circumferences of their orbits of $\sim$ tens to hundreds of thousands of light years.

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