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After watching a documentary about Einstein's theory of relativity, my mind was busy trying to comprehend how space bends when space is void and not made of matter. And additionally, does his theory involving gravity ignores Newton's law of gravity since Einstein thinks that we are being pushed down by space but not pulled down. I am looking for some answers since I am pretty confused right now.

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    $\begingroup$ Newton's law is a low speed, low energy density limit of Einstein's metric theory, but the derivations of the two laws start and end in different places $\endgroup$
    – Kyle Kanos
    Nov 18 '16 at 20:30
  • $\begingroup$ Perhaps related: physics.stackexchange.com/q/219171 $\endgroup$
    – userLTK
    Nov 18 '16 at 20:52
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When we talk about space bending, we often visualize it with matter, such as a rubber sheet. As you pointed out, space is not made of matter, so the reality has to be something different from the simple metaphor. One approach to understanding why we say "space bends" is to look at the effects of relativity that we can observe. We can then compare those who what happens when you stretch or bend a sheet of material, and see why that metaphor is so useful.

If I fire two laser beams perfectly parallel to each other, they should never intersect, right? That's the definition of parallel lines... or should I say is almost the definition of parallel lines. It's the definition of parallel in a Euclidean space, aka a flat space. In practice, this is not actually what happens. In real space, we find that two rays which appeared parallel when they started can intersect. We call this effect gravitational lensing:

Gravitational Lensing

This is a real, observable fact of nature (the picture above happens to be taken from the Hubble Space Telescope, and spectroscopy confirms that indeed we are seeing multiple images of the same galaxy and of the same quasar). It's just as real as the fact that light bends when it goes through a glass lens.

So when we try to model this effect, we find that the behaviors we see are almost identical in nature to what would happen if space-time was a "thing" and it was stretched. We actually capture this in a concept of "curvature" of space, but intuitively, the curvature has effects so similar to that of the curvature created when you stretch a rubber sheet that we like to say space "stretches." It's an effective metaphor that captures all of the behaviors we actually see.

Stretching spacetime

In this visual, we are trying to capture the 3d effect of the curvature of space with a 2d surface because that's easier for us to physically show. In case 'a', where there are masses, a photon can travel in a straight line which does not curve. In 'b', there is a mass distorting space time. Ignore the mass itself for a moment (we'll explain why it is shown that way in a few moments), just focus on the spacetime grid behind it. We can see that that distortion causes the path of the photon to deviate.

Now we typically show this sort of picture with a mass sitting on top of it, like we see here. What the heck does this mean?! The answer is merely that it is convenient to show it this way. It turns out that the distortions we see in spacetime are mathematically similar to the distortions we see when you put a mass on a rubber sheet and let it stretch the sheet out. For many, this is a natural image, and it helps them. It gives them something in their daily life that they can relate to. For others, it brings up existential questions, like your own.

Now as for "ignoring Newotnian gravity," it's not that we are ignoring it, its that we are explaining the observations in a different way. We never directly observe Netwtonian gravity. It isn't actually a physical thing; it's a model. For many observations (especially those of the planets), Newtonian gravity is very effective at modeling what motion we see.

Relativistic gravity explains the exact same behaviors using a different model. It explains it using the ideas of stretching space and distorting time. In simple cases (such as at slow speeds and planet-sized masses), the two equations yield almost exactly the same equations. They just arrived at those equations through different ways. However, the path Einstein's relativity takes also does a very good job of explaining what we see at high speeds. Newtonian gravity falters in this domain; it gives bad predictions.

The point of this is, Relativity and Newtonian gravity both model the same systems, and the same observations. Newton's model is valid in some domains, and relativity is valid in a larger domain. If you're doing simple things, you can typically use the simple model. If you're doing things which are very sensitive to gravity, you may need to use a more advanced model, such as the one put forth by relativity.

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  • $\begingroup$ Just as an aside, when we usually talk about Quantum Mechanics (QM) being in contradiction with general relativity(GR), on gravity that is, is it that QM tries to explain things using gravitons (I guess, in the Quantum Field theory regime)?? And this difference in the two pictures (of gravitons v/s space-time bending) give contradictory results?? I actually liked the simplicity of your answer, and thus thought that it would be great to learn more on where/why exactly these two theories are fundamentally so different. But sorry for the digression! $\endgroup$
    – seavoyage
    Nov 18 '16 at 20:58
  • $\begingroup$ @seavoyage The contradiction between QM and GR regarding gravity stems from the mathematical process of "normalizing" in QM. Normalizing is a tool used in QM to get rid of some pesky infinities in the math that we never observe in this universe. However, the techniques used to normalize fail to handle all of those infinities in the curved space-times relativity predicts. Thus a naive attempt to combine the two suggests that there are some observable infinite quantities, which most scientists agree is absurd. Thus, there is a search for a more complete theory that handles all domains. $\endgroup$
    – Cort Ammon
    Nov 18 '16 at 21:09
  • $\begingroup$ By "all of those infinities in the curved space-times", do you mean the singularities?? As in black holes, I guess :\ $\endgroup$
    – seavoyage
    Nov 18 '16 at 21:35
  • $\begingroup$ @seavoyage The hairy edge of QM is not my forte, so forgive me if I get this slightly wrong. From what I understand, QM has an issue with self-interaction. QM predicts behaviors that behave like self-interaction, but then that self-interaction interacts with itself, and so forth, in a nasty chain. In the end, you have an infinite number of interactions to consider. With normalization we recognize that the energy of each association gets fainter each time we go one rung up that strange chain, and the energy drops fast enough that we can "normalize" it based on the total energy and... $\endgroup$
    – Cort Ammon
    Nov 18 '16 at 21:48
  • $\begingroup$ ... sidestep the whole issue. However, the math which we choose to use to do this only works well in a flat spacetime. If you curve it, the equations stop playing nicely and you can't just hand-wave the infinite set of self-interactions away. I've always associated it with how the integral of 1/x^p dx from 0 to infinity is finite as long as p < 1, but becomes infinite if p=1 (similar in flavor... I don't believe those are the actual equations) $\endgroup$
    – Cort Ammon
    Nov 18 '16 at 21:49

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