Why do we fall, according to GR?

[TrickyDicky]

It's not showing time as a spatial dimension; it's simply representing time as an axis on the graph.

I'm making a (maybe too subtle) distinction between what it represents and what the literal picture shows in the context of the nuances of space vs. spacetime.
We all agree that this axis represents the time coordinate, but the graph cannot easily show the difference between a time coordinate and a space coordinate and so it depicts them in the same way, and we see a 2D SPATIAL surface that represents a 2D Lorentzian Spacetime that is basically a 1D space in time.

This might verge on the pedantic, but it could be a source of confusion for some (precisely those that are very familiar with seeing time plotted in graphs but not so much with Lorentzian manifolds and relativity) , and that is why I mention it.

 

[Fredrik]

Why would the Earth accelerate upwards?

What you're asking here is pretty much the same as "Why is GR a good theory?". You're focusing on a specific aspect of it, but I think the only good way to answer your question is to explain why the predictions of GR are accurate, and the only thing that can do that is a better theory of gravity.

Is it true spacetime push as back to Earth?

No, Earth is pushing you away from the part of space you'd have to be into do geodesic (i.e. non-accelerating) motion.

 

[kweba]

kweba I think it would help if you familiarised yourself with the concept of a 'four-velocity'. When you understand that concept you will see that there is no such thing as being 'at rest'. Every object has a nonzero four-velocity. Hence it has a unique geodesic that it follows - if it is free from non-gravitational forces. That geodesic is defined by its position and the direction in spacetime of its four-velocity.

ETA: looks like somebody has just explained this above, with diagrams too. Nice!

Yes, thank you. I admit I am not familiar with the concept (I never heard of it.) I guess this is what I'm missing all along. Thanks, I will read/study about it.

But I never realized, until now, that we are in a constant motion (velocity) through spacetime, that even though we may be at rest in space, we are still in motion through time.

It said that our four-velocity, with everything in the universe, is at the constant speed of light, but it just goes to the time direction. Is this true?

Yes I tried studying the graphs, but pardon me as I'm having a hard time getting it.

 

[Fredrik]

Yes, thank you. I admit I am not familiar with the concept (I never heard of it.) I guess this is what I'm missing all along. Thanks, I will read/study about it.

But I never realized, until now, that we are in a constant motion (velocity) through spacetime, that even though we may be at rest in space, we are still in motion through time.

It said that our four-velocity, with everything in the universe, is at the constant speed of light, but it just goes to the time direction. Is this true?

It's useful to understand four-velocity, but I don't think it will help you answer the question in the thread title. What you need to understand is that SR and GR both involve a mathematical thing called "spacetime", and that motion of particles is represented by curves in spacetime, called "world lines". There's a class of curves that really distinguish themselves from all the rest, in a way that doesn't depend on a choice of coordinate system. These curves are called geodesics.

Since they are the only world lines that are "special" in a coordinate-independent way, it's natural to define the proper acceleration of a world line in a way that ensures that the proper acceleration of a world line, at an event E on it, is a measure of much the curve "curves away" from a geodesic through E that's tangent to the world line. The force acting on the object can then be defined by F=ma. To find the geodesics near a spherical distribution of mass (like a planet), we must solve Einstein's equation for that distribution. When we do, we find that the curve representing the motion of an object on the surface isn't a geodesic, so the a (acceleration) of the object is non-zero. Since F=ma by definition, this means that there's a force acting on it. Since the surface of the Earth is preventing the object from doing geodesic motion, we say that the surface of the Earth is pushing it with force ma.

(What I wrote as F=ma is actually a four-vector equation. I'm trying to keep things simple, so I won't say anything more about that).

The four-velocity of a world line is defined as the normalized tangent vector of the world line. This makes the statement that everything moves through spacetime at speed c trivial and not interesting in my opinion.

 

[A.T.]

Yes I tried studying the graphs, but pardon me as I'm having a hard time getting it.

Falling from rest:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html

Vertical throw upwards (the second half is the same as falling from rest):
http://www.relativitet.se/spacetime1.html