Trouble wrapping my head around GR

[xcourrier]

So, I feel like I have a pretty good grasp of Special Relativity, but GR is a whole different beast. I am really stuck on where the acceleration comes from for Gravity. For instance, it's easy for me to understand how an orbit can form from bent space (an analogy for the way I think of it is a car driving straight down the middle of a road, a sort of geodesic, in a circular race track). But if that object were to halt its orbit, it would then begin to fall or accelerate, and that part gets me. I feel like it has to do with the bending of time part, but its not very clear. For that matter, if we are accelerating through curved space time by standing on the surface of the earth, where does the space time we just accelerated through go?

I hear a lot of analogies, like the bowling ball and the trampoline (which happens to be all I get when I try to look anything up), but that always requires real life gravity pulling down on the demo to accelerate objects towards the center. I also hear the phrase "Space-time tells mass how to move, mass tells space-time how to bend," which doesn't seem inherently a relativistic idea, since you could easily replace "space-time" with "the force of gravity" and "bend" with "pull" and get "Gravity tells mass how to move, mass tells gravity which way to pull," which seems like a perfectly Newtonian view of the same thing.

I apologize if that was too many questions, it was really only meant to be one tackled a few different ways. If its easier to direct me to somewhere else, I would appreciate that as well. I learned Special Relativity from the World Science U site, but they don't have a section for GR yet. Thanks.

 

[PeterDonis]

it's easy for me to understand how an orbit can form from bent space

It's not curved space, it's curved spacetime. There's a big difference. The space in the vicinity of the Earth, for example, is only very slightly curved--certainly not curved enough to make a circular path around the Earth "straight" in the spatial sense. The path of the orbiting object is only straight in spacetime--if you think of time as vertical, the object's path looks like a helix if we use coordinates centered on the Earth, but it's actually the straightest possible path given the initial condition that the object has a given tangential velocity.

if that object were to halt its orbit, it would then begin to fall or accelerate, and that part gets me.

There are two different kinds of acceleration, and it's important not to confuse them. The acceleration you are talking about is called "coordinate acceleration"; in coordinates centered on the Earth, the object is accelerating downward. But the object does not feel any acceleration; it is in free fall, just like the orbiting object. So its proper acceleration, which is the other kind of acceleration--the kind you actually feel--is zero. That means the falling object is also traveling on a straight path through spacetime--just a different straight path (because it starts out with a different initial condition) from the orbiting object.

if we are accelerating through curved space time by standing on the surface of the earth, where does the space time we just accelerated through go?

We are "accelerating" in the sense of proper acceleration--we feel acceleration. But we are not accelerating in the coordinate sense--in coordinates centered on the Earth, we are at rest, not moving at all. So we are not "accelerating through spacetime" in that sense; we are standing still.

 

[xcourrier]

Thanks for the reply, and for the clarification between coordinate and proper acceleration.

I understand the equivalence principle, but in the example given to illustrate the principle, a person in a rocket experiencing 1G can look back and point to the space he passed through. A person standing on Earth cannot, to the best of my understanding, do the same.

In the example you gave, a 3d graph with both the lateral axes being space and the vertical being time, you illustrated a helix as the straightest path thru curved spacetime. If we were to add a second object with no tangental velocity, it would follow something similar to a sign wave (assuming a hole through the Earth so that it won't experience any force). These objects are both experiencing the same curved spacetime, so how are their geodesics so different? And how does a straight path thru time and space = an acceleration in coordinates centered on Earth?

Thanks again.

 

[PeterDonis]

I understand the equivalence principle, but in the example given to illustrate the principle, a person in a rocket experiencing 1G can look back and point to the space he passed through. A person standing on Earth cannot, to the best of my understanding, do the same.

The equivalence principle is local; it only applies in a small region of spacetime. That means both space and time; once you are considering a long enough time that the person in the rocket can point to all the space he passed through, whereas the person standing on Earth can't, you are looking at a large enough region of spacetime that the EP no longer applies.

These objects are both experiencing the same curved spacetime, so how are their geodesics so different?

Because they start off with different initial conditions; one has a tangential velocity, the other does not. That means they start out pointing in different directions in spacetime (because that's what the different initial velocities means in spacetime terms), so of course they follow different paths.

how does a straight path thru time and space = an acceleration in coordinates centered on Earth?

Because that's how spacetime is curved in the vicinity of the Earth. It's not easy to visualize, and unfortunately I don't have a good image handy to post here; but Fig. 2.8 in this document is a fairly good illustration of how it works for the case of purely radial motion.

 

[xcourrier]

Right, I have been watching youtube and playing around with interactive graphs, namely these two:


http://www.adamtoons.de/physics/gravitation.swf

You don't necessarily have to click on them, they're basically the same thing you linked. I am just trying to understand how these graphs translate into the real world. I liked the graph you imagined because it was 2 dimensions of space, and it seems closer at least to the real world than the 'x' and 't' graph. So my question regarding the different geodesics was kind of trying to understand how a graph of that sort (2 space 1 time axes) could bend in such a way as to keep both of those lines straight, as well as maybe a 3rd that was a straight line up from the center of the Earth (e.g. the Earth's core).

In terms of what you said about the equivalence principle, that makes sense I suppose, though I thought the locality of it was something larger since you need a large object to feel tidal forces, not necessarily something that is gone the moment you pass through it. I'll prolly lose some sleep tonight mulling that over lol

 

[PeterDonis]

I liked the graph you imagined because it was 2 dimensions of space

No, it isn't. It's only showing one dimension of space (the radial direction) plus time. But there is curvature, so the two-dimensional surface shown is curved, so when we visualize it it takes three dimensions in the visualization.

 

[A.T.]

Right, I have been watching youtube and playing around with interactive graphs, namely these two:


http://www.adamtoons.de/physics/gravitation.swf

You don't necessarily have to click on them, they're basically the same thing you linked. I am just trying to understand how these graphs translate into the real world.

These only show radial motion: 1 space dimension. The flash applet has a representation of the "real world" bottom-left.

So my question regarding the different geodesics was kind of trying to understand how a graph of that sort (2 space 1 time axes) could bend in such a way as to keep both of those lines straight,

That is very hard to visualize in one graph. We can understand curved 2D graphs (1 space + 1 time or 2 space) by embedding them in flat 3D. But curved 3D (2 space + 1 time) would require more than 3 dimensional flat embedding space.

You could also show both graphs (1 space + 1 time, 2 space) side by side. But note that in general the free falling world lines will NOT be geodesics in either of these 2D slices, because they are geodesics in the total 4D spacetime. Only when the free falling motion is along 1 spatial dimension (radial fall), you also get geodesics on a 2D slice.

An alternative visualization is the loose analogy to light rays bending in a medium of varying optical density. In a 3D block of such a medium you can represent 2 space + 1 time.

 

[xcourrier]

No, it isn't. It's only showing one dimension of space (the radial direction) plus time. But there is curvature, so the two-dimensional surface shown is curved, so when we visualize it it takes three dimensions in the visualization.

I must have misunderstood your post, I was envisioning an orbit on a 2d plane stretched upwards to add a time component making it helical.

These only show radial motion: 1 space dimension. The flash applet has a representation of the "real world" bottom-left.

That is very hard to visualize in one graph. We can understand curved 2D graphs (1 space + 1 time or 2 space) by embedding them in flat 3D. But curved 3D (2 space + 1 time) would require more than 3 dimensional flat embedding space.

You could also show both graphs (1 space + 1 time, 2 space) side by side. But note that in general the free falling world lines will NOT be geodesics in either of these 2D slices, because they are geodesics in the total 4D spacetime. Only when the free falling motion is along 1 spatial dimension (radial fall), you also get geodesics on a 2D slice.

An alternative visualization is the loose analogy to light rays bending in a medium of varying optical density. In a 3D block of such a medium you can represent 2 space + 1 time.

So basically to see a curved 3d graph in 3d, we would have to accept that straight lines appear curved. I suppose that makes sense since the straight lines of any geodesic of an object with no initial velocity would have to intersect multiple times. Is there any other way to visualize how the space time of multiple objects relate to each other or does the locality of curved space time prohibit that?

Thank you both for your time

Edit: I am reading through the pdf that PeterDonis shared. I think that might have what I am looking for

 

[Orodruin]

a person in a rocket experiencing 1G can look back and point to the space he passed through.

Just to clear this up: "Space" is not a substance you can "point to". All he can say is "I was accelerating in that direction". Someone else might consider him to always be moving in the opposite direction, just decelerating. If this is unclear, you probably need to revisit your understanding of spacetime in special relativity.

 

[A.T.]

So basically to see a curved 3d graph in 3d, we would have to accept that straight lines appear curved.

Well, you cannot isometrically embed curved 3D in flat 3D. In the optical density analogy the light rays aren't straight, but still the quickest way for light between two nearby points.

I suppose that makes sense since the straight lines of any geodesic of an object with no initial velocity would have to intersect multiple times. Is there any other way to visualize how the space time of multiple objects relate to each other or does the locality of curved space time prohibit that?

In the radial fall visualizations you could add multiple objects, like done here with one just dropped, the other thrown upwards:



But if you want two free fall world-lines intersecting twice you need the non-local intrinsic curvature like shown here:
http://www.adamtoons.de/physics/gravitation.swf

 

[xcourrier]

Just to clear this up: "Space" is not a substance you can "point to". All he can say is "I was accelerating in that direction". Someone else might consider him to always be moving in the opposite direction, just decelerating. If this is unclear, you probably need to revisit your understanding of spacetime in special relativity.

My apologies, the course I took on special relativity showed a 3d grid with clocks at each intersection to represent spacetime, so I guess it seemed more real than that. Also, a documentary I watch had Michio Kaku explaining that space pushing down on us was what was causing gravity, but it appears that may have been an oversimplification for the sake of the documentary. We didn't deal any with acceleration so I thought maybe in an acceleration you could move through your own space-time grid, but that is apparently not the case even in accelerated motion.

 

[xcourrier]

Well, you cannot isometrically embed curved 3D in flat 3D. In the optical density analogy the light rays aren't straight, but still the quickest way for light between two nearby points.In the radial fall visualizations you could add multiple objects, like done here with one just dropped, the other thrown upwards:



But if you want two free fall world-lines intersecting twice you need the non-local intrinsic curvature like shown here:
http://www.adamtoons.de/physics/gravitation.swf

Thanks for the link. I was referring a bit more to 2 objects that didn't have the same starting location or lay on the same x axis, as if you were to take both those graphs and put them in the real world in a staggered manner because it seemed like they would interfere with each others spacetime, but its looking more and more as if that is not possible for the very reason that they would interfere with each other.

 

[Orodruin]

My apologies, the course I took on special relativity showed a 3d grid with clocks at each intersection to represent spacetime, so I guess it seemed more real than that. Also, a documentary I watch had Michio Kaku explaining that space pushing down on us was what was causing gravity, but it appears that may have been an oversimplification for the sake of the documentary. We didn't deal any with acceleration so I thought maybe in an acceleration you could move through your own space-time grid, but that is apparently not the case even in accelerated motion.

As with most popularised science, a healthy starting mindset is that what you will be told is an oversimplification intended only for the purpose of conveying a general idea. It usually is not something you can build further upon.

The grid you were shown was likely a coordinate grid, which is just a set of numbers used to describe the physics, it is not something physical in itself

 

[xcourrier]

As with most popularised science, a healthy starting mindset is that what you will be told is an oversimplification intended only for the purpose of conveying a general idea. It usually is not something you can build further upon.

The grid you were shown was likely a coordinate grid, which is just a set of numbers used to describe the physics, it is not something physical in itself

Thanks. In real life I am a pilot with an interest in physics, not an actual physics student, so most of my knowledge on the subject has come from documentaries and rather simplistic free online courses. Some of the subtleties still escape me, but thank you for sparing some time for my question. The documentaries seem to end their explanation of gravity at the trampoline and bowling ball analogy.

 

[PeterDonis]

I was envisioning an orbit on a 2d plane stretched upwards to add a time component making it helical.

Ah, ok. Yes, this is correct. I thought you were referring to the diagrams in the PDF I linked to, which all show only one spatial dimension, because they're showing purely radial motion.

 

[xcourrier]

Well, you cannot isometrically embed curved 3D in flat 3D. In the optical density analogy the light rays aren't straight, but still the quickest way for light between two nearby points.In the radial fall visualizations you could add multiple objects, like done here with one just dropped, the other thrown upwards:



But if you want two free fall world-lines intersecting twice you need the non-local intrinsic curvature like shown here:
http://www.adamtoons.de/physics/gravitation.swf

I guess another way to frame my question would be: Is there a graph similar to the one shown in this video that could show 2 equal mass objects out in the middle of space being pulled towards each other from a reference frame that favors neither of them? Or would you necessarily have to take the reference frame of one or the other?

Edit: on thinking it over, I suppose you must take one objects reference frame or the others for either line to remain straight, and a "neutral" reference frame would simply have both slightly curved. So then the bending of space time is felt differently at every point in space (and time?). Is that a technically correct statement to make?

 

[A.T.]

Is there a graph similar to the one shown in this video that could show 2 equal mass objects out in the middle of space being pulled towards each other from a reference frame that favors neither of them?

You mean two large masses, like stars? You can show it in principle, but note that most of each mass is not in free fall. Only one point of them is following a geodesic, the rest is subject to contact forces by the surrounding matter.

 

[m4r35n357]

I guess another way to frame my question would be: Is there a graph similar to the one shown in this video that could show 2 equal mass objects out in the middle of space being pulled towards each other from a reference frame that favors neither of them? Or would you necessarily have to take the reference frame of one or the other?

I doubt that a simple graph would be sufficient to describe such a complex situation, but there are some numerical simulations here.