# A rock falling towards the center of earth according to GR

• Ott Rovgeisha

#### Ott Rovgeisha

On youtube, I came across to one of many videos trying to explain General Relativity.
One of the comments was interesting to me:

"I understand how gravity causes an object with velocity to go in curves or to orbit, due to the curvature of space-time: the object thinks that it's traveling in a straight line. But how does gravity (and curved space-time) cause an object to ACCELERATE towards the gravity source? The object was stationary. Why is this g-acceleration a "straight line" to the object even though it was originally stationary. Wouldn't to remain stationary be more of a "continuous motion" for it?"

And then I realized that indeed: to my very limited knowledge, many if not every textbook in a subtle manner, slips around this very simple and basic aspect: why exactly and how, does a simple fall down with the acceleration occur.

Why? Is it SO OBVIOUS, that needs no words, or is it the fact, that actually many of people claiming to know RG do not actually grasp it?

Any ideas how to explain it?

It sounds as if you may be asking the same question as a recent thread called "How does anything change?" Read that and see if it provides any answers.

Remember that in GR it is not just space that is curved, but spacetime. A massive object may stop traversing space in some frame, but cannot stop traversing time.

There are some other threads on this topic here, so it is a question that does get some attention and is not obvious to everyone.

why exactly and how, does a simple fall down with the acceleration occur.

TEFLing, Nathanael and Nugatory
Hmm..
This fabric analogy bugs me personally actually, because, although they are talking about 3D space being bent, the analogy only bends a sheet, which is 2D.. In your opinion, doesn't this lead to nonsense, misconceptions?

Hmm..
This fabric analogy bugs me personally actually, because, although they are talking about 3D space being bent, the analogy only bends a sheet, which is 2D.. In your opinion, doesn't this lead to nonsense, misconceptions?

The fabric analogy bugs a lot of people here, but not because of the difference between 2D and 3D. What's important is the difference between 2d diagrams that are spatial diagrams with no time element, and 2d diagrams that are space-time diagrams. From your responses to date, I'm not sure if this point is being communicated or not, I haven't seen any response from you that would serve as an acknowledgment of this point. Without an acknowledgment of the point and in addition a response that appears to go off in a different direction, it's hard to tell how well communication is working :(.

It's reasonably easy to described curved 2D surfaces, such as a globe, intuitively. It is a bit harder to get across some of the more mathematical details of intrinsic 2D curvature (called Gaussian curvature), since I'm not sure how vital these details are I personally tend to gloss over them and address issues that may arise from this omission as needed.

Going beyond 2 dimensions, it becomes much harder to talk about curvature in spaces of higher dimension than 2D, I'm not aware of any treatment that doesn't require advanced mathematics and rather abstract and non-intuitive descriptions. There may be perils in going from 2D to higher dimension, but it's not clear to me what they are. I don't currently see this as a huge issue, if it does become an issue I don't see any easy solutions to the issue :(.

Sticking to 2 dimensions, though, it's fairly easy to talk about the difference between 2D surfaces which represent 1 space and 1 time dimension, and other 2D surfaces which represent 2 spatial dimensions. We have a name for a surface that is 2 dimensional and represents 1 space and 1 time dimension, this name is a "space-time diagram". While our visual representation of a space-time diagram is purely spatial, the point is what the diagram represents, not how we draw it.

The key difference between 2D space-time diagrams and 2D space-space diagrams in the fabric model are what AT and Dalespam attempted to communicate in answer to your original post. It's not clear to me from your responses whether or not you've noticed the distinction that is being made.

Note that the fabric model isn't actually "wrong" - it just doesn't answer your question, of "why things fall". If you were asking a different question, the fabric model might be an appropriate answer, because in GR pace is curved as well as space-time. But the fabric model is, as you noted, not helpful in answering the question you asked. It is also not the model used in AT's or Dalespam's response to your question.

The fabric analogy bugs a lot of people here, but not because of the difference between 2D and 3D. What's important is the difference between 2d diagrams that are spatial diagrams with no time element, and 2d diagrams that are space-time diagrams. From your responses to date, I'm not sure if this point is being communicated or not, I haven't seen any response from you that would serve as an acknowledgment of this point. Without an acknowledgment of the point and in addition a response that appears to go off in a different direction, it's hard to tell how well communication is working :(.

It's reasonably easy to described curved 2D surfaces, such as a globe, intuitively. It is a bit harder to get across some of the more mathematical details of intrinsic 2D curvature (called Gaussian curvature), since I'm not sure how vital these details are I personally tend to gloss over them and address issues that may arise from this omission as needed.

Going beyond 2 dimensions, it becomes much harder to talk about curvature in spaces of higher dimension than 2D, I'm not aware of any treatment that doesn't require advanced mathematics and rather abstract and non-intuitive descriptions. There may be perils in going from 2D to higher dimension, but it's not clear to me what they are. I don't currently see this as a huge issue, if it does become an issue I don't see any easy solutions to the issue :(.

Sticking to 2 dimensions, though, it's fairly easy to talk about the difference between 2D surfaces which represent 1 space and 1 time dimension, and other 2D surfaces which represent 2 spatial dimensions. We have a name for a surface that is 2 dimensional and represents 1 space and 1 time dimension, this name is a "space-time diagram". While our visual representation of a space-time diagram is purely spatial, the point is what the diagram represents, not how we draw it.

The key difference between 2D space-time diagrams and 2D space-space diagrams in the fabric model are what AT and Dalespam attempted to communicate in answer to your original post. It's not clear to me from your responses whether or not you've noticed the distinction that is being made.

Note that the fabric model isn't actually "wrong" - it just doesn't answer your question, of "why things fall". If you were asking a different question, the fabric model might be an appropriate answer, because in GR pace is curved as well as space-time. But the fabric model is, as you noted, not helpful in answering the question you asked. It is also not the model used in AT's or Dalespam's response to your question.

Thank you for answering. Both of those things bug me: no reference to temporal information AND the inability to explain the free fall towards the earth.
Especially I am bugged when one gives this fabric analogy and then says vaguely that "this basic principle explains gravity and that means that it explains why things falling down".
If this is a principle, then from this principle, it should be possible to deduce this simple thing as to why an apple or a rock or whatever is falling down, but as you mentioned, it seems not to do that.

Thank you for answering. Both of those things bug me: no reference to temporal information AND the inability to explain the free fall towards the earth. Especially I am bugged when one gives this fabric analogy and then says vaguely that "this basic principle explains gravity and that means that it explains why things falling down". If this is a principle, then from this principle, it should be possible to deduce this simple thing as to why an apple or a rock or whatever is falling down, but as you mentioned, it seems not to do that.

Well, if you think you understand how curved space could cause a moving particle to move along a curved spatial path, then it's conceptually not much different to understand how curved spaceTIME could cause a particle to move along a curved spacetime path. In the top figure below is a representation of spacetime as a two-dimensional manifold, with coordinates labeled $t$ for time and $r$ for space (distance from the center of the Earth, for example). In flat spacetime, which means spacetime without gravity, a particle initially at rest will stay at the same location in space, which means that $r$ doesn't change. But the particle continues to move forward in time. So the path of the particle is vertical when shown on a $t-r$ graph.

If instead spacetime is curved, then the path of the particle will not be straight as shown on a $t-r$ graph.

Thank you for answering. Both of those things bug me: no reference to temporal information AND the inability to explain the free fall towards the earth.
Especially I am bugged when one gives this fabric analogy and then says vaguely that "this basic principle explains gravity and that means that it explains why things falling down".
If this is a principle, then from this principle, it should be possible to deduce this simple thing as to why an apple or a rock or whatever is falling down, but as you mentioned, it seems not to do that.
Did you even read posts 3 and 4, both of which addressed both points. It is beyond irritating to answer a question and then have the next response be a complaint that the question is unanswered.

Did you even read posts 3 and 4, both of which addressed both points. It is beyond irritating to answer a question and then have the next response be a complaint that the question is unanswered.

There is no complaint. Where is the complaint that a question was unanswered? Please quote verbatim...

If there was a complaint that something is unanswered then there would have been a post saying this: " I asked something and it was left unanswered".

All I had done here, is expressing the idea, that there is a tendency of some people to explain certain things in a way that does not explain it in reality.

Where is the complaint that the question is unanswered? I was asked about the aspects that bug me about the analogy mentioned in the posts. I replied to it.

You may find this "beyond irritating", but before being "beyond irritated" one should consider if there really is a reason to be that way.

I must say that the fabric analogy STILL bugs me.
Gonna think and think and think...but so far, it still bugs me. I do not see an obvious connection with a falling object and still not entirely sure that there is not something fishy about it. I am reading and considering the posts 2 and 3.

All in all, analogies tend to be dangerous and misleading. A simple example is water in pipes analogy in relation to the electric current in metal wires. The water analogy is completely false because there is fundamental difference on how the electric field is created in the wires that "pushes" the electrons vs how pressure difference is created in the pipes.

Both of those things bug me: no reference to temporal information AND the inability to explain the free fall towards the earth.
"This bugs me" is a complaint.

Please respond directly to posts 3 or 4 (4 is probably better). It explains free fall and references temporal information. You should not be bugged after examining that material.

I must say that the fabric analogy STILL bugs me.
Gonna think and think and think...but so far, it still bugs me. I
Don't pay attention to the fabric analogy. It is not essential, and nobody here uses it to explain anything. The curvature of spaceTIME is the essential thing.

"This bugs me" is a complaint.

Please respond directly to posts 3 or 4 (4 is probably better). It explains free fall and references temporal information. You should not be bugged after examining that material.

Well, it is more like a statement... If it were a complaint, i would have described how much this thing that is bugging me, is actually driving me to beyond irritation...
All in all, if you want to call it a complaint, it is not a complaint towards you...rather towards the analogy ;) and I was genuinely interested in your thoughts on that.

3D space being bent, the analogy only bends a sheet, which is 2D.. In your opinion, doesn't this lead to nonsense, misconceptions?
It depends. When you want to show orbits you need at least 2 spatial dimensions, which together with time would require showing a curved 3D manifold. That is difficult to visualize in a single diagram, so you can only show certain aspects with curved 2D sheets. The common indented 2D sheet visualization shows only the spatial effect, which is not really what causes the planets to orbit, but merely what causes their orbits to precess. So yes that is misleading.

But for vertical (radial) fall you only need 1 spatial dimension, so it it perfectly OK to reduce 4D spacetime to 2D spacetime (time + 1D-space). See the videos in post #4.

Hmm..
This fabric analogy bugs me personally actually, because, although they are talking about 3D space being bent, the analogy only bends a sheet, which is 2D.. In your opinion, doesn't this lead to nonsense, misconceptions?

As A.T. points out, for a rock falling straight down, only one spatial dimension is relevant--the direction that the rock is falling. So 2D (one dimension for the only relevant spatial direction, and one dimension for time) is perfectly adequate to describe what's happening to the rock.

Ott Rovgeisha

As well as posts #4 and #8, you may find this relevant:

This is my own non-animated way of looking at it:

• A Two inertial particles, at rest relative to each other, in flat spacetime (i.e. no gravity), shown with inertial coordinates. Drawn as a red distance-time graph on a flat piece of paper with blue gridlines.
• B1. The same particles in the same flat spacetime, but shown with non-inertial coordinates. Drawn as the same distance-time graph on an identical flat piece of paper except it has different gridlines.

B2. Take the flat piece of paper depicted in B1, cut out the grid with some scissors, and wrap it round a cone. Nothing within the intrinsic geometry of the paper has changed by doing this, so B2 shows exactly the same thing as B1, just presented in a different way, showing how the red lines could be perceived as looking "curved" against a "straight" grid.
• C Two free-falling particles, initially at rest relative to each other, in curved spacetime (i.e. with gravity), shown with non-inertial coordinates. This cannot be drawn to scale on a flat piece of paper; you have to draw it on a curved surface instead. Note how C looks rather similar to B2. This is the equivalence principle in action: if you zoomed in very close to B2 and C, you wouldn't notice any difference between them.

Note the diagrams above aren't entirely accurate because they are drawn with a locally-Euclidean geometry, when really they ought to be drawn with a locally-Lorentzian geometry. I've drawn it this way as an analogy to help visualise the concepts.