<p>There are numerous ways to try to explain space-time by analogy, and the flat plane is one of them. Another is the inflating balloon. The are all only analogies. Either of your pictures above is equally valid, however the one &quot;the right way up&quot; is simpler to understand.</p>
<p>For humans, who can easily conceive of 3 dimensional space, an example that works in 3D is essential. If you have ever tried to visualise more than 3 space dimensions, you'll find it very tricky.</p>

CC BY-SA 4.0
2020-11-09T11:21:04.570Z
Thanks for your answer. But going about it that way has certain implications on how you think about it. And I haven't heard anybody address these issues.
Joshua
CC BY-SA 4.0
2020-11-09T13:23:32.700Z
What issues? Any analogy is by its very nature limited, and they are only guides to help you think. @PM2Ring's links on Physics.SE will be very useful for you.
Rory Alsop
CC BY-SA 4.0
2020-11-09T10:09:17.293Z
2020-11-09T10:09:17.293Z
Rory Alsop
<p>The rubber sheet is a good model of <em>Newtonian</em> gravity. It doesn't work at all as a model of curved spacetime in general relativity.</p>
<hr />
<p>Newtonian gravity can be described by a <a href="https://en.wikipedia.org/wiki/Gravitational_potential" rel="nofollow noreferrer">potential field</a> which has a numeric value at every point in space. Objects experience an acceleration that is equal to minus the gradient of the field. That is, they accelerate in the direction that the field around them is decreasing most rapidly, and the magnitude of the acceleration is proportional to the slope in that direction. Also, the field satisfies <a href="https://en.wikipedia.org/wiki/Poisson%27s_equation#Newtonian_gravity" rel="nofollow noreferrer">Poisson's equation</a>, which allows you to find its value at every point if you know the distribution of matter.</p>
<p>Objects on a hill on Earth accelerate downhill, and (subject to idealizing assumptions) the acceleration is proportional to the slope. Not only that, but the deformation of a horizontal elastic sheet in a uniform gravitational field with heavy objects resting on it is described (approximately) by Poisson's equation. The lower the mass density of the sheet, the vertical stretching, and the friction, the better this toy model approximates Newtonian gravity.</p>
<p>One big caveat is that a 2D rubber sheet is described by the 2D Poisson equation, which has different solutions than the 3D Poisson equation. With a 2D sheet you get 2D Newtonian gravity, which has a 1/r force law instead of 1/r². So in this sense it is far from correct. This problem only affects the elastic sheet part, not the rolling-downhill part. If you construct a rigid surface with the correct shape from the 3D Poisson equation, like the &quot;gravity wells&quot; found in some science museums, you get a pretty accurate simulation of test particles in a 1/r² gravitational field.</p>
<hr />
<p>In general relativity, the motion of objects depends on the intrinsic shape of spacetime.</p>
<p>You can embed hunks of curved space into Euclidean space. Like the rubber sheet analogy this is just a way of understanding them in terms of something we're more familiar with (in this case, space that isn't curved).</p>
<p>The shape of these embeddings is not correctly given by Poisson's equation. The embeddings also don't have any &quot;down&quot; direction. An ant crawling along the surface (which is a much better analogy to the effect of curved spacetime on test particles) doesn't care which way is down; it will follow the same path even if there is no background field at all.</p>
<p>It would make sense to always show GR embeddings as &quot;hills&quot; instead of &quot;valleys&quot;, since it makes no difference to the physics and it would avoid unnecessary confusion with gravity wells. But not only is it not standard to do that, it's almost unheard of in works for popular audiences. Quite often the surfaces presented as GR embeddings look like solutions to Poisson's equation, too, which they shouldn't. Either the authors want to invite confusion, or they don't understand the difference themselves. I suspect the latter.</p>

CC BY-SA 4.0
2020-11-09T17:20:49.683Z
2020-11-09T17:20:49.683Z
benrg
<p>Why is every representation of Einstein's theory of gravity depicted like this:</p>
<p><a href="https://i.sstatic.net/5rnwn.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/5rnwn.jpg" alt="enter image description here" /></a></p>
<p>and not like this:</p>
<p><a href="https://i.sstatic.net/6wlAb.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/6wlAb.jpg" alt="enter image description here" /></a></p>
<p>And many other ways it can be depicted. They suggest that the fabric of space-time is a flat plane</p>

CC BY-SA 4.0
2020-11-09T12:41:19.493Z
There are several problems with the rubber sheet analogy of gravity. One major flaw is that it doesn't really show the curvature of time. There are numerous questions on the Physics stack addressing the downsides of this analogy. Eg, https://physics.stackexchange.com/q/90592/123208  &  https://physics.stackexchange.com/q/7781/123208  & https://physics.stackexchange.com/q/176833/123208 and the various questions linked in those questions. Also see https://m.xkcd.com/895/
PM 2Ring
CC BY-SA 4.0
2020-11-09T12:47:57.157Z
And here's a recent related question https://physics.stackexchange.com/q/586835/123208 (to which I contributed an answer).
PM 2Ring
CC BY-SA 4.0
2020-11-09T13:56:40.080Z
Does this answer your question? [Have more recent LIGO/VIRGO gravitational wave measurements narrowed down the speed of gravity further?](https://astronomy.stackexchange.com/questions/33009/have-more-recent-ligo-virgo-gravitational-wave-measurements-narrowed-down-the-sp)
WarpPrime
CC BY-SA 4.0
2020-11-09T15:22:41.587Z
@fasterthanlight the question is about the speed of gravitational waves, mine is about the depiction used when discussing Einstein's theory of gravity.
Joshua
CC BY-SA 4.0
2020-11-09T15:41:12.223Z
I’m voting to close this question because this doesn't have much to do with astronomy is probably a better fit for the physics stackexchange (or maybe a physics educators stack exchange if there was one?)
usernumber
CC BY-SA 4.0
2020-11-10T23:47:06.363Z
@usernumber great idea,I think Physics SE has gotten [too large](https://physics.meta.stackexchange.com/a/12734/83380) and tries to serves several disparate communities and functions at the same time.
uhoh
<gravity><general-relativity>
Does Einstein's theory of gravity say anything about the direction of space
Joshua
# Does Einstein's theory of gravity say anything about the direction of space

(Asked by: username_0 on 2020-11-09)

Why is every representation of Einstein's theory of gravity depicted like this:

[![enter image description here](https://i.sstatic.net/5rnwn.jpg)](https://i.sstatic.net/5rnwn.jpg)

and not like this:

[![enter image description here](https://i.sstatic.net/6wlAb.jpg)](https://i.sstatic.net/6wlAb.jpg)

And many other ways it can be depicted. They suggest that the fabric of space-time is a flat plane

**Tags:** gravity, general-relativity

> **Comments:**
> **username_1**: There are several problems with the rubber sheet analogy of gravity. One major flaw is that it doesn't really show the curvature of time. There are numerous questions on the Physics stack addressing the downsides of this analogy. Eg, https://physics.stackexchange.com/q/90592/123208  &  https://physics.stackexchange.com/q/7781/123208  & https://physics.stackexchange.com/q/176833/123208 and the various questions linked in those questions. Also see https://m.xkcd.com/895/  
> **username_1**: And here's a recent related question https://physics.stackexchange.com/q/586835/123208 (to which I contributed an answer).  
> **username_2**: Does this answer your question? [Have more recent LIGO/VIRGO gravitational wave measurements narrowed down the speed of gravity further?](https://astronomy.stackexchange.com/questions/33009/have-more-recent-ligo-virgo-gravitational-wave-measurements-narrowed-down-the-sp)  
> **username_0**: @fasterthanlight the question is about the speed of gravitational waves, mine is about the depiction used when discussing Einstein's theory of gravity.  
> **username_3**: I’m voting to close this question because this doesn't have much to do with astronomy is probably a better fit for the physics stackexchange (or maybe a physics educators stack exchange if there was one?)  
> **username_4**: @username_3 great idea,I think Physics SE has gotten [too large](https://physics.meta.stackexchange.com/a/12734/83380) and tries to serves several disparate communities and functions at the same time.  

---

## 2 answers

### Answer by username_5 on 2020-11-09. Score: 2 | Accepted answer

There are numerous ways to try to explain space-time by analogy, and the flat plane is one of them. Another is the inflating balloon. The are all only analogies. Either of your pictures above is equally valid, however the one "the right way up" is simpler to understand.

For humans, who can easily conceive of 3 dimensional space, an example that works in 3D is essential. If you have ever tried to visualise more than 3 space dimensions, you'll find it very tricky.

> **Comments:**
> **username_0**: Thanks for your answer. But going about it that way has certain implications on how you think about it. And I haven't heard anybody address these issues.  
> **username_5**: What issues? Any analogy is by its very nature limited, and they are only guides to help you think. @PM2Ring's links on Physics.SE will be very useful for you.  

---

### Answer by username_6 on 2020-11-09. Score: 2

The rubber sheet is a good model of *Newtonian* gravity. It doesn't work at all as a model of curved spacetime in general relativity.

---

Newtonian gravity can be described by a [potential field](https://en.wikipedia.org/wiki/Gravitational_potential) which has a numeric value at every point in space. Objects experience an acceleration that is equal to minus the gradient of the field. That is, they accelerate in the direction that the field around them is decreasing most rapidly, and the magnitude of the acceleration is proportional to the slope in that direction. Also, the field satisfies [Poisson's equation](https://en.wikipedia.org/wiki/Poisson%27s_equation#Newtonian_gravity), which allows you to find its value at every point if you know the distribution of matter.

Objects on a hill on Earth accelerate downhill, and (subject to idealizing assumptions) the acceleration is proportional to the slope. Not only that, but the deformation of a horizontal elastic sheet in a uniform gravitational field with heavy objects resting on it is described (approximately) by Poisson's equation. The lower the mass density of the sheet, the vertical stretching, and the friction, the better this toy model approximates Newtonian gravity.

One big caveat is that a 2D rubber sheet is described by the 2D Poisson equation, which has different solutions than the 3D Poisson equation. With a 2D sheet you get 2D Newtonian gravity, which has a 1/r force law instead of 1/r². So in this sense it is far from correct. This problem only affects the elastic sheet part, not the rolling-downhill part. If you construct a rigid surface with the correct shape from the 3D Poisson equation, like the "gravity wells" found in some science museums, you get a pretty accurate simulation of test particles in a 1/r² gravitational field.

---

In general relativity, the motion of objects depends on the intrinsic shape of spacetime.

You can embed hunks of curved space into Euclidean space. Like the rubber sheet analogy this is just a way of understanding them in terms of something we're more familiar with (in this case, space that isn't curved).

The shape of these embeddings is not correctly given by Poisson's equation. The embeddings also don't have any "down" direction. An ant crawling along the surface (which is a much better analogy to the effect of curved spacetime on test particles) doesn't care which way is down; it will follow the same path even if there is no background field at all.

It would make sense to always show GR embeddings as "hills" instead of "valleys", since it makes no difference to the physics and it would avoid unnecessary confusion with gravity wells. But not only is it not standard to do that, it's almost unheard of in works for popular audiences. Quite often the surfaces presented as GR embeddings look like solutions to Poisson's equation, too, which they shouldn't. Either the authors want to invite confusion, or they don't understand the difference themselves. I suspect the latter.


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