<p>When talking about curvature in (General) Relativity one usually refers to the metric interval and how this depends on the space-time coordinates.</p>

<p>For example, the metric interval in Minkowski space can be written
$$ ds^2 = c^2\ dt^2 - dr^2$$</p>

<p>This is an example of both flat space and flat space-time. It is flat space because if I want to find the distance between two events happening at the same time $t$, it is simply the integral of $dr$.</p>

<p>Similarly I can ask what the time interval is between two events and this is just given by the integral of $dt$.</p>

<p>A counterexample is seen when using the Schwarzschild metric, which is appropriate in the space outside a spherically symmetric mass. Considering only the radial and time components, the metric is
$$ds^2 = c^2 (1 - r_s/r)\ dt^2 - (1- r_s/r)^{-1}\ dr^2,$$
where $r_s = 2GM/c^2$ is the Schwarzschild radius associated with the mass $M$.</p>

<p>Now there is no simple relationship between $dr$ and the spatial separation between two events that occur at the same $t$, we must do an integral. Similarly, to work out the time interval between two events at the same $r$ we must do a more complicated integral. Both space and space-time are curved.</p>

<p>The evolution of the universe as a whole is governed by the <a href="https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" rel="nofollow noreferrer">Robertson-Walker metric</a>.
$$ds^2 = c^2\ dt^2 - a(t)^2\frac{dr^2}{1 - kr^2},$$
where $k$ is a curvature parameter that may be -1, +1 or zero.</p>

<p>At the moment, the best measurements are consistent with a zero curvature - that the universe is spatially flat. That does not mean it is flat on much large scales than we can observe - that is still possible. A curved universe that was exponentially "inflated" shortly after the big bang may well mimic a flat universe even though the curvature is finite.</p>

<p>Note though that because the universe is inherently "lumpy", this spatial flatness is only true on scales much larger than planets, stars, galaxies and even clusters of galaxies.</p>

<p>Whether you could call space-time flat in a spatially flat Robertson-Walker universe I doubt. The reason for this is the time-dependent scale factor $a(t)$, which ensures there is a coupling between the $r$ and $t$ coordinates. It is this coupling that is responsible, for example, for the cosmological redshift of light. </p>

<p>A technical answer would need to look at the <a href="https://en.wikipedia.org/wiki/Einstein_tensor" rel="nofollow noreferrer">Einstein Tensor</a> which characterises the curvature of space-time caused by mass and energy. In the Robertson-Walker metric, this tensor definitely has non-zero components, so you would have to say that even if space is flat, space-time is curved.</p>

CC BY-SA 4.0
2018-05-18T16:11:26.223Z
2018-05-18T16:11:26.223Z
ProfRob
<blockquote>
  <p>Is the shape of observable universe and shape of space same?</p>
</blockquote>

<p>Nobody knows for sure. I imagine the answer is <em>yes</em>, and they're both spherical, but I don't actually know this.  </p>

<blockquote>
  <p>I have read that Euclidean space is considered as flat space and Minkowski space is flat space-time. So when we say our observable universe is flat are we saying that the space is flat or space-time is flat? </p>
</blockquote>

<p>Space. Space-time is not flat because the universe is expanding. To get a handle on this imagine you dob a big glob of treacle down, and spray a (non-radial) line of cream across it. As the glob of treacle flattens and expands the line of cream will tend to curve. It's similar for a light beam in an expanding universe. If the universe wasn't expanding, the light would go straight. Then space-time would be flat. Note that space-time models space at all times, so there's no motion through space-time. The treacle represents space. Space is expanding, not space-time. Unfortunately people tend to get confused between space and space-time.          </p>

<blockquote>
  <p>I know the Planck data has proved the space to be flat but is it only space or space-time? </p>
</blockquote>

<p>It's space. The paper concerned is <a href="https://arxiv.org/abs/1303.5086" rel="nofollow noreferrer">Planck 2013 results. XXVI. Background geometry and topology of the Universe</a>. They found no evidence of any "higher dimensional curvature". So if you could go <em>thataway</em> at some phenomenal speed, you won't end up coming back <em>thisaway</em>.    </p>

<blockquote>
  <p>Since in general relativity matter curves space-time and not just space.</p>
</blockquote>

<p>See what <a href="http://www-history.mcs.st-andrews.ac.uk/Extras/Einstein_ether.html" rel="nofollow noreferrer">Einstein said</a>. A concentration of energy usually in the guise of a massive star "conditions" the surrounding space, rendering it inhomogeneous in a non-linear fashion. Then people call it curved spacetime. Note however that there's no overall gravity in the universe. As userLTK said  the surface of the Earth is "lumpy", and in similar vein there are  gravitational fields all over the universe. But to the best of our knowledge the universe is not like the surface of the Earth. On the largest scale, to the best of our knowledge, the universe is flat. </p>

CC BY-SA 3.0
2018-04-18T13:31:07.833Z
2018-04-18T13:37:25.947Z
2018-04-18T13:37:25.947Z
John Duffield
<p>Space time is curved by gravity and there's gravity everywhere in the observable universe and perhaps everywhere in the (entire) universe or just "universe" is the proper term.   So, pretty much everywhere, there should be some local space-time curvature.   </p>

<p>As an analogy, the surface of the Earth is lumpy.  It's got mountains and valleys and high and low regions everywhere on it's surface.  No area of the Earth is completely smooth and perfectly zero sea-level, but despite that lumpiness, the Earth is round (an oblate spheroid but round is close enough).   </p>

<p>Earth isn't a perfect example, but the Earth's surface is a representation of a two dimensional surface wrapped around a 3 dimensional spheroid and Earth's curvature is measurable from it's surface.   One method of doing this is to draw a triangle and measure the sum of the angles and the variation from 180 degrees relative to the size of the triangle gives you what you need to calculate the curvature.  Using a different and more practical method, Eratosthenes measured <a href="https://en.wikipedia.org/wiki/Eratosthenes#Measurement_of_the_Earth&#39;s_circumference" rel="nofollow noreferrer">Earth's curvature and size over 2000 years ago</a>.  Eratosthenes's two measurements were taken about 500 miles apart was far enough to reduce any variation in elevation to a small error.</p>

<p>The Earth is about a thousand times as far across as the tallest mountain or deepest valley and it's got some gravitational variation, but it's still generally spheroid, both in gravitation and in shape.  The Universe is incalculably large compared to the local ripples and curves caused by gravity and the Universe, as far as we can tell, appears quite uniform, so the local pockets of high gravity and local curvature or quite irrelevant to the overall shape or potential curvature of the Universe, just as the mountains and valleys on Earth are irrelevant to it still being an oblate spheroid.</p>

<p>The Universe may not be flat, it just appears to be flat to our best measurements.   It might be curved, but so large, modern measuring techniques aren't able to measure the curvature.   This raises questions as to what a curved universe actually means (distinct from curved space-time that's easily measurable).   That can make some fun reading, but it's all theoretical.  Nobody knows the true shape or curvature of the universe though the universe being infinite and flat appears to be the more accepted idea today, but it's still an unanswerable question. </p>

CC BY-SA 3.0
2018-03-18T10:52:07.537Z
Space time is curved. Space here is flat by we don't fly away.
Alchimista
CC BY-SA 3.0
2018-03-18T20:52:14.720Z
I don't follow your point, @Alchimista and I'm open to correction and willing to delete if I got this wrong.   I thought 3D space was curved by gravity and space-time is just a 4 dimensional representation (usually drawn on two dimensions for ease of representation).
userLTK
CC BY-SA 3.0
2018-03-18T21:01:53.170Z
I am not a specialist but space is right euclidean even around a big mass as, say, Earth. You can't say space OR space time. This is the same error popular magazine do when saying mass curves the space around while forgetting -time. Space is flat and even more so locally. Even more universally, even if the global curvature differ from zero (at least it seems very close to it) at local scale space will stay flat. It is why we can use euclidean geometry not only to build geometrical shape on earth, but also to measure astronomical distances and sizes.
Alchimista
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2018-03-19T08:59:20.503Z
@Alchimista fair enough.  I removed all the lines saying curved space.
userLTK
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2018-03-19T09:12:39.220Z
Let me add to the end of my comment: ....but also to measure astronomical distances and sizes at least within and near our galaxy.
Alchimista
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2018-05-18T15:33:36.560Z
@Alchimista is incorrect. For example the space defined by the Schwarzschild metric is curved. The circumference around a spherically symmetric gravitating mass is not $2\pi r$.
ProfRob
CC BY-SA 4.0
2018-05-18T15:38:56.660Z
You need to define what you mean by curvature.
ProfRob
CC BY-SA 4.0
2018-05-18T15:39:20.980Z
@Rob Jeffries. Do you mean that observable universe and the probably larger space as in the Q shall be described by the Schwarzschild metric?  Is to a sort of global shape that OP ask for . Not the geometry around a black hole.
Alchimista
CC BY-SA 4.0
2018-05-18T15:46:27.553Z
@Alchimista As I'm sure you know the evolution of the universe if assumed to be governed by the Robertson-Walker metric. This has a spatial curvature term in it which appears to be close-to-zero. It was you who claimed space was Euclidean  "even around a big mass", which I am pointing out is incorrect. There is also a clear difference in a metric between flat space and flat space-time.
ProfRob
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2018-05-18T15:49:05.077Z
Is exactly what I meant.  Not to confuse space with space time.
Alchimista
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2018-03-18T00:06:11.343Z
2018-03-19T08:58:02.303Z
2018-03-19T08:58:02.303Z
userLTK
<p>I have read that Euclidean space is considered as flat space and Minkowski space is flat space-time. So when we say our observable universe is flat are we saying that the space is flat or space-time is flat? I know the Planck data has proved the space to be flat but is it only space or space-time? Since in general relativity matter curves space-time and not just space.</p>

CC BY-SA 3.0
2018-03-16T17:04:34.907Z
space is spacetime.
A. C. A. C.
CC BY-SA 3.0
2018-03-17T14:10:08.950Z
So you are saying that the space-time is flat for observable universe? Because it would be weird if mass causes space-time to curve as a local curvature but the large scale curvature is only for space!
shivani
CC BY-SA 3.0
2018-03-18T10:51:09.213Z
Space is not spacetime. Curvature and shape are a bit confusing the Q tough. As we know observable universe is a sphere. How the curvature of its interior could modify this spherical shape complicates the situation.  I suggest is the curvature of the observable U the same as that of U. Obviously we don't know. Also the question in the body  (what's flat/curved ? Just space or space time?) not the same as in the title
Alchimista
CC BY-SA 4.0
2018-05-18T15:38:55.357Z
If there aren't stars, or at least nebula, how would we tell?
Wayfaring Stranger
<space-time><space><observable-universe>
Is the shape of observable universe and shape of space same?
shivani
# Is the shape of observable universe and shape of space same?

(Asked by: username_0 on 2018-03-16)

I have read that Euclidean space is considered as flat space and Minkowski space is flat space-time. So when we say our observable universe is flat are we saying that the space is flat or space-time is flat? I know the Planck data has proved the space to be flat but is it only space or space-time? Since in general relativity matter curves space-time and not just space.

**Tags:** space-time, space, observable-universe
> **Comments:**
> **username_1**: space is spacetime.  
> **username_0**: So you are saying that the space-time is flat for observable universe? Because it would be weird if mass causes space-time to curve as a local curvature but the large scale curvature is only for space!  
> **username_2**: Space is not spacetime. Curvature and shape are a bit confusing the Q tough. As we know observable universe is a sphere. How the curvature of its interior could modify this spherical shape complicates the situation.  I suggest is the curvature of the observable U the same as that of U. Obviously we don't know. Also the question in the body  (what's flat/curved ? Just space or space time?) not the same as in the title  
> **username_3**: If there aren't stars, or at least nebula, how would we tell?  

---

## 3 answers

### Answer by username_4 on 2018-05-18. Score: 3

When talking about curvature in (General) Relativity one usually refers to the metric interval and how this depends on the space-time coordinates.

For example, the metric interval in Minkowski space can be written
$$ ds^2 = c^2\ dt^2 - dr^2$$

This is an example of both flat space and flat space-time. It is flat space because if I want to find the distance between two events happening at the same time $t$, it is simply the integral of $dr$.

Similarly I can ask what the time interval is between two events and this is just given by the integral of $dt$.

A counterexample is seen when using the Schwarzschild metric, which is appropriate in the space outside a spherically symmetric mass. Considering only the radial and time components, the metric is
$$ds^2 = c^2 (1 - r\_s/r)\ dt^2 - (1- r\_s/r)^{-1}\ dr^2,$$
where $r\_s = 2GM/c^2$ is the Schwarzschild radius associated with the mass $M$.

Now there is no simple relationship between $dr$ and the spatial separation between two events that occur at the same $t$, we must do an integral. Similarly, to work out the time interval between two events at the same $r$ we must do a more complicated integral. Both space and space-time are curved.

The evolution of the universe as a whole is governed by the [Robertson-Walker metric](https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric).
$$ds^2 = c^2\ dt^2 - a(t)^2\frac{dr^2}{1 - kr^2},$$
where $k$ is a curvature parameter that may be -1, +1 or zero.

At the moment, the best measurements are consistent with a zero curvature - that the universe is spatially flat. That does not mean it is flat on much large scales than we can observe - that is still possible. A curved universe that was exponentially "inflated" shortly after the big bang may well mimic a flat universe even though the curvature is finite.

Note though that because the universe is inherently "lumpy", this spatial flatness is only true on scales much larger than planets, stars, galaxies and even clusters of galaxies.

Whether you could call space-time flat in a spatially flat Robertson-Walker universe I doubt. The reason for this is the time-dependent scale factor $a(t)$, which ensures there is a coupling between the $r$ and $t$ coordinates. It is this coupling that is responsible, for example, for the cosmological redshift of light.

A technical answer would need to look at the [Einstein Tensor](https://en.wikipedia.org/wiki/Einstein_tensor) which characterises the curvature of space-time caused by mass and energy. In the Robertson-Walker metric, this tensor definitely has non-zero components, so you would have to say that even if space is flat, space-time is curved.


---

### Answer by username_5 on 2018-04-18. Score: 1

> Is the shape of observable universe and shape of space same?

Nobody knows for sure. I imagine the answer is *yes*, and they're both spherical, but I don't actually know this.

> I have read that Euclidean space is considered as flat space and Minkowski space is flat space-time. So when we say our observable universe is flat are we saying that the space is flat or space-time is flat?

Space. Space-time is not flat because the universe is expanding. To get a handle on this imagine you dob a big glob of treacle down, and spray a (non-radial) line of cream across it. As the glob of treacle flattens and expands the line of cream will tend to curve. It's similar for a light beam in an expanding universe. If the universe wasn't expanding, the light would go straight. Then space-time would be flat. Note that space-time models space at all times, so there's no motion through space-time. The treacle represents space. Space is expanding, not space-time. Unfortunately people tend to get confused between space and space-time.

> I know the Planck data has proved the space to be flat but is it only space or space-time?

It's space. The paper concerned is [Planck 2013 results. XXVI. Background geometry and topology of the Universe](https://arxiv.org/abs/1303.5086). They found no evidence of any "higher dimensional curvature". So if you could go *thataway* at some phenomenal speed, you won't end up coming back *thisaway*.

> Since in general relativity matter curves space-time and not just space.

See what [Einstein said](http://www-history.mcs.st-andrews.ac.uk/Extras/Einstein_ether.html). A concentration of energy usually in the guise of a massive star "conditions" the surrounding space, rendering it inhomogeneous in a non-linear fashion. Then people call it curved spacetime. Note however that there's no overall gravity in the universe. As userLTK said the surface of the Earth is "lumpy", and in similar vein there are gravitational fields all over the universe. But to the best of our knowledge the universe is not like the surface of the Earth. On the largest scale, to the best of our knowledge, the universe is flat.


---

### Answer by username_6 on 2018-03-18. Score: 0

Space time is curved by gravity and there's gravity everywhere in the observable universe and perhaps everywhere in the (entire) universe or just "universe" is the proper term. So, pretty much everywhere, there should be some local space-time curvature.

As an analogy, the surface of the Earth is lumpy. It's got mountains and valleys and high and low regions everywhere on it's surface. No area of the Earth is completely smooth and perfectly zero sea-level, but despite that lumpiness, the Earth is round (an oblate spheroid but round is close enough).

Earth isn't a perfect example, but the Earth's surface is a representation of a two dimensional surface wrapped around a 3 dimensional spheroid and Earth's curvature is measurable from it's surface. One method of doing this is to draw a triangle and measure the sum of the angles and the variation from 180 degrees relative to the size of the triangle gives you what you need to calculate the curvature. Using a different and more practical method, Eratosthenes measured [Earth's curvature and size over 2000 years ago](https://en.wikipedia.org/wiki/Eratosthenes#Measurement_of_the_Earth's_circumference). Eratosthenes's two measurements were taken about 500 miles apart was far enough to reduce any variation in elevation to a small error.

The Earth is about a thousand times as far across as the tallest mountain or deepest valley and it's got some gravitational variation, but it's still generally spheroid, both in gravitation and in shape. The Universe is incalculably large compared to the local ripples and curves caused by gravity and the Universe, as far as we can tell, appears quite uniform, so the local pockets of high gravity and local curvature or quite irrelevant to the overall shape or potential curvature of the Universe, just as the mountains and valleys on Earth are irrelevant to it still being an oblate spheroid.

The Universe may not be flat, it just appears to be flat to our best measurements. It might be curved, but so large, modern measuring techniques aren't able to measure the curvature. This raises questions as to what a curved universe actually means (distinct from curved space-time that's easily measurable). That can make some fun reading, but it's all theoretical. Nobody knows the true shape or curvature of the universe though the universe being infinite and flat appears to be the more accepted idea today, but it's still an unanswerable question.

> **Comments:**
> **username_2**: Space time is curved. Space here is flat by we don't fly away.  
> **username_6**: I don't follow your point, @Alchimista and I'm open to correction and willing to delete if I got this wrong.   I thought 3D space was curved by gravity and space-time is just a 4 dimensional representation (usually drawn on two dimensions for ease of representation).  
> **username_2**: I am not a specialist but space is right euclidean even around a big mass as, say, Earth. You can't say space OR space time. This is the same error popular magazine do when saying mass curves the space around while forgetting -time. Space is flat and even more so locally. Even more universally, even if the global curvature differ from zero (at least it seems very close to it) at local scale space will stay flat. It is why we can use euclidean geometry not only to build geometrical shape on earth, but also to measure astronomical distances and sizes.  
> **username_6**: @Alchimista fair enough.  I removed all the lines saying curved space.  
> **username_2**: Let me add to the end of my comment: ....but also to measure astronomical distances and sizes at least within and near our galaxy.  
> **username_4**: @Alchimista is incorrect. For example the space defined by the Schwarzschild metric is curved. The circumference around a spherically symmetric gravitating mass is not $2\pi r$.  
> **username_4**: You need to define what you mean by curvature.  
> **username_2**: @Rob Jeffries. Do you mean that observable universe and the probably larger space as in the Q shall be described by the Schwarzschild metric?  Is to a sort of global shape that OP ask for . Not the geometry around a black hole.  
> **username_4**: @Alchimista As I'm sure you know the evolution of the universe if assumed to be governed by the Robertson-Walker metric. This has a spatial curvature term in it which appears to be close-to-zero. It was you who claimed space was Euclidean  "even around a big mass", which I am pointing out is incorrect. There is also a clear difference in a metric between flat space and flat space-time.  
> **username_2**: Is exactly what I meant.  Not to confuse space with space time.  

---

