<p>&quot;Mass must curve the universe but the pressure due to energy density may curve the universe in the opposite direction, leading to a flat universe&quot;</p>
<p>This is not correct. What produces curvature is the total energy density of the universe (adding up the contributions from mass, radiation, <em>and</em> dark energy) relative to the <a href="https://en.wikipedia.org/wiki/Friedmann_equations#Density_parameter" rel="nofollow noreferrer">critical density</a>. If the total is <span class="math-container">$&lt;$</span> the critical density, then the curvature is negative; if it is <span class="math-container">$&gt;$</span> than the critical density, then the curvature is positive (and if it is <em>equal to</em> the critical density, then the curvature is zero and the universe is flat).</p>
<p><em>Expansion</em> is a completely separate issue. Mass (and radiation) will slow down the expansion, while dark energy will increase it, regardless of what the curvature is.</p>

CC BY-SA 4.0
2021-06-08T08:16:14.180Z
Dark energy has the property to give negative curvature (while normal energy gives positive curvature. When dark energy supplants mass energy in the universe the overall structure of spacetime will have negative curvature (as in the early universe). That's the weird thing about dark energy. It works oppositely to normal energy. It's connected with negative mass which makes objects repel from each other (tese masses give spacetime a negative curvature, related to expansion).
Deschele Schilder
CC BY-SA 4.0
2021-06-08T09:17:56.950Z
@Barbierium you take expansion as dictated by curvature, but it is not like that in the Friedman equation,which has a cosmology constant
Alchimista
CC BY-SA 4.0
2021-06-08T09:32:59.747Z
@Alchimista A negatively curved spacetime can only expand. Negative pressure (caused by negative, or imaginary energy, or negative mass) causes expansion. As in the early universe (the inflaton field contained a lot of these hypothetical particles). It cannot exist stationary. So, expanding spacetime has a negative curvature (what else would be the difference with a flat spacetime).
Deschele Schilder
CC BY-SA 4.0
2021-06-08T09:45:54.070Z
@Barbierium It's possible you are confusing *spacetime* curvature with *spatial* curvature; it's the latter that is being asked about and is what is meant by "curvature" in cosmological discussions. The spatial curvature is determined by the energy density, and the energy density of dark energy is positive (otherwise, it cannot provide acceleration), so it provides the *same* kind of contribution to the curvature as ordinary matter. See, e.g., https://physics.stackexchange.com/questions/96551/why-does-dark-energy-produce-positive-space-time-curvature
Peter Erwin
CC BY-SA 4.0
2021-06-08T10:52:05.313Z
@PeterErwin But the acceleration caused by dark energy is contrary to that of normal energy. Like negative mass causes acceleration also, but oppositely directed to that of normal masses. You can say that this acceleration away from each other is caused by an expanding universe, and indeed it expands. But only negatively curved space can expand.
Deschele Schilder
CC BY-SA 4.0
2021-06-08T11:08:55.780Z
@PeterErwin can you succinctly tell me if when saying Sun curve the space near it it is meant that the space is curved? Or spacetime? The latter it what I think but in this site I was recently been I am wrong.
Alchimista
CC BY-SA 4.0
2021-06-08T11:39:57.133Z
@Alchimista Most of the curvature of spacetime near the Sun is in the time direction, but there *is* some curvature in the purely spacelike geometry (i.e., the spatial geometry is not Euclidean). See here: https://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_massive/index.html
Peter Erwin
CC BY-SA 4.0
2021-06-08T11:41:32.453Z
@Barbierium "Only negatively curved space can expand" -- this is, simply, wrong, both as a matter of GR and observationally (i.e., our universe is expanding, and is spatially flat).
Peter Erwin
CC BY-SA 4.0
2021-06-08T12:26:59.217Z
@PeterErwin Well, I meant negatively curved space*time*. Sorry for that. If it expands spacetime simply can't be flat. The observations seem to imply that space is flat but this as seen locally. Who knows what's going on behind the horizon? It can look flat everywhere, but this doesn't tell you anything about the global structure. So an *observed* flat space can go hand in hand with a global negatively curved spacetime. *If* spacetime expands than it has to have negative curvature overall. So it can locally be described as flat expanding space.
Deschele Schilder
CC BY-SA 4.0
2021-06-08T13:47:51.430Z
@PeterErwin ok thanks, I was understanding that curving light rays is possible in an Euclidean space too....
Alchimista
CC BY-SA 4.0
2021-06-07T21:31:55.273Z
2021-06-07T21:31:55.273Z
Peter Erwin
<p>Mass curves spacetime and so does energy density. They both find their place in the time (energy) component of the stress-energy tensor, which functions as the source of the gravity field (like charge for the electromagnetic field). So positive mass and energy curve spacetime in the same way.<br />
The only way to give the universe a negative curvature is therefore to introduce negative energy (dark energy). This makes mass-energy move away from each other. In the inflation era there was supposed to be an inflaton field which made spacetime grow. The field is thought to consist of particles with imaginary mass or imaginary momentum, because <span class="math-container">$E^2=m^2c^4+p^2$</span>. If the particles are massless then <span class="math-container">$E^2=p^2$</span> so the equality holds if the particles have negative momentum.<br />
If you consider these particles to have negative mass and negative momentum <span class="math-container">$E^2$</span> will be positive, but this is just as imaginary as imaginary mass and momentum.<br />
The same argument holds for dark matter although this has not a comparable energy as the energy contained in the inflaton field.</p>
<p>So the curvature of spacetime can be negative while there is positive mass-energy in it. Positive mass-energy curvature is canceled by negative mass-energy curvature. Positive pressure can be compensated by negative pressure. The state of the early universe was one of hugely negatively curved spacetime. When real matter appeared this curvature was inversed in a positive curvature, but in the future, the underlying negative curvature will reappear (as it is already now because of the diminishing mass-energy density) and drive matter apart again. Due to the strong negatively curvature that will come to be in the future, there will be a new possibility for virtual particle fields to become real again, so in the future a new big bang will happen in our spacetime (this implies a universe that is infinite in space as well in time).</p>

CC BY-SA 4.0
2021-06-07T21:23:00.043Z
Dark energy is *not* "negative energy" -- it is positive energy density with negative *pressure*. *All* forms of energy -- normal matter, radiation, dark energy -- contribute in the same way to the curvature of the universe. The way to get negative curvature is to have the total energy density (of all kinds) be less than the critical density.
Peter Erwin
CC BY-SA 4.0
2021-06-07T21:24:58.297Z
There's a lot of confusion and error in this answer: e.g., "imaginary mass", "imaginary momentum", "negative mass", "positive mass-energy is canceled by negative mass-energy", etc.
Peter Erwin
CC BY-SA 4.0
2021-06-07T21:31:40.563Z
@PeterErwin The problem is how to make  $E^2=m^2 c^4+p^2$ hold if the pressure is negative. Imaginary mass is just another way of saying that it's negative, but the conclusion for energy will be the opposite.
Deschele Schilder
CC BY-SA 4.0
2021-06-07T21:44:05.467Z
I think you're trying to treat dark energy as though it's made up of individual particles, which is probably not a good idea.
Peter Erwin
CC BY-SA 4.0
2021-06-07T21:50:10.393Z
@PeterErwin Isn't the inflaton field a particle field? Or the dark energy field (virtual particles). Or is it spacetime itself that contains the energy (as in quantum loop gravity)?
Deschele Schilder
CC BY-SA 4.0
2021-06-07T21:53:27.677Z
@PeterErwin When positive mass-energy is canceled by negative mass-energy then curvature (globally) can be zero or negative.
Deschele Schilder
CC BY-SA 4.0
2021-06-07T21:55:56.770Z
Dark energy has the *same* basic effect on curvature as normal matter; there is no "canceling out".
Peter Erwin
CC BY-SA 4.0
2021-06-08T21:26:51.177Z
@PeterErwin Doesn't dark energy make the curvature negative (as it in an expanding spacetime has to be)? If not then what's the difference with normal energy? You can say negative pressure, but what causes negative pressure? How can the pressure be negative and the energy positive (sorry for "pushing" a bit, but I just want to know). :)
Deschele Schilder
CC BY-SA 4.0
2021-06-08T22:48:43.280Z
w in the equation of state for energy is usually negative. This leads to a negative pressure and so a negative force I presume - negative because the force due to gravity is positive. I think the force due to pressure (from energy density) counters the force of gravity.
John Hobson
CC BY-SA 4.0
2021-06-08T22:55:20.577Z
@JohnHobson Is w potential energy?
Deschele Schilder
CC BY-SA 4.0
2021-06-15T09:13:51.293Z
Pressure x volume is potential energy. w is dimensionless because pressure and energy density have the same units. There is a lot of talk here about positive or negative energy but surely the important question is whether the resultant pressure is positive or negative and that also depends on the sign of w in the equation of state. If dark energy is positive and w is negative , we get a negative pressure and force. Negative because it is opposite to the force of gravity. I don't see why dark energy has to be negative to counter the force of gravity.
John Hobson
CC BY-SA 4.0
2021-06-07T13:57:46.260Z
2021-06-09T13:30:54.827Z
2021-06-09T13:30:54.827Z
Deschele Schilder
<p>I may know the answer to my question. Mass must curve the universe but the pressure due to energy density may curve the universe in the opposite direction, leading to a flat universe, if they exactly cancel. But this leads to a supplementary question.</p>
<p>It seems to me that in a universe with positive curvature, gravity is a force encouraging the contraction of the universe while the pressure due to the density of energy is a force encouraging the expansion of the universe. But in a universe with negative curvature, the opposite should apply. Gravity should encourage the expansion of the universe while pressure should encourage its contraction. And in a flat universe neither gravity nor pressure should have any effect on the expansion of the universe. So if our universe is flat, dark energy can not be encouraging accelerating expansion. Any thoughts?</p>

CC BY-SA 4.0
2021-06-07T14:21:29.870Z
Gravity $\neq$ force in General Relativity.
ProfRob
CC BY-SA 4.0
2021-06-08T08:18:06.060Z
@tfb If space is flat time is also flat. So spacetime is also flat.
Deschele Schilder
CC BY-SA 4.0
2021-06-08T09:01:38.937Z
@tfb Why is this simply false?
Deschele Schilder
CC BY-SA 4.0
2021-06-08T09:11:58.313Z
It's not known if the universe is flat (globally). There is evidence for dark energy, which leads to an accelerated expansion (if this dark energy was less than the amount of mass-energy, spacetime would stop expanding). This means that spacetime has a negative overall curvature. This kind of overall curvature is one that has an expanding structure. Any two pieces of mass energy in this spacetime will move away from each other (if their mutual gravity is overcome).
Deschele Schilder
CC BY-SA 4.0
2021-06-08T12:15:58.553Z
@tfb I'm not sure if I get the metric you have given. What is $d^2$? The point is though that a spacetime with the metric you describe is flat too (this is not how real spacetime developed though). So if a null geodesic is reversed in this (at al times) flat spacetime (with the metric above) they will all meet. So even in flat spacetime al time- and null-like reversed geodesics can meet (you will see that the curvature tensor for this spacetime is zero).
Deschele Schilder
CC BY-SA 4.0
2021-06-08T14:43:33.207Z
@tfbThat's exactly what I meant. Thanks. The confusion was indeed due to conflating space and spacetime... The RW metric has indeed negative curvature. Because of which the spacetime expands. This negative curvature is caused by negative pressure (or negative mass). Everything clear now!:)
Deschele Schilder
<cosmology><expansion><dark-energy>
How is it possible that a universe with mass does not have curvature?
John Hobson
# How is it possible that a universe with mass does not have curvature?

(Asked by: username_0 on 2021-06-06)

I may know the answer to my question. Mass must curve the universe but the pressure due to energy density may curve the universe in the opposite direction, leading to a flat universe, if they exactly cancel. But this leads to a supplementary question.

It seems to me that in a universe with positive curvature, gravity is a force encouraging the contraction of the universe while the pressure due to the density of energy is a force encouraging the expansion of the universe. But in a universe with negative curvature, the opposite should apply. Gravity should encourage the expansion of the universe while pressure should encourage its contraction. And in a flat universe neither gravity nor pressure should have any effect on the expansion of the universe. So if our universe is flat, dark energy can not be encouraging accelerating expansion. Any thoughts?

**Tags:** cosmology, expansion, dark-energy
> **Comments:**
> **username_1**: Gravity $\neq$ force in General Relativity.  
> **username_2**: @tfb If space is flat time is also flat. So spacetime is also flat.  
> **username_2**: @tfb Why is this simply false?  
> **username_2**: It's not known if the universe is flat (globally). There is evidence for dark energy, which leads to an accelerated expansion (if this dark energy was less than the amount of mass-energy, spacetime would stop expanding). This means that spacetime has a negative overall curvature. This kind of overall curvature is one that has an expanding structure. Any two pieces of mass energy in this spacetime will move away from each other (if their mutual gravity is overcome).  
> **username_2**: @tfb I'm not sure if I get the metric you have given. What is $d^2$? The point is though that a spacetime with the metric you describe is flat too (this is not how real spacetime developed though). So if a null geodesic is reversed in this (at al times) flat spacetime (with the metric above) they will all meet. So even in flat spacetime al time- and null-like reversed geodesics can meet (you will see that the curvature tensor for this spacetime is zero).  
> **username_2**: @tfbThat's exactly what I meant. Thanks. The confusion was indeed due to conflating space and spacetime... The RW metric has indeed negative curvature. Because of which the spacetime expands. This negative curvature is caused by negative pressure (or negative mass). Everything clear now!:)  

---

## 2 answers

### Answer by username_3 on 2021-06-07. Score: 3

"Mass must curve the universe but the pressure due to energy density may curve the universe in the opposite direction, leading to a flat universe"

This is not correct. What produces curvature is the total energy density of the universe (adding up the contributions from mass, radiation, *and* dark energy) relative to the [critical density](https://en.wikipedia.org/wiki/Friedmann_equations#Density_parameter). If the total is $<$ the critical density, then the curvature is negative; if it is $>$ than the critical density, then the curvature is positive (and if it is *equal to* the critical density, then the curvature is zero and the universe is flat).

*Expansion* is a completely separate issue. Mass (and radiation) will slow down the expansion, while dark energy will increase it, regardless of what the curvature is.

> **Comments:**
> **username_2**: Dark energy has the property to give negative curvature (while normal energy gives positive curvature. When dark energy supplants mass energy in the universe the overall structure of spacetime will have negative curvature (as in the early universe). That's the weird thing about dark energy. It works oppositely to normal energy. It's connected with negative mass which makes objects repel from each other (tese masses give spacetime a negative curvature, related to expansion).  
> **username_4**: @Barbierium you take expansion as dictated by curvature, but it is not like that in the Friedman equation,which has a cosmology constant  
> **username_2**: @Alchimista A negatively curved spacetime can only expand. Negative pressure (caused by negative, or imaginary energy, or negative mass) causes expansion. As in the early universe (the inflaton field contained a lot of these hypothetical particles). It cannot exist stationary. So, expanding spacetime has a negative curvature (what else would be the difference with a flat spacetime).  
> **username_3**: @Barbierium It's possible you are confusing *spacetime* curvature with *spatial* curvature; it's the latter that is being asked about and is what is meant by "curvature" in cosmological discussions. The spatial curvature is determined by the energy density, and the energy density of dark energy is positive (otherwise, it cannot provide acceleration), so it provides the *same* kind of contribution to the curvature as ordinary matter. See, e.g., https://physics.stackexchange.com/questions/96551/why-does-dark-energy-produce-positive-space-time-curvature  
> **username_2**: @PeterErwin But the acceleration caused by dark energy is contrary to that of normal energy. Like negative mass causes acceleration also, but oppositely directed to that of normal masses. You can say that this acceleration away from each other is caused by an expanding universe, and indeed it expands. But only negatively curved space can expand.  
> **username_4**: @PeterErwin can you succinctly tell me if when saying Sun curve the space near it it is meant that the space is curved? Or spacetime? The latter it what I think but in this site I was recently been I am wrong.  
> **username_3**: @Alchimista Most of the curvature of spacetime near the Sun is in the time direction, but there *is* some curvature in the purely spacelike geometry (i.e., the spatial geometry is not Euclidean). See here: https://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_massive/index.html  
> **username_3**: @Barbierium "Only negatively curved space can expand" -- this is, simply, wrong, both as a matter of GR and observationally (i.e., our universe is expanding, and is spatially flat).  
> **username_2**: @PeterErwin Well, I meant negatively curved space*time*. Sorry for that. If it expands spacetime simply can't be flat. The observations seem to imply that space is flat but this as seen locally. Who knows what's going on behind the horizon? It can look flat everywhere, but this doesn't tell you anything about the global structure. So an *observed* flat space can go hand in hand with a global negatively curved spacetime. *If* spacetime expands than it has to have negative curvature overall. So it can locally be described as flat expanding space.  
> **username_4**: @PeterErwin ok thanks, I was understanding that curving light rays is possible in an Euclidean space too....  

---

### Answer by username_2 on 2021-06-07. Score: -5

Mass curves spacetime and so does energy density. They both find their place in the time (energy) component of the stress-energy tensor, which functions as the source of the gravity field (like charge for the electromagnetic field). So positive mass and energy curve spacetime in the same way.  
The only way to give the universe a negative curvature is therefore to introduce negative energy (dark energy). This makes mass-energy move away from each other. In the inflation era there was supposed to be an inflaton field which made spacetime grow. The field is thought to consist of particles with imaginary mass or imaginary momentum, because $E^2=m^2c^4+p^2$. If the particles are massless then $E^2=p^2$ so the equality holds if the particles have negative momentum.  
If you consider these particles to have negative mass and negative momentum $E^2$ will be positive, but this is just as imaginary as imaginary mass and momentum.  
The same argument holds for dark matter although this has not a comparable energy as the energy contained in the inflaton field.

So the curvature of spacetime can be negative while there is positive mass-energy in it. Positive mass-energy curvature is canceled by negative mass-energy curvature. Positive pressure can be compensated by negative pressure. The state of the early universe was one of hugely negatively curved spacetime. When real matter appeared this curvature was inversed in a positive curvature, but in the future, the underlying negative curvature will reappear (as it is already now because of the diminishing mass-energy density) and drive matter apart again. Due to the strong negatively curvature that will come to be in the future, there will be a new possibility for virtual particle fields to become real again, so in the future a new big bang will happen in our spacetime (this implies a universe that is infinite in space as well in time).

> **Comments:**
> **username_3**: Dark energy is *not* "negative energy" -- it is positive energy density with negative *pressure*. *All* forms of energy -- normal matter, radiation, dark energy -- contribute in the same way to the curvature of the universe. The way to get negative curvature is to have the total energy density (of all kinds) be less than the critical density.  
> **username_3**: There's a lot of confusion and error in this answer: e.g., "imaginary mass", "imaginary momentum", "negative mass", "positive mass-energy is canceled by negative mass-energy", etc.  
> **username_2**: @PeterErwin The problem is how to make  $E^2=m^2 c^4+p^2$ hold if the pressure is negative. Imaginary mass is just another way of saying that it's negative, but the conclusion for energy will be the opposite.  
> **username_3**: I think you're trying to treat dark energy as though it's made up of individual particles, which is probably not a good idea.  
> **username_2**: @PeterErwin Isn't the inflaton field a particle field? Or the dark energy field (virtual particles). Or is it spacetime itself that contains the energy (as in quantum loop gravity)?  
> **username_2**: @PeterErwin When positive mass-energy is canceled by negative mass-energy then curvature (globally) can be zero or negative.  
> **username_3**: Dark energy has the *same* basic effect on curvature as normal matter; there is no "canceling out".  
> **username_2**: @PeterErwin Doesn't dark energy make the curvature negative (as it in an expanding spacetime has to be)? If not then what's the difference with normal energy? You can say negative pressure, but what causes negative pressure? How can the pressure be negative and the energy positive (sorry for "pushing" a bit, but I just want to know). :)  
> **username_0**: w in the equation of state for energy is usually negative. This leads to a negative pressure and so a negative force I presume - negative because the force due to gravity is positive. I think the force due to pressure (from energy density) counters the force of gravity.  
> **username_2**: @JohnHobson Is w potential energy?  
> **username_0**: Pressure x volume is potential energy. w is dimensionless because pressure and energy density have the same units. There is a lot of talk here about positive or negative energy but surely the important question is whether the resultant pressure is positive or negative and that also depends on the sign of w in the equation of state. If dark energy is positive and w is negative , we get a negative pressure and force. Negative because it is opposite to the force of gravity. I don't see why dark energy has to be negative to counter the force of gravity.  

---

