If you Open a 1-m Radius Portal Between Earth and Deep-Space, Will Equilibrium Be Reached?

Question

After I looked through this question, I had one of my own.

I can think of no real reason why a portal does not also allow the gravitational field through. So we suppose that the portal does, in fact, let gravitational field lines through. Since the portal connects the two pieces of space together, the other side of the portal is inside a gravity well, just like on the surface of Earth.

Because the air on Earth is bound to the Earth by the gravity of Earth, I'd imagine that if you opened a portal in deep-space, air would rush out and form a bubble around the portal. When the air has reached the height of the atmosphere, the pressure at both sides of the portal will equalize, resulting in equilibrium.

Will this actually happen? What happens at equilibrium?

Addendum: Deep space can be considered to be where Earth's gravitational effects are minimal compared to the gravitational effect through the portal.

Answer 1

Equilibrium will be reached when the atmosphere of a square of 25 km is transported through the portal

I will make some general assumption to make the question answerable. So as stated in the comment if you don't have gravity air will simply dissipate into the void. You can't have gravity projected as a vector through the portal since you would have a weird gravity edge and in any case if you have that air will just fall over the edge into the void, never getting equilibrium.

So if you project gravity through the portal it should behave as a spherical mass. So basically, I would surmise, you create an object with the radius of your portal and the mass of the earth. From that assumption you can calculate the amount of atmosphere that will be held around the portal.

So very rough calculation: So the troposphere (lowest layer of Earth's atmosphere) extends on average 12 km and contains 80% of the mass of Earth's atmosphere. Since you are dealing with the same gravity the atmosphere of your portal will be about the same. That means you get a bubble of atmosphere around your portal of 4/3 x pi x 12^3 = 7238 km^3 of air. As soon as roughly that amount of air has gone through the portal you will have equilibrium. This is quite a rough calculation since no density gradient of the atmosphere or any other of probably millions of atmosphere effects have been taken into account.

To put in in perspective, equilibrium will be reached after sucking all the atmosphere out of a region of 25 x 25 km. (25 ≈ sqrt(7238/12)) So don't open you portal in the middle of town but a remote forest is fine.

Answer 2

Earth will become a barren rock.

Assuming gravity works by line of sight (if two point masses $m_1$ and $m_2$ are on opposite sides of the portal, and there is a line segment through the portal that connects them, each will experience gravity as if, instead of going through the portal, the line segment continued going through space and the other mass was on the other end of the line segment). This gets rid of some of the discontinuities in the strength of gravity that would happen if just the field lines passed through the portal.

Then, using the equations $\color{#328396}{a=\frac{-1}{\rho}\frac{dP}{dz}}$ and $PV=nRT$, we get $a=\frac{-kT}{P}\frac{dP}{dz}$. Assuming that some equilibrium is reached, $a_{net} = 0$, so $a_g=\frac{kT}{P}\frac{dP}{dz}$ for some positive constant $k$.

Next, assume that when equilibrium is reached, the air pressure on earth is non-zero, and consider the line that if you were on and looked through the portal, you would be looking straight at the sky, directly away from Earth. From that line, none of Earth is in view, so nothing on that line will experience gravity. Therefore, on that line $a_g=0$ meaning that $\frac{dP}{dz} = 0$. Also, at any point next to the portal, the pressure will be equal to the pressure where the portal is on Earth. That means that on our line, the pressure at all points must be the same pressure as on earth's surface. However, this is impossible. Because the line extends to infinity, it would require an infinite amount of gas to maintain a positive pressure on the whole line. This contradiction means that the pressure on Earth's surface must be 0.

In reality, the gas that passed through the portal would eventually have enough mass that its gravity would be significant, preventing the contradiction above from arising. However, because Earth has so little gas (only about $5*10^{18}$kg), the pressure on Earth's surface would likely be very very approximately a millionth of an atmosphere once equilibrium is reached.