# How gravity could be created by energy (and by what amount of energy)?

Firstly let me explain that I'm a physics newbie, but interested in all things gravity at the moment!

My question relates to some of my understanding of Einstein's general relativity. This is just from my personal reading, I'm not in any kind of academia (and probably too old to start!).

There are a couple of key concepts that seems to contradict themselves.

Firstly, I understand that energy and mass are the same things. And also that gravity is really the distortion of spacetime caused by the presence of mass, and the latent energy therein. So it follows, that energy should also distort spacetime and cause us to observe gravity. Other forums I've read seem to confirm this.

My first question is - this being the case, what form of non-latent energy could be used to prove this?

Secondly, from reading other forums, it's been speculated that enormous amounts of energy would be required for a measurable amount of gravity to be observed if the above assumption about energy creating gravity is true. This is because (as I understand it) the latent energy in mass is huge, so therefore in pure energy form, the equivalent amount of energy would also be huge to create a measurable distortion in spacetime to observe gravity - right?

This seems to contradict Einstein's principle that gravity and acceleration are also the same things. By that I mean, Einstein said that a person in a closed box, being accelerated in space, would have no way of proving (or disproving) whether the were being accelerated or were in fact at rest e.g. on the planet Earth. This is because acceleration and gravity are the same things.

So my second question is: Surely it follows that the amount of energy required to accelerate a person in space e.g at the equivalent of 1g Earth gravity, is the same amount of energy that would be required to simulate the equivalent amount of gravity using energy instead?

I'm clearly missing something. My ultimate question is trying to understand how realistic it is theoretically to have artificial gravity, using energy as a source to warp spacetime, and whether the amount of energy required is already a fundamental blocker or not.

Many thanks for any insight and I hope I've been clear. It's a very interesting topic!

Regards, Glen.

Yes, GR says that spacetime curvature is proportional to the stress-energy density in the vicinity. So energy in any form can be used to "create gravity". However, matter is a particularly dense form of energy, and it has a convenient tendency to stay where you put it, so it's not easy to warp spacetime as efficiently or conveniently with energy in other forms.

For example, if you could somehow capture the entire energy output of the Sun for 13 months and trap it inside a ball with a diameter of 1 km, that ball would have approximately the same surface gravity as the Earth. Clearly, this is not a practical way to generate artificial gravity. :)

[FWIW, I got that figure from Google Calculator using: (5.972E24 kg) * (1km/6371km)^2 * c^2 / (3.86E26 watts) in days]

But all is not lost. As you mention in your question, acceleration is equivalent to gravity. (I should mention that the equivalence is only local: if you're in a very tall spaceship and perform delicate measurements both at the top and bottom of the ship you'll be able to tell if you're on a planet or freely accelerating in space: on a planet, the gravity will be slightly lower at the top of the ship).

If the ship is going on a long voyage to other stars, then you simply need to continuously accelerate the ship at 1$g$ for the duration of the trip, flipping the ship around at the midpoint. After a year of travel your speed relative to your starting frame will be around 0.77$c$ and you will have travelled 0.56 light-years (as measure in that frame). But as you can imagine, continuous acceleration does consume quite a lot of fuel. To fly past Alpha Centauri at $1g$ acceleration (without stopping, or doing that mid-point flip-over) takes around 10kg of energy per kg of payload. That's quite a lot of energy: it's the amount of energy that'd be released if you annihilated 5kg of normal matter with 5kg of antimatter. Using $E = mc^2$, 10 kg of mass is $9 \times 10^{17}$ joules.

For more details on this, please see the classic Usenet article The Relativistic Rocket.

The cheap way to do artificial gravity is to use the centrifuge effect. The beauty of this method is that once our centrifuge is in motion it doesn't take much energy to keep it spinning, due to conservation of angular momentum.

The equation for acceleration in uniform circular motion is

$$a_c = \omega^2 r$$

where $a_c$ is the centripetal acceleration, $\omega$ is the angular velocity (in radians per second), and $r$ is the radius.

So if we have a centrifuge with a radius of 100 metres, we can get an acceleration of $g = 9.81m/s^2$ from an angular velocity of

$$\omega = \sqrt{9.81 m/s^2/ 100m} \approx 0.3132 \, \text{radians / second}$$ which is around 3.00 RPM.

A smaller radius would certainly work, I chose 100 metres because we want the floor to be reasonably flat, and we don't want our astronauts to get dizzy. :)

• Good answer, I just corrected your RPM number (it was a bit too large ;-)) – magma Jun 12 '17 at 8:36
• @magma Oh dear. I multiplied by $2\pi$ instead of dividing. :oops: Thanks for that. – PM 2Ring Jun 12 '17 at 8:58

This seems to contradict Einstein's principle that gravity and acceleration are also the same things. By that I mean, Einstein said that a person in a closed box, being accelerated in space, would have no way of proving (or disproving) whether the were being accelerated or were in fact at rest e.g. on the planet Earth. This is because acceleration and gravity are the same things.

Not quite. Your central sentence ("... a person in a closed box .... on the planet Earth") is a good statement of Galileo's relativity principle in this situation, and it is true. The other parts not quite.

In general relativity, the equivalence principle is a statement about the tangent space to the spacetime manifold - in other words, about the first derivatives of fields. It amounts essentially to the statement that spacetime can be described as a geometric thing that has tangent space at all and is locally like a flat space (i.e. a manifold), so that there are always freefall frames relative to which an observer's tests will tell him or her that spacetime in their immediate neighborhood looks just like flat Minkowski spacetime.

For the equivalence notion to work, you have to be looking at a portion of spacetime that is small enough that the first order description - that involving only the first derivatives of co-ordinates and fields - of physics is accurate.

So the second order and higher behavior of the spacetime manifold doesn't enter the equivalence principle, by definition. Two situations, with quite different physics, can set up the same first order behavior. The equivalence principle then says that no experiment measuring these first order effects can tell the difference. Your elevator guy is accelerating though flat Minkowski spacetime. He or she can use a Rindler Chart with all kinds of funky properties, but one can still do a co-ordinate transformation to show that the spacetime is perfectly flat. Someone pinned to the Earth's surface, on the other hand, sees the same first order behavior of spacetime around them as the elevator guy, but the global picture is wholly different: spacetime is curved by the Earth's presence. That's why you need only a few liters of hydrazine for the first scenario and more stone and liquid iron than you can imagine for the second. They are scenarios of totally different physics requiring totally different amounts of energy to set up.

I'm not sure I fully understand your question on latent energy, but one can certainly say there is energy in the gravitational field. It's a subtle and tricky concept though, because there's no easy way to define gravitational energy density: you can't say that the field has energy density at this point here. This is because one can always choose freefall co-ordinates to annul the so-called Christoffel symbols at any given point. No Christoffel co-efficients means no gravitational energy density ascribable to that point, so the existence of a localizable "gravitational energy density" would seem to tell against the equivalence principle. So your "latent" energy is not comparable to anything else you already know.

Welcome to Physics SE, Glen! The way I understand your question is this:

If acceleration (with respect to a local inertial frame) and gravity are the same things then does it follow that the energy required to create that acceleration and the energy required to create the equivalent gravity are also equal?

Now, the energy required to accelerate a particle with an acceleration $a$ with respect to a local inertial frame is not a definite quantity. It depends on for what time you are accelerating the object and what was the initial velocity of the object. In addition, it most certainly depends on the mass of the object.

On the other hand, the energy required to produce certain gravity at a certain place is also not a definite quantity. It depends on how far away you are putting the energy, how that energy is distributed, and so on... Also, it is not only the energy that can produce gravity, the momentum also does it. So, the energy required to produce certain gravity at a certain place is not a unique thing.

Since both the quantities that you want to compare have an indefinite status, there is no way we can compare them. More importantly, the energy to accelerate an object for some time starting from some velocity would always depend on its mass. On the other hand, the energy (which can possibly be calculated if we restrict the form of its distribution, where it should be put and so on) required to produce certain gravity at a certain place would never depend on the mass of the object that is going to experience that gravity. So, the two quantities that you want to compare seems inherently incomparable in any meaningful sense.

So, it is clear that it is not what Einstein's insight that acceleration (with respect to the local inertial frame) and gravity are the same thing mean. Rather the way it should be interpreted is a very kinematical approach. The acceleration that is being talked about here is the acceleration of a frame of reference. It doesn't need to refer to the acceleration of any physical object. It simply means that if you were to analyze things from a frame of reference accelerating at some acceleration with respect to the local inertial frame then the physics would look like as if there is this certain amount of gravity present. It is actually not even the case that you can always ascribe some energy (or momentum) as the source of this gravity. In General Relativity, some forms of gravity arise purely out of our choice of frame and do not need any energy (or momentum) as their origin.

Now, can gravity be created by energy that doesn't have mass? Yes. Photons do bend spacetime and they gravitate things. At what extent such effects would be measurable by our current technology and what amounts of photon should be gathered to make them detectable by our current technology are rather cultural questions. In principle, energy without mass does create gravity.