On gravity and the conservation of energy

[Catreece]

Alright, so this has been bothering me for awhile, and the more I think about it, the more it bugs me. I'm almost guaranteed wrong about what I'm going to say here, but I'm not sure why I'm wrong, so hopefully someone here can explain my wrongness to me. Unfortunately, this is going to take awhile to explain, so bare with me a bit here if you could, please.

First off, gravity's a bit odd compared to the other forces. The others all have a fairly limited range, they're many times stronger than gravity, and they each have a defined carrier particle (or two) that we're aware of. Conservation of energy is pretty straightforward with these, so no big deal.

Then we get to gravity. Gravity does... some weird things. One of the things that really annoys me about it, is that, the more I look at it, the more it looks like it breaks the basics of the conservation of energy.

Lemme explain here for a moment.

If you have an object, say a rock for argument's sake, and lift it off the ground, no big deal! You're applying energy to the rock to lift it up, and then that energy is converted into acceleration when you let go of said rock. The planet and the rock both accelerate, but there's nothing special here.

The thing is, what happens when you take something which doesn't really have energy applied to it in the manner of "lifting" the rock? The gravity well compresses and warps spacetime so that objects are continually drawn down the gravitational incline. There's no limit to how far that gravity well extends, nor how many objects can be affected at a time. If we gave a random asteroid from the asteroid belt just a bit of a nudge so it was moved into the Earth's orbit in such a way that the asteroid gets pulled into our gravity well and slingshots off into space, we run into a really weird problem all of a sudden... there's no energy that was really put into that asteroid, yet it was able to accelerate, which requires energy to be expended. In fact, we can't even really deal with this by just saying there's an opposite, but equal reaction of the Earth being accelerated towards the asteroid, either, because that just means we now have two distinct objects which gained energy from somewhere... but from where?

Gravity isn't magic. It doesn't somehow conjure up energy out of nowhere. But neither does gravity mean that objects are losing mass by having it be converted from mass into energy. That would be handy if it were the case, as it'd explain where this energy's coming from, but it doesn't.

Instead I get stuck thinking on the concept of relativity and the whole idea of gravity being a compression of spacetime. In The Elegant Universe, Brian Greene described Einstein's concept of gravity as not being a force at all, but as gravity and acceleration being essentially the same thing. The issue I have with this, is that acceleration requires energy to exist. Energy has to come from somewhere. If gravity is a compression of spacetime, this works great towards explaining the motion of such objects interacting with one another, but it doesn't deal with the issue of where this energy is coming from.

In our example of the asteroid that gets slingshotted past the Earth, it accelerates because it's passing through a region of compressed space. This makes perfect sense for why you need a certain amount of velocity to escape the Earth's gravity to reach orbit, or why light can be bent by gravity - you might be traveling in a straight line, but the paper the line is drawn upon is being bent. The problem comes in when you realize that, A: there had to be some kind of energy expended to bend space in the first place, and B: that a continual amount of energy is required to maintain bending space in that manner for this to continue to persist. We can explain A easily enough, but B is a problem because there doesn't appear to be anything continually feeding energy into the system to maintain this distortion of spacetime which allows for such.

The most obvious answer would be that it's in equilibrium - a balance of forces, so that for all of the distortion in one area, it's being equally distorted elsewhere. My immediate assumption would be that, since extremes of gravity or acceleration, such as a black hole or the speed of light, are able to distort time, then the guess would be that the equilibrium is being made where the "energy" of gravity to maintain its effect is being balanced out by warping time by the same amount that it's warping space. The limitation of the speed of light would suggest this is the case.

However... there's a problem with that, too. The outer and inner horizons of a black hole kind of call this into question, because the outer horizon loops space back in on itself in a circle, but the inner horizon loops time in a similar manner... but these are two, distinctive and separate amounts of gravity. This kind of tells me that the 100% bending of space is not equal to the 100% bending of time. As such, it implies there should be a remainder left over. The acceleration our planet can produce upon the asteroid is too much for it to be just time alone being compressed. So... where's that extra energy coming from? What balances it out to 0 again?

I might be wrong here. This is already my first concern. I may have already made a horrible mistake at this point - in fact, I probably have. If I have, do please feel free to stop me now and explain why. Otherwise, feel free to continue reading this probable nonsense. =P

So, with the assumption that I'm not completely missing something immensely important here, and that the acceleration produced by gravity is using more energy than should be possible, and there's no "new" energy being added to the system... this implies that every time an object is affected by the gravity of another object, that it has to create a replacement somehow. What kind of a replacement would this be, and what would it look like?

Well... I have a guess, but it's getting into science-fiction territory at this point. Soooo I'm probably wrong. But I don't know enough to know why I'm wrong, so let's take a look at this idea so you can tell me why this is horribly broken of a hypothesis.

The thing is, gravity is the compression of spacetime... so what would the equal and opposite version of such be? Well, the stretching of spacetime. If this were the case, where would that stretching occur? Well... we'd expect that the compression can only exist in relation to other compressing elements, so you wouldn't see the stretching to occur immediately next to such or it'd cancel each other out directly and there'd be no compression in the first place. Instead, you'd expect to see a bunch of compression next to each other, such as solar systems or galaxies and whatnot, with the stretching taking place at the first convenient location where compression isn't taking place. So... right around the outside of a galaxy. Which is oddly right around the location we see consistent halos of dark matter.

Funny thing about that, things like gravitational lensing look pretty much identical, regardless of whether it's a compressed area of space, or a stretched area of space. You might see an inversion of red/blue shifts, but if you didn't know it was inverted as a stretch rather than an area of compression, you'd get incorrect answers as to how far away something is but you wouldn't get an indication that the shift itself is wrong without several different points of reference to compare.

So... this leads me to wonder, is there any reason why "dark matter" is not just the displaced stretching of gravity shoved off to the outer rim of a galaxy? I mean, it'd immediately explain not only my initial question about the issue with gravity seeming to break the conservation of energy, but it'd also instantly do away with dark matter and dark energy, and the expansion of the universe accelerating for that matter. Individual galaxies would remain intact, held together not just by their own gravity, but the halo of inverted gravity which would stretch space to push the galaxies inward and stretching the space between galaxies apart. In fact, it would also mean that, the more gravity affects things, the more this stretching occurs as well, and since gravity was shown by Einstein not to be instantaneous in effect, and can only spread out from its source at the speed of light, this means that over time, there will be a greater stretching force as the gravity sources come into contact with more sources to affect.

I'm not a physicist. I'm not even a physics student. I don't even remotely know where to begin looking at the math to even consider where I'm wrong with this. From a purely qualitative standpoint and what I know of physics, it makes sense and explains a lot of things all in one go. From a quantitative perspective, I dun' got nothin'.

So yeah, I'm almost guaranteed wrong here. I just have no idea why I wrong. I have a sneaking suspicion that it's related to the bit about the distortion of space and time being unequal when it comes to gravity's distortion of the two, but I'm not really sure. Anyway, the point is, I'm almost guaranteed wrong, and I know juuuust enough to come up with this stupid idea, but not enough to know why it's a stupid idea. Any help explaining what I'm missing would be greatly appreciated. =P

 

[PeterDonis]

First, welcome to PF!

Second, as you admit in your thread title, this post is long. It would be better to break it up into shorter, simpler questions and take them one at a time. That's what I will be doing in my responses; they will be broken up into several posts, each on one question.

The gravity well compresses and warps spacetime so that objects are continually drawn down the gravitational incline.

The "gravity well" image only describes space in the presence of a gravitating body, not spacetime. It doesn't actually give a good explanation of why objects are "continually drawn down the gravitational incline", because the whole idea of being drawn "down an incline" depends on gravity--which is what this whole analogy was supposed to be explaining. In other words, it's really circular. Sometimes it helps as an image, but it's not really an explanation.

For an ordinary gravitating body like the Earth, and for ordinary effects of gravity like falling rocks, the effect of gravity is really on time, not space. The reason objects take particular paths when they are freely falling in the presence of a gravitating mass is that those paths maximize the proper time of the objects ("proper time" just means time according to a clock that is carried along with the object as it moves). The presence of the gravitating body curves time in such a way that the paths that maximize proper time fall towards the body. Unfortunately this is harder to visualize than the common "gravity well" image. (A PF member, @A.T., has made some good images that help with this; searching for posts by him in this forum might find them.)

gravity being a compression of spacetime

"Compression" is not a good description. "Curvature" is better, since curvature can be positive (which more or less corresponds to "compression") or negative (which more or less corresponds to "expansion"--but both of those correspondences have caveats). Also, spacetime curvature can be different in different directions; spacetime curvature is really tidal gravity. For example, near the Earth, tidal gravity is positive ("compression"--it makes freely falling objects converge) in tangential directions, but negative ("expansion"--it makes freely falling objects diverge) in the radial direction.

In our example of the asteroid that gets slingshotted past the Earth, it accelerates because it's passing through a region of compressed space

This is not a good description. A better one is that it accelerates because the Earth is curving time so that the path of maximal proper time is the one that accelerates towards the Earth and then slingshots around it.

However, you are also missing a more important point: once the asteroid has passed the Earth, it is decelerating, not accelerating. And once it reaches the same distance from Earth as it originally started from, it will have decelerated by just as much as it accelerated while falling, so its speed will now be the same as it started with. So energy is conserved in this whole process. What happened during the slingshot process was that gravitational potential energy (due to height above the Earth) got converted into kinetic energy (speed of the asteroid) and then back again. The gravitational potential energy due to height above the Earth, which is due to the spacetime curvature produced by the Earth, is the missing piece that must be restored to properly understand energy conservation in this scenario.

 

[PeterDonis]

A: there had to be some kind of energy expended to bend space in the first place

This is true (at least on a charitable interpretation); the presence of stress-energy (of which the dominant component is mass for an object like the Earth--but that's not true of all objects, other components include pressure and other stresses) is required to produce spacetime curvature. The precise relationship is given by the Einstein Field Equation.

a continual amount of energy is required to maintain bending space in that manner for this to continue to persist.

This is not true, unless by "continual amount of energy" you simply mean that the source object (like the Earth) has to remain there. The Earth doesn't have to do anything to continue producing spacetime curvature.

 

[PeterDonis]

The outer and inner horizons of a black hole kind of call this into question, because the outer horizon loops space back in on itself in a circle, but the inner horizon loops time in a similar manner...

I don't know where you're getting this from. First, only a charged or a rotating black hole has an inner horizon at all. Second, your statements about "looping space" and "looping time" make no sense as descriptions of either horizon. I think you have some serious misconceptions about black holes; but that discussion really belongs in a separate thread.

 

[PeterDonis]

is there any reason why "dark matter" is not just the displaced stretching of gravity shoved off to the outer rim of a galaxy?

The short answer is yes. The longer answer would require considerable discussion, but before that is even advisable, I think you first need to clear up your more basic misconceptions about how gravity works. So I would keep this thread focused on that. Dark matter discussion can be had in a new thread once the more basic points are cleared up.

 

[PeterDonis]

I have a guess

As my comments in previous posts should illustrate, your guess is, unfortunately, "not even wrong"; the whole reasoning that led you to it is based on misconceptions, and you need to clear those up first.

It's natural to want to guess, but if you really want to learn about science, the temptation to guess is a temptation that has to be resisted.

 

[Catreece]

First, welcome to PF!

Thanks! This helped a lot with some fundamental misunderstandings! Going to cover a few things while I'm here though. =3

The presence of the gravitating body curves time in such a way that the paths that maximize proper time fall towards the body. Unfortunately this is harder to visualize than the common "gravity well" image. (A PF member, @A.T., has made some good images that help with this; searching for posts by him in this forum might find them.)

Thanks for this in particular; I'd not encountered this description of gravity before and it's a bit oddly worded, though I think that's more of an issue with the concept being odd than anything on your part. I'll definitely be looking up the posts you suggested to see if I can get a better grasp on such!

For example, near the Earth, tidal gravity is positive ("compression"--it makes freely falling objects converge) in tangential directions, but negative ("expansion"--it makes freely falling objects diverge) in the radial direction.

This is also really handy. I've never even encountered the term tidal gravity, except in relation to things like moons being compressed (like Neptune's) or the more obvious ocean tides. Even searching for it, it took half a google page to even come across something relevant. With an idea of where to look though, I've got a few new sources to read up on now though!

However, you are also missing a more important point: once the asteroid has passed the Earth, it is decelerating, not accelerating.

I'd considered such actually, but given how long the post was getting already, I didn't bother with it since I'd (incorrectly) ruled it out. The thought had been that, as it was moving slowly at first, it would spend more time at the front of the Earth accelerating, and then once it'd slingshot past, it would be moving much quicker and spend less time nearby, so it wouldn't have an equal value of net force due to gravity falling off once it was farther away, but that it would eventually catch up and surpass the limit just due to gravity not having a finite range. So the assumption was that it wouldn't immediately cancel out the force, and when it did, it would continue to keep decelerating to the point of eventually reversing (assuming no other gravity sources affecting it, as just a purely 2-body problem to simplify matters) so the reversal would mean that it'd invariably come back again like a comet and keep making passes until the orbits weren't quite right and they collided. Which doesn't really add anything to the problem so I ignored it. I was apparently wrong to have done so, so thanks for pointing this out as well!

Anyway, thank you for your patience and the very helpful answers! Not just for what you covered, but giving me enough of a foothold to know where to look next!

 

[PeterDonis]

thank you for your patience and the very helpful answers!

You're welcome! Feel free to post new threads with other questions as you investigate.

The thought had been that, as it was moving slowly at first, it would spend more time at the front of the Earth accelerating, and then once it'd slingshot past, it would be moving much quicker and spend less time nearby

As you have realized, this is incorrect; but a good starting point for seeing why in more detail is to consider Kepler's laws of planetary motion, particularly the second law. It's usually used to describe elliptical orbits, but it works for hyperbolic orbits (the kind of "slingshot" trajectories you are considering) as well. Kepler's laws are still perfectly valid in GR, as approximations when gravity is weak and all of the objects are moving slowly compared to the speed of light; more generally, Newtonian gravity is a valid approximation in this regime (and it's straightforward to derive Kepler's laws from Newton's).

 

[Catreece]

Ah, there's more. XD

This is not true, unless by "continual amount of energy" you simply mean that the source object (like the Earth) has to remain there. The Earth doesn't have to do anything to continue producing spacetime curvature.

Alright, like if you take the Earth away, obviously the gravity's gone. Or at least so long as the mass is gone. ...Normally. Though as I understand it (and likely am off on this as well), a black hole basically has no actual mass as it tends to tear apart massive particles until they no longer have mass directly any longer, and it's just left as a spacetime curve. I'm still a bit iffy on how that works.

The short answer is yes. The longer answer would require considerable discussion, but before that is even advisable, I think you first need to clear up your more basic misconceptions about how gravity works. So I would keep this thread focused on that. Dark matter discussion can be had in a new thread once the more basic points are cleared up.

Fully reasonable! I think that's something I may need awhile to get to asking again later on, or I may very well figure out when working my way through the bits I clearly don't understand. =P

I don't know where you're getting this from. First, only a charged or a rotating black hole has an inner horizon at all. Second, your statements about "looping space" and "looping time" make no sense as descriptions of either horizon. I think you have some serious misconceptions about black holes; but that discussion really belongs in a separate thread.

I'd recently been updated a bit on the concept of black holes with information about the inner/outer horizons. Like the outer horizon is the point at which space curves to a point that it essentially loops back in on itself. The inner horizon was described as doing a similar thing, but with time, where it supposedly has small almost "loops" of time but it wasn't made very clear beyond that. I'd assumed it was very similar to the way the event horizon worked, but it's very likely such was a bad assumption to make and that the "loops" that were described were probably just dumbed down explanations to try to get the rough point across, but that the metaphors don't really work overly well. I've found this happens a tooooon in almost anything related to physics, which is more than a little frustrating, because it winds up leaving a lot of misinformation out there and just as soon as I think I've got the grasp of something, all the explanations turn out to be themselves wrong. So when I finally correctly understand what they were saying, it turns out what they were saying wasn't correct because they oversimplified it.

Hence that thing about getting to the point of knowing enough to come up with stupid ideas, but not knowing enough to know why they're stupid. =P

As my comments in previous posts should illustrate, your guess is, unfortunately, "not even wrong"; the whole reasoning that led you to it is based on misconceptions, and you need to clear those up first.

It's natural to want to guess, but if you really want to learn about science, the temptation to guess is a temptation that has to be resisted.

I don't know, there is that whole cunningham's law thing about the fastest way to get the correct answer to something is to provide the wrong one. I know it's bad form for science, and speculation without comprehension is bad in such a setting, but in terms of trying to learn, I find the act of trying to understand what's wrong with a bad idea often teaches me a lot more than not hazarding a guess. I'd rather be curious, screw up, and learn from the mistakes quickly than not guess at all. Case in point - I learned not only was my guess wrong, but by putting it out there, I got to learn all sorts of stuff tangentially that I didn't even know I misunderstood in the process. I now know where to direct my next batch of reading, whereas before this interaction, I wasn't even sure where my next step was because I didn't know what it was that I didn't know.

So yeah, it's not good for scientific study, but it's pretty effective for learning what you don't know very quickly. I'll try to be cautious about doing such on here in the future, but I can't guarantee I won't do so again. =P

 

[PeterDonis]

a black hole basically has no actual mass

A black hole is a vacuum solution of the Einstein Field Equation, yes. That means there is no stress-energy (no "stuff"--no atoms, gas, etc.) inside it. But it does have mass, in the same sense that any other gravitating body has mass: it curves spacetime (in fact a black hole is basically "made of" spacetime curvature), and you can put objects in orbit around the hole and measure its mass by measuring the orbital parameters, just as for any other gravitating body.

the outer horizon is the point at which space curves to a point that it essentially loops back in on itself

No, that's not correct. The outer horizon is the point at which radially outgoing light no longer moves outward; it stays at the same radial coordinate. If you think in terms of a spacetime diagram, with time going upwards and radial "space" going outward to the right, in ordinary flat spacetime an outgoing light ray would be a 45 degree line going up and to the right. But at the horizon of a black hole, the light ray goes straight up: it "moves" in time but not in space. (Note that this viewpoint has limitations, but it's better than thinking of space as looping back on itself.)

The inner horizon was described as doing a similar thing, but with time

Can you give a reference to the source where you got this? It does not sound reliable. In a rotating black hole, it is true that there are closed timelike curves inside the inner horizon, but the inner horizon itself is not such a curve. (And a charged black hole has an inner horizon, but there are no closed timelike curves anywhere.)

 

[PeterDonis]

the fastest way to get the correct answer to something is to provide the wrong one

That in itself is fine, and if your initial post had just been about the asteroid slingshot scenario, guesses and all, I wouldn't have had to say anything about speculation. The part that prompted that comment was the latter half of your post, speculating about dark matter. You'll note that I didn't give any really useful information beyond "no" in response to that part.

 

[CWatters]

Then we get to gravity. Gravity does... some weird things. One of the things that really annoys me about it, is that, the more I look at it, the more it looks like it breaks the basics of the conservation of energy.

Lemme explain here for a moment.

If you have an object, say a rock for argument's sake, and lift it off the ground, no big deal! You're applying energy to the rock to lift it up, and then that energy is converted into acceleration when you let go of said rock. The planet and the rock both accelerate, but there's nothing special here.

The thing is, what happens when you take something which doesn't really have energy applied to it in the manner of "lifting" the rock?

The big bang created a lot of gravitational potential energy by separating matter. Some has since come back together to make stars that have exploded and "lifted a few rocks" that later crash into other rocks called planets. The energy the asteroid had did come from somewhere.

 

[CWatters]

In our example of the asteroid that gets slingshotted past the Earth, it accelerates because it's passing through a region of compressed space.

If it leaves the Earth going faster then the acceleration of the asteroid is at the expense of the Earth slowing down.

Look up how we used the sling shot effect to speed up the Viking probes. When they went passed a planet they speeded up and the planet slowed down ever so slightly. So overall energy was conserved.

 

[PeterDonis]

The big bang created a lot of gravitational potential energy by separating matter.

This is not correct. The concept of gravitational potential energy is not well defined for the universe as a whole, because it is expanding. Gravitational potential energy is only well-defined for a stationary spacetime.

If it leaves the Earth going faster then the acceleration of the asteroid is at the expense of the Earth slowing down.

Note that this viewpoint is coordinate-dependent; you're adopting coordinates in which the Earth is moving. This is commonly done for spacecraft in the solar system, for example. In these coordinates, the relevant gravitational potential energy is not that due to the field of the Earth, but due to the field of the Sun (or more precisely that of the total mass of the solar system, considered as a whole--but more than 99 percent of that is the Sun's mass, so treating the Sun as the only source of potential energy is a good approximation for most purposes).

However, for a simple analysis of the two-body problem, just the Earth and the asteroid, it might be easier for the OP to use coordinates in which the Earth is at rest. That makes it easier to see how energy conservation works in that simple example.

 

[JMz]

Thanks for this in particular; I'd not encountered this description of gravity before and it's a bit oddly worded, though I think that's more of an issue with the concept being odd than anything on your part.

I believe @PeterDonis is using this wording because it is related to what's known as Hamilton's principle, or the principle of Least Action. (Action is a term of art for a particular mathematical construct, not a synonym for "movement".) It's one of the deepest principles in physics. But it's not typically invoked in popular accounts of physics, so it's not surprising that you haven't seen it before.

 

[Monsterboy]

Imagine there are only two massive particles in the universe, both of them are separated and are at rest with respect to each other. Now, because of gravity they start moving towards each other, accelerating as they come closer. Now, where did this kinetic energy come from ? Initially there was nothing but two particles at rest, now all of a sudden there is kinetic energy without any other form of energy being converted to kinetic energy ?

Gravity field itself must be providing this energy. Where does the gravity field get it's energy from ? If it is from mass then the mass of the particles must decrease overtime, but that doesn't happen either correct ?

 

[PeterDonis]

where did this kinetic energy come from ?

From the gravitational potential energy between them, which decreases as they get closer together.

Where does the gravity field get it's energy from ?

It doesn't have to get it from anywhere. Gravitational potential energy is a property of spacetime (more precisely, of certain types of spacetimes).

 

[Monsterboy]

more precisely, of certain types of spacetimes

How many types are there ? what are those ?

 

[PeterDonis]

How many types are there ?

For purposes of the remark you quoted, there are two: stationary (gravitational potential energy is well-defined) and non-stationary (it isn't well-defined).

 

[russ_watters]

I'll agree that being more concise/compartmentalized would help. I'm not going to guarantee that I'll get all the way through this:

Then we get to gravity. Gravity does... some weird things. One of the things that really annoys me about it, is that, the more I look at it, the more it looks like it breaks the basics of the conservation of energy.
...
The thing is, what happens when you take something which doesn't really have energy applied to it in the manner of "lifting" the rock? The gravity well compresses and warps spacetime so that objects are continually drawn down the gravitational incline. There's no limit to how far that gravity well extends, nor how many objects can be affected at a time. If we gave a random asteroid from the asteroid belt just a bit of a nudge so it was moved into the Earth's orbit in such a way that the asteroid gets pulled into our gravity well and slingshots off into space, we run into a really weird problem all of a sudden... there's no energy that was really put into that asteroid, yet it was able to accelerate, which requires energy to be expended. In fact, we can't even really deal with this by just saying there's an opposite, but equal reaction of the Earth being accelerated towards the asteroid, either, because that just means we now have two distinct objects which gained energy from somewhere... but from where?

I don't see a problem. The asteroid was in orbit and had a potential and kinetic energy with respect to Earth, which was changed by however much it was "nudged". It almost seems like you are ignoring the starting potential and kinetic energy and then saying the later kinetic energy came from nowhere. Obviously it didn't: the new kinetic energy came from the potential energy and the total energy is conserved. It appears to me that everything else you said is attempts to try to confuse this basic principle.

 

[Monsterboy]

For purposes of the remark you quoted, there are two: stationary (gravitational potential energy is well-defined) and non-stationary (it isn't well-defined).

"non-stationary" means in an expanding universe ?

http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

First, unlike with ordinary matter fields, there is no such thing as the density of gravitational energy. The thing you would like to define as the energy associated with the curvature of spacetime is not uniquely defined at every point in space. So the best you can rigorously do is define the energy of the whole universe all at once, rather than talking about the energy of each separate piece. (You can sometimes talk approximately about the energy of different pieces, by imagining that they are isolated from the rest of the universe.) Even if you can define such a quantity, it’s much less useful than the notion of energy we have for matter fields.

I didn't get this part. For the system in my example of two massive particles, can we not say that the density of gravitational energy decreases as the particles move away from each other ? maybe I didn't get what "density" of gravitational energy means.

The second reason is that the entire point of this exercise is to explain what’s going on in GR to people who aren’t familiar with the mathematical details of the theory. All of the experts agree on what’s happening; this is an issue of translation, not of physics. And in my experience, saying “there’s energy in the gravitational field, but it’s negative, so it exactly cancels the energy you think is being gained in the matter fields” does not actually increase anyone’s understanding — it just quiets them down. Whereas if you say “in general relativity spacetime can give energy to matter, or absorb it from matter, so that the total energy simply isn’t conserved,” they might be surprised but I think most people do actually gain some understanding thereby.

I didn't get what "matter field" actually means, is it like saying (with respect to my example) kinetic energy is positive and gravitational potential energy is negative so they cancel each other out and so energy is conserved ?

 

[PeterDonis]

"non-stationary" means in an expanding universe ?

That is an example of a non-stationary spacetime, yes.

For the system in my example of two massive particles, can we not say that the density of gravitational energy decreases as the particles move away from each other ?

No. You can say that the potential energy increases (not decreases) as the particles move apart, but you can't convert that potential energy to an energy density, because it isn't localized. It's a property of the global configuration of the two particles. It doesn't "belong" to any particular point in space.

I didn't get what "matter field" actually means

It means anything that isn't spacetime/gravity.

 

[PeterDonis]

is it like saying (with respect to my example) kinetic energy is positive and gravitational potential energy is negative so they cancel each other out and so energy is conserved ?

It's similar in some respects, but not the same, because, as already noted, an expanding universe is not a stationary spacetime, so the concept of gravitational potential energy is not well-defined. The "energy in the gravitational field" that Carroll is talking about in the expanding universe case isn't the same as the gravitational potential energy in the case of the two masses.

 

[JMz]

Gravity field itself must be providing this energy. Where does the gravity field get it's energy from ?

From you: You started the two particles off with that potential energy, because of where you placed them. They were "born" with it. Whatever energy you gave them in the beginning, they will keep forever (in this two-particle universe). The energy you gave them is indeed conserved, and there is no other energy in this universe.

 

[Monsterboy]

From you: You started the two particles off with that potential energy, because of where you placed them. They were "born" with it. Whatever energy you gave them in the beginning, they will keep forever (in this two-particle universe). The energy you gave them is indeed conserved, and there is no other energy in this universe.

Yea, the potential energy is because of the presence of a gravity field right ?

Where does the gravity field originate from ? does it originate from the masses or is it present everywhere even before there were any particles with mass ?

 

[JMz]

Yea, the potential energy is because of the presence of a gravity field right ?

Where does the gravity field originate from ? does it originate from the masses or is it present everywhere even before there were any particles with mass ?

Yes, from the gravitational field.

"Originate" is either a deep philosophical question (Why does gravity exist at all?) or a simple one: Because the Einstein field equations, or something like them (even Newton's), hold, and they say that mass creates gravity.

 

[jbriggs444]

"Originate" is either a deep philosophical question (Why does gravity exist at all?) or a simple one: Because the Einstein field equations, or something like them (even Newton's), hold, and they say that mass creates gravity.

One can adopt a less causal interpretation. They say that mass (or, more generally, the stress energy tensor) is associated with gravity. Not necessarily that it creates gravity.

Correlation is not the same as causation.

 

[JMz]

One can adopt a less causal interpretation. They say that mass (or, more generally, the stress energy tensor) is associated with gravity. Not necessarily that it creates gravity.

Correlation is not the same as causation.

Quite so, but I'm not sure that bringing the semantics of cause into an I-level thread is philosophically simpler. ;-) And the "creates" verb is at least consonant with MTW's "[mass] generates [gravity]".

 

[nikkkom]

If you have an object, say a rock for argument's sake, and lift it off the ground, no big deal! You're applying energy to the rock to lift it up, and then that energy is converted into acceleration when you let go of said rock. The planet and the rock both accelerate, but there's nothing special here.

The thing is, what happens when you take something which doesn't really have energy applied to it in the manner of "lifting" the rock? The gravity well compresses and warps spacetime so that objects are continually drawn down the gravitational incline. There's no limit to how far that gravity well extends, nor how many objects can be affected at a time. If we gave a random asteroid from the asteroid belt just a bit of a nudge so it was moved into the Earth's orbit in such a way that the asteroid gets pulled into our gravity well and slingshots off into space, we run into a really weird problem all of a sudden... there's no energy that was really put into that asteroid, yet it was able to accelerate, which requires energy to be expended. In fact, we can't even really deal with this by just saying there's an opposite, but equal reaction of the Earth being accelerated towards the asteroid, either, because that just means we now have two distinct objects which gained energy from somewhere... but from where?

The behavior of bodies moving in gravitational fields like these are very similar to how electrically charged objects interact. A very light negatively charged "asteroid" will swing by a very light, positively charged "Earth" exactly the same (the attractive forces will have the same form, 1/r^2, and magnetic interactions due to the fact that charges are not stationary will be very small).

But somehow that interaction would not look strange to you:

"The others all have a fairly limited range, they're many times stronger than gravity, and they each have a defined carrier particle (or two) that we're aware of. Conservation of energy is pretty straightforward with these, so no big deal."

Why gravity looks stranger that electromagnetism to you?

 

[JMz]

the other forces ... all have a fairly limited range,

BTW, this seemed an odd thing to say: Electromagnetism has exactly the same range as gravity. This is usually stated to be "infinite", but the point is that both drop off as 1/r^2 (cosmological corrections aside), so their ratio is constant. As @nikkkom points out, you seem to be asking, for gravity, about something you already accept for electromagnetism.

 

[Demystifier]

"Compression" is not a good description. "Curvature" is better

In the Newtonian limit all gravity is encoded in $g_{00}(x)$, which is more like "compression" (of time) than like curvature.