Science a GoGo's Home Page Forums General Discussion Physics Forum Gravity

It is easy to visualise a situation in which, once a gradient has been established, moving objects will follow it, but how much of this is due to our familiarity with gravitational influences on Earth? We expect things to move downhill.

Is a geodesic a gradient, or simply a curve in spacetime?

We live in 4-dimensional spacetime, so why does a geodesic look curved to us, if in fact it is not? Are there other dimensions involved?

Well, "A geodesic is a locally length-minimizing curve." Does
that help? That's from some mathematical dictionary, I didn't
pay much attention to where it came from. My use of gradient
doesn't have much real meaning to it. As I said I was
getting myself a bit confused in my last answer.
I'm sure you must have seen the bowling ball on a rubber
sheet analogy. The warping of the sheet represents the
warp in spacetime which I kind of thought of as the gradient.
The path followed by a marble placed on the sheet would be
the geodesic. In that case if you just place a marble on the
sheet, not moving with respect to the bowling ball, it
will follow the gradient to the ball. The gradient is
equivalent to the warpage of spacetime. The path it follows
would be the geodesic. If the marble is started with some
initial movement the line it follows will be a geodesic, but
the velocity will determine just what geodesic it will follow
as it traverses the warp in the rubber sheet. I'm not
sure that will help much but maybe it will.

Bill Gill

No!!! Well, actually I suppose it does, if one interprets it as saying that a geodesic is the shortest distance between two points, but it is not a straight line. I think I can live with that.

I have wrestled with the rubber sheet analogy for a long time. It makes the basic idea of distorted spacetime reasonably clear, but I find it raises some other interesting problems about any energy that might be involved.

Well as far as energy is concerned, basically you just have to
remember the law of conservation of energy. Energy can not
be created or destroyed. As an object approaches a gravity
well it will gain energy, which is taken from the gravitational
field, and as it leaves the well it will lose energy to the
gravitational field. This of course leaves open the question
of the slingshot method of giving space vehicles a boost by
looping them around a planet. I'm not absolutely sure how
that works. Except that it does. I'm sure if I understood
the math, which doesn't require GR, just Newton, I could work
it out. I had plenty of math in college, but when I got out I
immediately forgot it, because I didn't need any of it in
my work.

One thing that may be mildly confusing is the way that a
gravitational well is often represented. It is often shown
as a cone with curved sides. When it is drawn in a wireframe
drawing the sides are shown as circles which decrease in size
as they approach the main gravitational mass. Well, that view
won't quite be true. After all the mass of the approaching
object will also have its own gravitational well. It will
be small in comparison with the gravitational well of a "large"
object such as a planet or a sun, but it will still be there.
So the actual shape of the well as the object approaches
will be distorted. The distortion will be constantly changing
as the particle passes. Remember that GR describes spacetime
as being dynamic. There is nothing static about it.

Well, that last paragraph may not really be germain to the
question, but I hope it all helps.

Bill Gill

How does this equate with the idea that as an object is raised from the Earth's surface it gains gravitational potential energy, which it loses as it falls back to Earth?

Let's think about the classical energy equation E = mV^2. So the energy of the object is not really there when it is stationary since V = 0. But when it is released it gains speed, and therefore energy. The amount of energy it has when it hits the ground will essentially be the same amount that was used to raise it off the ground. That is what we speak of as potential energy. So the potential energy an object has is the amount of energy it can gain by falling into a nearby energy well. In the case of an object in free space approaching a gravitational well it will have its own energy of motion. That energy will be increased as it approaches the well, but if its geodesic takes it past the well it will lose the same amount as it gained on the approach. I think that is about the way it works. As I have said, I don't know enough math to work out just exactly what is going on.

Bill Gill

So far, so good. You have anticipated my next line of thought.



This is a thought experiment, so let's not worry about the practicalities.

An object from space is on a collision course with the Earth. You place something in its path that stops it. Its kinetic energy is lost. It is stationary, so, as you pointed our, V = 0, so E = 0. It has not gained GPE, because was not raised from Earth.

Now you remove the thing that stopped the object, and it falls towards the Earth. If gravity is not a force, where does the energy come from that re-starts the object's motion?

Bill S.

I wish you wouldn't ask things like that, it makes me have to think. Not my most favorite thing to do. But then I have to go ahead and try to answer. You may be aware that graduate students are required to teach a number of courses in their discipline. One reason for this is that the university gets cheap teachers. But one other reason is that teaching a course makes you actually think about what you have learned. When you get so you can explain it to others you have a pretty good handle on the material yourself.

Who said that gravity isn't a force? Well, in a way Einstein did. He said that gravity is the result of warped space. Let me think how to say this. To us gravity appears as a force. This is because in general things try to get to the lowest energy level. A massive object creates a warp that has its lowest energy at the center of the mass, so that anything entering the area of warpage will try to get to the center of the mass. This is the same as the way your car will roll down a hill if you park it with the brakes off and the transmission in neutral. The steeper the hill, the faster it will roll. So the deeper the gravity well, that is the more massive the object, the steeper the slope of the warp. The slope of course isn't constant, it varies inversely as the square of the distance between the objects. That's why gravitation "force" varies with distance.

As far as the GPE (gravitational potential energy) of an object from space is concerned, the initial example was a stone raised from the surface of the Earth, GPE doesn't really require that the object be raised from the Earth. GPE is really the amount of energy that the object can gain in falling to the Earth.

Hope this helps.

Bill Gill

Thanks, Bill, I'm always on the lookout for someone to do some thinking for me.



This is more or less how I thought of it until I started reading PS books, and, even worse, thinking about what I read.

Looking back through my notes, I find myself arguing that the GPE of (eg) a stone resting on a shelf 3m above the ground cannot simply be the result of my having picked it up and placed it there, because if I dig a pit under the shelf then push the stone off, it will fall to the bottom of the pit.

Thinking along these lines did bring me to a sort of conclusion, but I'm going to find my notes, and see if they still make sense to me before say anything I might regret.

I’ve found the appropriate notes. I should explain that I wrote these notes as though I were trying to explain the issues to someone with little or no knowledge of the subject, so please don’t be offended by the tone.

It appears that the degree of curvature of spacetime is directly related to both the mass and density of the body causing the curvature. For example, a body of the mass and density of the sun will cause relatively gentle curvature over a large area. If this mass were compressed to the size of the Earth, the curvature of spacetime around it would be much more severe. In terms of the rubber sheet model, the depression in the sheet becomes deeper, and steeper sided, either as a result of an increase of the mass within it, or as a result of the compression of that mass. Given a situation in which an enormous mass, such as the total mass of the Universe, is compressed into an unthinkably small “speck”, we might just be forgiven for referring to the resulting curvature of spacetime as “infinite”. This, we are told, approximates to the state of the Universe at the instant of the Big Bang. If this is the case, it follows that every particle of matter and energy in the Universe, at the start of its life – or of this cycle of its life – occupied the same point in spacetime. The energy, whatever its source, that caused this infinitesimal, primordial speck to expand, transforming itself into billions of light years of spacetime would also have caused the curvature of spacetime to expand as well, and to “soften”, but, it would always remain curved, thus it would always tend to return to its original condition, like the rock falling back to Earth once the restraining force has been removed. This would mean that the energy which drives gravitational attraction is the potential energy imparted to every particle in the Universe by the Big Bang. Thus, there is sufficient potential energy within the Universe to bring every particle back to an infinitesimally small speck. In this scenario every particle in the Universe contains enough gravitational potential energy to bring it back into contact with every other particle. Every particle distorts spacetime around it to a minute degree. As particles clump together, not only are their masses added together, but so is their power to distort spacetime. What is more, without a continued expenditure of energy to prevent this clumping from taking place, it must continue until all the matter and energy in the Universe has returned to its starting point. This would imply that the real mystery is not where the energy of gravity comes from, or why it seems to be inexhaustible, but rather where the energy comes from that is causing the expansion of the Universe not just to continue, but to accelerate, as modern observations assure us that it is.

Bill S. Now you are getting into some deeper stuff. When it comes to the Big Bang I think I will offer you a reference to a science blog I have been following

Starts With A Bang is written by a cosmologist who tries to explain the universe. There are a number of places in his old blog entries that give a pretty good idea of how the Big Bang worked, along with dark matter and dark energy.

Bill Gill

Thanks for the link, Bill, I shall have to find some time to study it.

Did you spot any howlers in my previous post that I should re-think?

Bill S.

I have a few questions about some of your assumptions, but I'm not going to comment on them right now. Instead I am going to give you a different link to Starts With A Bang.

The Greatest Story Ever Told.

This is a link to the first installment of the story of the life of the universe on Starts With a Bang. There are at least 7 installments, if there isn't a link from this one to the next one just go to the archives page and search for Greatest Story Ever Told. That should locate all of them.

When you have finished those I hope you will have a fairly good understanding of how the Big Bang and the expanding universe work.

Bill Gill

Bill G,

Time's a bit limited at present, but I have dipped into the material you recommended. Looks good.

I think I already had a fairly good understanding of the BB, at a Pop. Sci. level, but this gives the clearest explanation of inflation I have seen.

I intend returning to do more justice to the material when time permits, but in the meantime, are there any particular references to the origin of gravity I could take a short cut to?

Bill S.

I don't really have any thing that I can completely recommend, but I did a little research by Google, and came up with a couple.

http://en.wikipedia.org/wiki/General_relativity
http://www.ws5.com/spacetime/

The Wiki article seems to be in pretty good depth, but may be a bit technical. I would have to work on it to figure it all out myself. The second one may be a good starting point. It has a long list of links that may provide a lot of answers. I didn't check out all of either one of them. Now I may have to go back and start checking there.

Bill Gill

Thanks again for the links, Bill. I hope to have a chance to follow them up in the next day or so.