Exploring Geometric Shapes of Gravity

[Hoku]

I'm trying to gain a deeper understanding of gravity from a purely geometric point of view (as opposed to the more Newtonian "force" point of view). This thread is the result of a discussion that began in another thread. However, that should not cause a problem for people who are just joining.

The essential question is, why (or how ) matter "chooses" one particular geodesic path over another. For example, what is the explanation for why an apple follows a geodesic back to Earth rather than following a different geodesic to somewhere else?

DaleSpam has already provided some answers to get this moving. He has emphasized that I'm asking about local geodesics. A local geodesic is not concerened with a "destination" but only with maintaining a purely straight path. To define a local geodesic, all we need to know is an objects starting point and it's direction ("tangent vector"). I now understand this simple concept and can differentiate it from a global geodesic. Global geodesics are paths not defined by tangent vectors, but by the straightest way between two points. (I think global geodesics can still be defined locally if you're looking at the properties of a manifold?).

At any rate, this should catch everyone up with the issue. I will make a separate post in which I've singled out one of DaleSpam's answers that will help us sort through the confusion of gravity.

 

[dx]

Like you said, it depends on the initial velocity of the apple. Given the initial position and velocity of the apple, the path that it follows is uniquely determined. So if the velocity was greater than the escape velocity of Earth, it will not fall back to earth. It still moves on a geodesic, but this geodesic will not fall back to Earth.

You can imagine this in space rather than spacetime. Imagine the surface of a sphere for example. At any given point on the sphere, there will be many geodesics passing through that point. Given a point and also a direction at that point however, a unique geodesic is determined.

 

[Hoku]

I think I understand escape velocity. That is to say, I always thought I understood it. Unfortunately, the way I always understood it was that it was the force needed to escape another force. But I'm trying to understand this geometrically, which means "escape velocity" becomes more of a challenge for me. I run into trouble when it gets to the "limit" of a geodesic. I guess I'm not understanding why speed should affect the shape or size of a geodesic.

DaleSpam says:
"That is due to the direction of the curvature around the Earth (Schwarzschild metric). Spefically, time runs slower at your feet than at your head[...] This curvature in the time direction geometrically means that timelike geodesics curve downwards."

This confuses me on two points:

1) Circular cross-sections of spacetime are the same in all directions. So, if I were to compare it to water pressure, I would think this would even things out. Water pressure doesn't affect us when we dive because it is uniform around us. So why can't we just go straight in spacetime??

2) It seems that there should be an inverse proportion between space and time. So, time may be slower at our feet, but our feet aren't trying to get as far as our heads are, either. So I don't understand how this would make us curve "down". Shouldn't this balance out and give us the ability to go straight (I guess I mean "straight" in a more global sense)?

I know my questions and ideas about this may seem stupid, but that's why I'm here. Please maintain patience.

 

[Dale]

1) Circular cross-sections of spacetime are the same in all directions.

Hi Hoku, I am not sure what you mean by this. Could you elaborate?

 

[guy_incognito]

Water pressure doesn't affect us when we dive because it is uniform around us. So why can't we just go straight in spacetime??

I don't think that's true. If you dive head first your head will be 6 feet lower than your feet, and will be experiencing a different pressure than your feet. But a cross section of your body where every point is the same depth would have the same pressure (is that what you mean?). So really the water pressure varies at each depth, but is uniform spreading outward in concentric spheres. Its like an onion where the outside is low pressure, but each layer inside is incrementally higher in pressure.

 

[yuiop]

Take a look at this image representing time and space as a curved surface:

The green line represents the path of a particle rising to its maximum height before falling back down again, while following the straightest possible path through spacetime.

In this second image:

the purple line indicates the path of an orbiting particle (constant height) and the light blue path is (possibly) the path of a particle with escape velocity. Again, the paths are the straightest paths through curved spacetime given an initial positon and velocity.

(I am not sure if to be exact, each particle would require its own unique curved surface calculated from its initial position and velocity. Anyone know?)

See http://www.rpi.edu/dept/phys/Dept2/Courses/ASTR2050/CurvedSpacetimeAJP.pdf"

 

[Hoku]

@ DaleSpam: I see where this would be confusing because a "circular cross-section" implies something that is "flat", yet, spacetime is not flat. However, I don't think this changes my analogy. If we visualize spacetime as having color changes that correlate with it's curve, my analogy still holds (for now). When you take those "flat" cross-sections, with the line of your body perpendicular to the Earth as the center, the color gradient is the same in all directions. So why can't we go "straight" in a more global sense, if the inclination to fall is the same in all directions?

@ guy_incognito: You're right that the water pressure increases as you go deeper, but that water pressure doesn't affect you (air spaces aside, of course) because it's the same in all directions. In other words, I can remain motionless or make "global" beeline to the surface, as I choose, without water pressure inclining me in any other direction.

 

[yuiop]

...
1) Circular cross-sections of spacetime are the same in all directions. So, if I were to compare it to water pressure, I would think this would even things out. Water pressure doesn't affect us when we dive because it is uniform around us. So why can't we just go straight in spacetime??

When you dive, the pressure on the lower half of your body is greater than the pressure on the top part of your body and this difference in pressure is what causes the upwards buoyancy force. If you put a bubble of air in a tank of water that is far from any gravitational source, the air bubble won't rise because there is no gravity to create a pressure gradient in the water. Divers often have weighted belts so that they have neutral buoyancy, so that the buoyancy force acting upwards is matched by the gravitational force acting downwards. Its not quite accurate to say that the differential pressure acting on the diver has no effect, because without the upward buoyancy force, he would sink like a skydiver jumping out of a plane.

...
2) It seems that there should be an inverse proportion between space and time. So, time may be slower at our feet, but our feet aren't trying to get as far as our heads are, either.

.. but your head is trying to get to where your feet are as you will find out if you relax your muscles when standing. As Dalespam hinted at, things tend to gravitate from where clocks tick fast to where clocks tick relatively slower.

 

[Hoku]

I'm not trying to compare this with buoyancy. I know it can be difficult to separate buoyancy from physical pressure, but I'm thinking of it in terms of having 3-atmospheres of pressure on one side of your body, say with a really strong firehose, vs. 3-atmospheres of pressure ALL OVER your body, say 60+ feet underwater. A fire hose will knock you down and hurt you, but 3-atmospheres underwater won't "crush" or hurt you in any way (again, air spaces aside). That's how I mean the analogy to be understood.

kev, thanks for including the nice graph! Unfortunately, the reason this graph doesn't work for me is because it's geometry is NOT the same as the geometry of real-life spacetime. This means that the geodesics in this graph cannot be compared to the geodesics of an object in spacetime. This graph let's space be infinite but the "time" portion of it is limited and closed back on itself, which is the ultimate reason why the line comes back to 0.0m. I think the graphs on these links are better, but they still don't clarify the issues I'm trying to understand.

http://www.adamtoons.de/physics/gravitation.swf
http://www.adamtoons.de/physics/relativity.swf

 

[Dale]

@ DaleSpam: I see where this would be confusing because a "circular cross-section" implies something that is "flat", yet, spacetime is not flat. However, I don't think this changes my analogy. If we visualize spacetime as having changing color changes that correlate with it's curve, my analogy still holds (for now). When you take those "flat" cross-sections, with the line of your body perpendicular to the Earth as the center, the color gradient is the same in all directions. So why can't we go "straight" in a more global sense, if the inclination to fall is the same in all directions?

OK, I am not 100% sure I am getting your point. Are you somehow under the impression that the curvature of spacetime requires some spherical asymmetry in gravitation such that falling objects should be deflected in some horizontal direction? I'm not trying to change your analogy, I am just trying to understand what you are asking here.

Before we continue let me ask you a very quick question about worldlines in flat spacetime. When we are doing geometry in the flat spacetime of special relativity it is traditional to draw a horizontal axis and label it x for space and a vertical axis and label it ct for time (usually with using units where c=1). In such a spacetime diagram do you understand how a vertical line represents an object at rest, a 45º line represents a pulse of light, a straight line at some angle inbetween represents an inertially moving object, and a curvy line represents a non-inertial object? Is that all clear to you?

 

[yuiop]

This graph let's space be infinite but the "time" portion of it is limited and closed back on itself, which is the ultimate reason why the line comes back to 0.0m.

The time portion is not limited, it just cyles around repeatadly, so n revolutions is n seconds and n can be infinite. If the light blue path has escape velocity it could cycle around the time dimension an infinite number of times as it heads towards spatial infinity and the orbiting particle can also cycle around an infinite number of times.

 

[jfy4]

I had a really great thought just yesterday about that same question. I was in class and i had come to a conclusion that the reason a particle travels a particular geodesic is multiplicity. Much like entropy, where why does a system with half the gas on one side come to equilibirum, its becuase of the multiplicity of the system to be in that macrostate.

So the reason a particle travels that geodesic is that it is overwhelmingly more likely to take that path, and that is the path that maximizes entropy.

So I immediately went online to see if anyone had thought of that idea. And behold... a recent popular paper almost has the same idea, I was 2 months too late... its fairly cool. The man re derives both Newtonian mechanics and Einsteins equations from entropic force. but there are things to critique for sure. See if you can find them!

http://arxiv.org/abs/1001.0785

 

[Hoku]

Are you somehow under the impression that the curvature of spacetime requires some spherical asymmetry in gravitation such that falling objects should be deflected in some horizontal direction?

My impressions are forming as we speak. Here's the best I can do right now:

My impression is that spacetime curves around the Earth with uniformity and symmetry. Any "asymmetry" would have more to do with OUR asymmetrical position in relation to spacetime. For example, if I'm not symmetrical in spacetime, I will fall either to the left, right, front or back. But if I maintain symmetry I don't fall.
EDIT: So, applying this to my analogy, why can't we maintain this symmetry and keep going "up" without altering this "symmetrical" orientation to the Earth. Does this make sense??

[...]do you understand how a vertical line represents an object at rest, a 45º line represents a pulse of light, a straight line at some angle inbetween represents an inertially moving object, and a curvy line represents a non-inertial object? Is that all clear to you?

Not even kind of. The vertical line is easy to understand but the other two are not even remotely intuitive for me. I can't even pretend. Are there some keywords you can refer me to so I can research it, ala google??

 

[yuiop]

I had a really great thought just yesterday about that same question. I was in class and i had come to a conclusion that the reason a particle travels a particular geodesic is multiplicity. Much like entropy, where why does a system with half the gas on one side come to equilibirum, its because of the multiplicity of the system to be in that macrostate. ... http://arxiv.org/abs/1001.0785

Roger Penrose touches on a similar idea in his "Road To Reality" book (p706/7). He points out a cloud of gas particles in one half of a box expanding to fill the box is an example of increasing Entropy, as is a cloud of hot gas mixing with cold gas until they come into thermal equilibrium. A cloud of gas particles collapsing to a clump under gravity, which tends to heat the gas and create thermal gradients, appears to contadict the laws of thermodynamics, unless gravitational collapse is taken to be a form of increasing entropy. This is supported by the discoveries of Bekenstein and Hawking, that black holes have huge entropy associated with them. I think this is an interesting area for research, but personally I do not like the holographic approach, as taken by Verlinde (and others).

 

[DaveC426913]

When you dive, the pressure on the lower half of your body is greater than the pressure on the top part of your body and this difference in pressure is what causes the upwards bouyancy force. If you put a bubble of air in a tank of water that is far from any gravitational source, the air bubble won't rise because there is no gravity to create a pressure gradient in the water.

I am not sure what you're saying here. Are you claiming that the cause of bouyancy is that the lower part of his body is deeper, and therefore under greater pressure than the upper half? i.e there is a pressure gradient along his body?

Does this not imply that a diver that orients himself horizontally will have less bouyancy than a diver who orients himself vertically?


No, a diver experiences bouyancy the same reason everything else experiences bouyancy - his mass is less than the volume of water he displaces.

 

[yuiop]

I am not sure what you're saying here. Are you claiming that the cause of bouyancy is that the lower part of his body is deeper, and therefore under greater pressure than the upper half? i.e there is a pressure gradient along his body?

Does this not imply that a diver that orients himself horizontally will have less bouyancy than a diver who orients himself vertically?

No, a diver experiences bouyancy the same reason everything else experiences bouyancy - his mass is less than the volume of water he displaces.

More accurately, he experiences buoyancy because his weight is less than the weight of the water his body displaces. That is true, but it only one way of looking at it.

When the diver is orientated horizontally, the pressure differential is smaller and the horizontal cross section it is acting on is greater, than when he is orientated vertically. Since the buoyant force is the product of the differential pressure and horizontal cross sectional area it is acting on, the end result is that the upwards force due to the pressure gradient, is the same which ever way the diver is orientated.

See this PF Library entry for "buoyant force" by TIny Tim. https://www.physicsforums.com/library.php?do=view_item&itemid=123 and for non uniform shapes see http://www.ce.utexas.edu/prof/kinnas/319LAB/Lab/lab 2-HydroStatic Forces/Lab2Statics.htm and http://fluid.itcmp.pwr.wroc.pl/~znmp/dydaktyka/fundam_FM/Lecture6.pdf

 

[yuiop]

My impression is that spacetime curves around the Earth with uniformity and symmetry. Any "asymmetry" would have more to do with OUR asymmetrical position in relation to spacetime. For example, if I'm not symmetrical in spacetime, I will fall either to the left, right, front or back. But if I maintain symmetry I don't fall.
EDIT: So, applying this to my analogy, why can't we maintain this symmetry and keep going "up" without altering this "symmetrical" orientation to the Earth. Does this make sense??

Nobody likes the rubber sheet analogy and it does have its limitations, but it might be helpful to use it here. Imagine the depression created by a heavy ball on a horizontal rubber sheet. This represents a "gravity well". The depression is uniform and symmetrical in the sense that at a given distance from the heavy ball in any direction the gradient is equal. However the fact that there is a gradient is important and it means that although the gravitational potential to the left and right (horizontally) of a given point is equal, the potential above and below the same point is not equal and this presents an asymmetry in the vertical direction. Now if you give a small marble an initial velocity away from the larger ball it moves away and gradually slows down until it stops and falls back towards the larger ball. This slow down and reversal is caused by the gradient. If the marble has too high an initial velocity (simulating exceeding the escape velocity) it will keep going away indefinitely and not return, (if we ignore friction slowing the marble down). Horrible as the rubber sheet analogy is, it is quite good for approximately simulating circular and elliptical orbits too and it does make the gradient and the asymmetry of the gravitational potential visually obvious. However, the rubber sheet demonstrates curvature of space and not curvature of space AND time, so it can not for example demonstrate precession of elliptical orbits.

With a bit of abstraction you could imagine the gradient of the rubber sheet to represent the atmosphere around the Earth, with the steeper gradient near the large ball representing the greater density and pressure of the air near the Earth's surface and the shallow gradient further away from the ball representing the lower density and lower pressure high up in the Earth's atmosphere. Most of the time this air pressure gradient seems unnoticeable in everyday situations, but it is significant and is delectable in your own home between the floor and the ceiling of a room. You can make a simple manometer out of a glass U tube containing some water, connected to an otherwise sealed glass bottle. Place the contraption on the floor and mark the water levels on the U tube. Raise the bottle towards the ceiling and you will see a change in the water levels due the the change in pressure. You might even notice the change just moving from your feet to your head is to me is quite surprising. That might help change your conception that the pressure acting on you in air or water is symmetrical, if you can measure the difference in air pressure between you feet and head using a simple device.

Not even kind of. The vertical line is easy to understand but the other two are not even remotely intuitive for me. I can't even pretend. Are there some keywords you can refer me to so I can research it, ala google??

To get you started there is a convention that spacetime diagrams are drawn such that the vertical time axis is scaled so that c*t = x where x is the horizontal space axis. For example the time axis might use units of seconds and the space axis uses units of light-seconds. Scaled like this, a particle with velocity c moves a distance x that is equal in magnitude to the time t, and is therefore a diagonal line at 45 degrees. This 45 degree line is called a lightlike vector. Anything moving slower than the speed of light has a vector that is orientated closer to the vertical is called a timelike vector and it possible for a particle with rest mass to follow such a trajectory. Two events on a timelike path are causally connected (in principle one event can be the cause of the other event) and the order of those events can not be reversed from the point of view of any observer with any relative velocity, so the relationship between cause and event can not be reversed. All paths at less than 45 degrees from the vertical time axis and above the x axis, are said to be the future light cone or absolute future of the event at the origin. All paths at less than 45 degrees and below the x-axis are said to be in the absolute past or past light cone of the event at the origin. Any events outside the light cone can not possibly be caused by the particle at the origin. Nothing with rest mass can follow a path that is more than 45 degrees from the vertical as that is exceeding the speed of light and such a path is called a spacelike vector. A n example of a spacelike vector is a measure of length when the ends of the rod are measured simultaneously. Two events on a spacelike vector can not be causally related and it is possible for observers with different velocities to disagree on the order of spacelike connected events. All objects with constant velocity are inertial and follow a straight line in spacetime and therefore anything following a curved path in spacetime must be accelerating and be non-inertial. That is enough waffle from me for now. Keywords that might be useful to Google for might be, spacetime diagram chart Minkowski space light cone spacelike timelike Lorentz transformation.

P.S. Congrats to Greg and Jessica on their wedding!

 

[Frame Dragger]

This is fascinating; I never thought I could get or needed a deeper understanding of buoyancy!

@Kev: I truly wish you had been there years ago with that explantion for me. It would have saved me time and a headache! Maybe you can help me with one another type of diagram...

I'm poring over Gravitation (MTW) right now, and I'm trying to learn how to interpret Kruskal Diagrams (Kruskal-Szekeres vs. Schwarzschild). Given time I can do it, while referring to MTW, but to blunt... it's heavy! Can you recommend and article or book that might help with this?

P.S. I don't know them, but I saw the pics and read the article; that is a GREAT wedding, and congrats on the incoming kidlet! Mazeltov!

 

[Dale]

My impression is that spacetime curves around the Earth with uniformity and symmetry. Any "asymmetry" would have more to do with OUR asymmetrical position in relation to spacetime. For example, if I'm not symmetrical in spacetime, I will fall either to the left, right, front or back. But if I maintain symmetry I don't fall.
EDIT: So, applying this to my analogy, why can't we maintain this symmetry and keep going "up" without altering this "symmetrical" orientation to the Earth. Does this make sense??

There is spherical symmetry, meaning that there is no tendency to fall left or right (azimuthal or polar angles), but there is asymmetry in the radial direction, meaning that up is asymmetrical from down. Specifically, time is always more curved in the down direction than in the up direction.

Not even kind of. The vertical line is easy to understand but the other two are not even remotely intuitive for me. I can't even pretend. Are there some keywords you can refer me to so I can research it, ala google??

Here are a couple of good starting places (I prefer the first):
http://othello.alma.edu/~jensens/teaching/classes/spacetime.pdf
http://zope.mpiwg-berlin.mpg.de/living_einstein/teaching/1905_S03/pdf-files//Minkowski.pdf

Please look at these as a start and, if you still find something confusing, then I can try to help. This part is reasonably important because these flat-spacetime diagrams are really the beginning of the geometric understanding of spacetime, and in a small enough local region you can always approximate a curved spacetime as locally being flat.

 

[Hoku]

Scaled like this, a particle with velocity c moves a distance x that is equal in magnitude to the time t, and is therefore a diagonal line at 45 degrees.

Your description of this (not just the part I quoted) was very simple and easy to understand. It gave me a much better foundation to understand what the other websites were describing. I would've had a much greater struggle were it not for your post. Thank you!

[...]a vertical line represents an object at rest, a 45º line represents a pulse of light, a straight line at some angle inbetween represents an inertially moving object, and a curvy line represents a non-inertial object?

I understand the difference between inertial and non-inertial movement and I CAN visualize them as straight and curvy lines, respectively. I can't say I have a FIRM grasp on it, but I'm reasonable confident and I think it's sufficient to move forward with.

There is spherical symmetry, meaning that there is no tendency to fall left or right (azimuthal or polar angles), but there is asymmetry in the radial direction, meaning that up is asymmetrical from down.

I think I understand this. This "up/down" asymmetry is why I was using the orientation of a standing person to define symmetry from asymmetry.

Specifically, time is always more curved in the down direction than in the up direction.

This is where we consistently hit a wall. This single point is a serious hang-up for me. I keep trying to explain why I'm confused but, because my understanding is SOOO undeveloped, I think it becomes difficult for others to understand my confusion.

I think it can be boiled down to the question I posed in post #3. Why does speed put limits on the shape or size of a geodesic? Again, I can easily understand this if gravity is a force, but I'm having trouble reconciling it from a purely geometric point of view.

I don't understand why time dilation should make us turn back. Clocks may be moving slower, but they're still moving! Why should that make us turn around? It seems that if we have the correct direction in space, we can compensate for any curvature in time.

 

[DaveC426913]

I don't understand why time dilation should make us turn back. Clocks may be moving slower, but they're still moving! Why should that make us turn around? It seems that if we have the correct direction in space, we can compensate for any curvature in time.

Can you elaborate? What you you mean by "make us turn back"? It sounds like you're confusing unrelated concepts.

 

[Dale]

I think it can be boiled down to the question I posed in post #3. Why does speed put limits on the shape or size of a geodesic?

OK, this is probably the only point where I will try to get you to consider geometry on something besides a sphere. The shape of a geodesic depends on 3 things, first is the shape of the underlying spacetime itself, second is the initial position, and third is the tangent vector from that initial point (i.e. the four-velocity).

Here is a page that has several examples of different geodesics on a torus:
http://www.rdrop.com/~half/math/torus/geodesics.xhtml

Note that most of the radically different shapes that you see cross the inner equator at some point, so all of those only differ in the tangent vector at the inner equator. Therefore, a simple change in direction can radically change the shape of the geodesic because the geodesic then crosses different regions with different curvatures at different angles than it would have crossed had another tangent been chosen.

 

[Hoku]

Can you elaborate? What you you mean by "make us turn back"? It sounds like you're confusing unrelated concepts.

I know that if you go in a circle, you are not "going back", however, you DO go from 0m, 1m, 2m, 3m and then BACK to 2m, 1m, 0m. So you ARE passing through the "same" places in that sense. If I jump up to reach a branch, I end up, essentially back where I started; not in time, but certainly in space. I guess another way to approach this confusion is to ask, "why must movement in time restrict us to one direction in space (mass-wards)?"

If we are playing soccer, we can stop a ball with our foot. Now that it's stopped, we can kick it any other direction we want and not expect it to "boomerang" back. WE are stopped on Earth's surface but time moves to the future no matter how far from the surface we go. So why can't we keep going? How does time boomerang us back?

Here is a page that has several examples of different geodesics on a torus:
http://www.rdrop.com/~half/math/torus/geodesics.xhtml

I can't seem to open the xhtml file. I tried finding software to help but my computer isn't letting me use it. I do have a few torus pictures with lines in the book I got, "The Shape of Space", but I'm not sure which, if any, would help.

The shape of a geodesic depends on 3 things, first is the shape of the underlying spacetime itself, second is the initial position, and third is the tangent vector from that initial point (i.e. the four-velocity).

Ok, I don't mean to be letting thoughts about "forces" into my head, but I keep coming back to it because it's the only thing that makes sense to me. Don't get upset, just help push the idea back out. You said, "The shape of a geodesic depends on 3 things[...]" So I'm thinking these are the "mysterious" initial conditions that we accept as a starting point. Is this right? But it really seems to me that all three of these conditions can be defined by the internal energies that things have. In the other thread, we began discussing the orbit of a planet. I asked if the shape of its geodesic is the result of the lesser mass' energy opposing the greater mass' energy trying to pull it in. It seems to me that the "push/pull" of these forces is what defines all three of the initial conditions.

 

[DaveC426913]

WE are stopped on Earth's surface but time moves to the future no matter how far from the surface we go. So why can't we keep going? How does time boomerang us back?

I do not understand what you are saying. Why are you saying time brings us back? Back to where?

What have you been reading that is causing you to express ideas like this?

 

[Hoku]

I do not understand what you are saying. Why are you saying time brings us back? Back to where? What have you been reading that is causing you to express ideas like this?

I appreciate you pursuing this, DaveC, because I think this is important. A few weeks ago, gravity wasn't confusing to me at all. Back then, gravity was a force. But, as a result of the thread, "spacetime...3+1" (relevant discussion begins on page 4), I became aware that many people don't see it as a foce at all, only as geometry. This brought up all sorts of questions for me that were never an issue when gravity was a "force". The first meaningful question I had from this new view of gravity is quoted below:

"So, if gravity is described as geodesics, then light/matter, whatever, must have momentum in order to follow those geodesics. right?

Humans are stuck to the Earth because of gravity, but what is the momentum that we have that makes use of the geodesics? As far as I can tell, we are at rest relative to gravity at the Earth. So why do we keep following the geodesic? Why are we actually stuck to the Earth?"

This question brought lots of insults from some people, but others were more willing to help me through the problem. I will list three answers provided by three different people addressing this question.

1) The apple is at rest in space relative to Earth. But it still advances in space-time, and that is where it follows a geodesic, once it detaches from the tree.
2) The curvature is that of spacetime, and we are always moving in spacetime, because we are always moving in time.
3) This curvature in the time direction geometrically means that timelike geodesics curve downwards.

Now, I could be misunderstanding these, but they seem to be saying that we are stuck to the Earth because we are moving in "time". So, my question is, why does time care whether we go "Earthwards" or "upwards"?

 

[atyy]

1) The apple is at rest in space relative to Earth. But it still advances in space-time, and that is where it follows a geodesic, once it detaches from the tree.
2) The curvature is that of spacetime, and we are always moving in spacetime, because we are always moving in time.

Yes. Note that there is more than one time direction at each point in spacetime - there is a whole cone of them.

3) This curvature in the time direction geometrically means that timelike geodesics curve downwards.

Now, I could be misunderstanding these, but they seem to be saying that we are stuck to the Earth because we are moving in "time". So, my question is, why does time care whether we go "Earthwards" or "upwards"?

Some curve downwards, others upwards. Which one you take depends on your "initial condition". The Earth (and any other mass-energy in the universe) determines how each and every geodesic curves.

 

[Dale]

I can't seem to open the xhtml file. I tried finding software to help but my computer isn't letting me use it. I do have a few torus pictures with lines in the book I got, "The Shape of Space", but I'm not sure which, if any, would help.

Try this link instead, there is a lot of math but you can start with the pictures of geodesics on pages 10-12: http://www.rdrop.com/~half/math/torus/torus.geodesics.pdf

 

[Dale]

But it really seems to me that all three of these conditions can be defined by the internal energies that things have.

It is more than just the energies involved. The stress-energy tensor is what relates the distribution of matter and energy to the curvature of space. It has 1 term for energy, and 15 terms for other things including pressure, momentum, energy and momentum flux, stress, etc. All of these things together determine the spacetime curvature which is, in turn, one of the initial conditions.

 

[A.T.]

So, my question is, why does time care whether we go "Earthwards" or "upwards"?

Why does time care?

We simply assume time to be a curved dimension, because the resulting geometry of the geodesics in space-time fits the observation of falling objects.

 

[Naty1]

I guess I'm not understanding why speed should affect the shape or size of a geodesic.

Spacetime curves according to (a) mass, (b) energy, (c) pressure...it's NOT necessarily intuitive that I can tell...unlike a fixed geodesic, say a great circle on a sphere, a spacetime geodesic is ever changing according to the influences a,b,c,.

On the other hand, you can see how a path in space varies according to speed vairations by firing cannon balls of fixed mass with different powder charges.

Time changes "intuitively" with speed via special relativity.

 

[Dale]

So, my question is, why does time care whether we go "Earthwards" or "upwards"?

Let's consider a somewhat related question that may be easier for you to visualize where we are considering curved coordinates on a flat space. Specifically, let's consider geodesics in a plane (straight lines) from the perspective of polar coordinates. Now, in terms of polar coordinates all geodesics get deflected "outwards". Why do polar coordinates care whether a geodesic goes outwards or inwards? Do you have an intuitive feel for that?

 

[Hoku]

I'm optimistic right now because it seems like the geometric understanding, as well as the "force" reconcilation, that I'm looking for is close. Then again, I could be painfully wrong.

I've been having trouble finding time to finish this post. I should be able to get back to it later today.

 

[Hoku]

I want to be sure too many people don't get hung up on the use of the word "care" as it relates to space or time. It's used metaphorically. I don't think time or space has "emotions". These kinds of metaphors are helpful for me and I hope people can find the grace to let me use them.

Specifically, let's consider geodesics in a plane (straight lines) from the perspective of polar coordinates. Now, in terms of polar coordinates all geodesics get deflected "outwards". Why do polar coordinates care whether a geodesic goes outwards or inwards?

I think the only thing time "cares" about is that it isn't repeated. Otherwise, I don't think either coordinate cares what happens on the "canvas". Time requires movement "upwards" on the graph, but I don't think it cares about direction in space.

I've been up and down with this all day (no graph pun intended). Sometimes optimistic that progress is happening (thanks to the torus diagrams) and sometimes worried that I'm still missing it. Here's how I've organized the rest of this post:
First, I've re-establish the primary problem. Second, I've brought in answers that have been provided. Third, I've paraphrased and elaborated on these answers in case there's a flaw in my interpretation. Fourth and finally, I've described exactly why these answers confuse me. For the first time, I think I can do this effectively and that's where my greatest hope lies.

1) Primary problem:
If gravity is not a force, then how are we stuck to the Earth?

2) Answer:
Because we are following a geodesic into it. We are never at rest because we are always moving in the "time" direction, which curves down (Earthwards).

3) Answer Paraphrased:
A geodesic is a path in spacetime. Because space and time are inseparable, following a geodesic in time necessitates movement in space in some way. This is why we "stick" to the Earth. We're trying to follow the geodesic spatially, but electromagnetism prevents our movement, which leaves us pressed into the earth.

4) Why this doesn't make sense:
Part 1: Even when we're stationary on the Earth's surface, we're still SUCCESSFULLY moving through space. We're stationary relative to the Earth, but not the sun, or stars. Our actual movement through space is congruent with the spinning of the Earth and because the Earth is spherical, it's also in the "downward" direction. Therefore, we have an outlet for movement in space. This makes "pressing" into the Earth seem redundant.

Part 2: It seems like there are two geodesics that are simultaneoulsly in effect when stationary on the Earth's surface. The first is a local geodesic and it's the one I described in "Part 1". The other is more "global-like".

In the global-like geodesic, we have a beginning point and and ending destination. The beginning point is the spot on which we are standing. The destination is the center of the Earth. The radius from Earth's surface to Earth's center is NOT curved and it's the fastest way. Part of what I've been trying to ask is why this radius can't be extended through and past the atmosphere. This would be a "straight" geodesic that shouldn't require our return to Earth if we're looking at it from a purely gemetric viewpoint. Of course, because the Earth is spinning, it wouldn't actually be straight but I'd think it could be simulated.

I also wanted to address posts #28 and #30, but I think I've made a big enough mess.

 

[yuiop]

This is fascinating; I never thought I could get or needed a deeper understanding of buoyancy!

@Kev: I truly wish you had been there years ago with that explantion for me. It would have saved me time and a headache! Maybe you can help me with one another type of diagram...

I'm poring over Gravitation (MTW) right now, and I'm trying to learn how to interpret Kruskal Diagrams (Kruskal-Szekeres vs. Schwarzschild). Given time I can do it, while referring to MTW, but to blunt... it's heavy! Can you recommend and article or book that might help with this?

Hi Dragger,
I think it would be great to discuss the KS metric with you. I should warn you that my personal interpretation is not the conventional one, but in keeping with rules of this forum I will try and keep to the conventional interpretation. I think you should also start a new thread to avoid hijacking this thread and specify the aspects of the KS metric that you find puzzling.

I know that if you go in a circle, you are not "going back", however, you DO go from 0m, 1m, 2m, 3m and then BACK to 2m, 1m, 0m. So you ARE passing through the "same" places in that sense. If I jump up to reach a branch, I end up, essentially back where I started; not in time, but certainly in space. I guess another way to approach this confusion is to ask, "why must movement in time restrict us to one direction in space (mass-wards)?"

Hi Hoku,
Have another look at the link you posted earlier http://www.adamtoons.de/physics/gravitation.swf

Click play and watch the motion of the particle on the rectangular time and space chart on the left. The particle rises to its maximum height and falls back down again. Now imagine playing the animation backwards (reversing time) and it is easy to see that the particle still rises and falls back even when time is going backwards! So it is not the direction of time that determines this rising and falling path. It is the way the time dimension curves with respect to the space dimensions (together with the initial location and velocity of the particle) that determines the path followed by the particle. This curving of the time and space dimension is illustrated by the bulbous cylindrical surface in the 3D diagram on the right. Here time is the circular cross section of the distorted cylinder and wraps around, as you can see by watching the particle cycle around. If the animation was longer, the particle would oscilate to and fro indefinitely (assuming a tunnel through the massive body and no friction). This is the natural state of the particle with NO forces acting on it and it is following a geodesic. If the tunnel is blocked the particle is no longer following a geodesic and it feels a force acting on it (the same force you feel acting on your feet when you are standing) and so it no longer has inertial motion, which is defined as the motion of a particle with no forces acting on it.

If you mentally cut off the the right third of the distorted cylinder in the adamtoons animation, you essentially get the curved spacetime funnel shape in the link I gave earlier here http://www.rpi.edu/dept/phys/Dept2/Courses/ASTR2050/CurvedSpacetimeAJP.pdf"

4) Why this doesn't make sense:
Part 1: Even when we're stationary on the Earth's surface, we're still SUCCESSFULLY moving through space. We're stationary relative to the Earth, but not the sun, or stars. Our actual movement through space is congruent with the spinning of the Earth and because the Earth is spherical, it's also in the "downward" direction. Therefore, we have an outlet for movement in space. This makes "pressing" into the Earth seem redundant.

You should not get too hung up on the effect of the stars and the Sun in this situation. You will still be pressed to the Earth even if the Sun and stars are absent and if the Earth is not spinning. There is no requirement to have an "outlet" for movement in space, but as long as you are not exactly following a geodesic, you will feel a force acting upon you. Put another way, if you require that a particle must have inertial motion (and feel no forces acting on it), then the particle must follow its geodesic though spacetime and any movement in the space diirections that is not exactly the same as the spatial components of the geodesic path, will not meet that requirement and so the spin of the Earth abouts its own axis and the orbit of the Earth around the Sun will not do the trick.

 

[Frame Dragger]

@kev: Understood, I'll do that this weekend or monday. Thanks!

@Hoku: Remember that Earth, along with everything including our local cluster, is hurtling "outwards" from the presumed BB point of origin, at a fairly impressive rate relative to that presumed origin point.

Even if we were not, moving in time-like geodesics is what we do. We could be alive, utterly still in a vacuum, and we wouldn't REALLY be utterly still. To be alive, even on the (biological) microscropic level, requires constant motion and exchanges of molecules and charges, etc. All of that follows a time-like geodesic too. To meet our definition of LIFE, you must follow a time-like geodesic (for starters only) to participate in thermodynamic processes' which underly life (we eat, we radiate heat, and create material waste with less chemical energy than what "went in").

Kev is right in forgetting celestial bodies, but you can really just think of a single cell in a medium (agar) being alive. If we can call it alive, it's experiencing the passage of time, does not exactly follow a space-like geodesic, and will experience a ficticious force called gravity, even if it is only from the medium, and its own mass. Reduce to absuridity as needed!

Anyway, remember the parable of the apple, but now apply it to curved time. To "get" from one moment to the next requires that we follow time-like geodesics, or we're not getting anywhere at all.