Spacetime curvature implies gravitation

[md65536]

Imagine a volume with flat spacetime, inside of which is several particles whose movement is described as a random walk.

We would expect that the particles would approach even distribution, just as gas pressure approaches equilibrium in a container.

 

 

Now imagine that the same volume consists of highly curved spacetime.

For this thought experiment, let us suppose that the volume looks to us, as outside observers, as a long uniform cylinder,

but from an inside perspective, the volume looks more like a cone, where one end of the cylinder is wide and the other narrow.

 

From an inside perspective, we would expect that the particles would tend to evenly distribute, so that more would be found in the wider larger side.

From an inside perspective, we see nothing weird... no "force"... just particles drifting randomly.

 

From outside, we see more particles migrating toward the side that looks wider from an inside perspective.

From here, we see a weird force: Something is drawing the particles to one side of the cylinder, where they remain with greater density than on the other side.

 

Is this exactly what would happen with a cylinder in a gravitational field? Say on earth, with an upright cylinder. The spacetime curvature is very very slight, but then so is the pressure difference between the 2 ends of a small container.

 

 

 

The force on these particles would be very slight in slightly curved space. The force on an apple would only be apparent due to the many many particles comprising the apple. With this interpretation, the particles do not see a "gravitational pull" as we see from an external perspective. They "see themselves" drifting forcelessly through a space that happens to open up on one side (with more room to randomly wander into) more than on the other side.

 

One problem with this is that the spacial curvature caused by the Earth, say, is very small, yet the density of matter in the Earth's core vs upper atmosphere is very different. So the gas pressure example would only be an example, and not an analogy for all particles. This part requires some more thought...

 

Basically, it seems intuitive that if space was "completely open" in one direction and "completely closed" in the opposite direction, then particles would move at c in the direction of open space (there would be no room for the matter to move in the opposite direction, even if that movement just involved subatomic mass energy oscillation). So, I would think that there is some relationship between the ratios of our Earthly "fairly flat" spacial curvature vs. a "maximum curvature", and of the "fairly slow" acceleration due to gravity vs. some instantaneous acceleration to a speed of c.

 

 

Questions:

- Is this a fair interpretation of "how gravity works" according to GR? Is there even an accepted answer to "how" according to GR, as it relates to this discussion?

- Any problems with the assumptions or inferences herein?

 

 

[Spyman]

First I wonder why you are asking here and not in the Relativity subforum?

(IMHO, I think you would get more qualified answers by asking there.)

 

Secondly I think you are mixing a lot of things:

¤ A uniform cylinder would continue to appear uniform for those inside it, when it gets "submerged" in gravity.

¤ An upright cylinder on Earth is more curved on the downside, so particles are moving toward the narrow side.

¤ Gravity itself is not a pressure/equilibrium action difference in a media, it acts individually on all particles.

 

 

Like a moving object have a trajectory through space, all objects have a path they follow through spacetime and without gravity these paths are straight but gravity curve them towards its source. Objects in free fall follows those paths and objects forced out of their natural paths feel an acceleration.

 

Particles in this uprised cylider have natural free fall paths that are curved downward by the gravity field of Earth and unless something is preventing them they will follow their paths. Since they are locked inside a cylinder their overall paths will be down towards the center of Earth, which will eventually get prevented by the bottom or other particles already resting on it or other particles stuck inbetween, causing an upright acceleration of 9.8g. All particles will stack on the bottom and up, leaving the upper part empty and putting lower parts under higher pressure. If the particles are part of a fluid or gas then the stacking procedure will be more complicated than for several individual solid particles, but the general tendency caused by gravity will be evident.

 

If you want more detailed information you will need to read and learn about Geodesics and ask relativity experts.

[md65536]

First I wonder why you are asking here and not in the Relativity subforum?

(IMHO, I think you would get more qualified answers by asking there.)

Yes, that was a mistake I realized long after posting. The ideas are speculative but all my questions are about GR. Discussing implications of GR is really what I'm interested in.

 

Secondly I think you are mixing a lot of things:

¤ A uniform cylinder would continue to appear uniform for those inside it, when it gets "submerged" in gravity.

¤ An upright cylinder on Earth is more curved on the downside, so particles are moving toward the narrow side.

¤ Gravity itself is not a pressure/equilibrium action difference in a media, it acts individually on all particles.

Now we're getting somewhere!

 

On point 1: One of my assumptions is that "curved space" must always appear to be a different shape from some different perspectives, because locally it appears flat (which is the same as saying that all geodesics that pass through a point appear straight when viewed from that point???). If curved space looked the same (ie flat) from everywhere, then it couldn't be called curved.

So I tend to talk about curved space looking different from "inside" (flat?) and "outside".

HOWEVER, I think that if the space in the cylinder has significant curvature from one end to the other, then a viewpoint that moves around inside the cylinder would see the cylinder change shape as the viewpoint moved. So... would the cylinder always appear uniform, but as you move from one side to the other it appears to change size? (It still seems intuitive that one side would always be bigger than the other.)

 

On point 2: Yes, I think that this is a better way to put it. Okay so if we instead say the cylinder IS uniform, and appears uniform in flat space, then from outside it appears narrower on the side closer to a gravitational mass. From inside, particles look uniformly distributed in a uniform(ish) volume, and from outside they appear slightly more dense in the narrower part.

It seems certain that with significant enough curvature, the cylinder would look distorted from any perspective.

 

On point 3: Yes, there's something I'm missing there. I think I'm failing to consider the effect of acceleration. Curvature causes acceleration -- I think I'm showing that in a hand-wavy way -- but the equilibrium state of gas in a volume depends on its acceleration, not just on the shape (as affected by curvature) of the volume. The particles would *accelerate* due to a random drift through changing space, but the momentum they gain would cause a non-random motion. In equilibrium, it is only gas pressure that counteracts gravitational force and allows the particle's motion to appear to be a "random drift". Perhaps using a single particle in an essentially endless tube would be a better example, and then only its inertial motion would be considered, but accelerated relative motion would have to be accounted for.

 

 

So I'm wrong that the particles would appear uniformly distributed from inside the volume. They would still "feel" that the cylinder is accelerating around them, and would still experience the pressure difference within the cylinder.

Like a moving object have a trajectory through space, all objects have a path they follow through spacetime and without gravity these paths are straight but gravity curve them towards its source. Objects in free fall follows those paths and objects forced out of their natural paths feel an acceleration.

 

Particles in this uprised cylider have natural free fall paths that are curved downward by the gravity field of Earth and unless something is preventing them they will follow their paths. Since they are locked inside a cylinder their overall paths will be down towards the center of Earth, which will eventually get prevented by the bottom or other particles already resting on it or other particles stuck inbetween, causing an upright acceleration of 9.8g. All particles will stack on the bottom and up, leaving the upper part empty and putting lower parts under higher pressure. If the particles are part of a fluid or gas then the stacking procedure will be more complicated than for several individual solid particles, but the general tendency caused by gravity will be evident.

 

If you want more detailed information you will need to read and learn about Geodesics and ask relativity experts.

That all makes sense but I can't figure out how all the details work together. I think I have to simplify my example, and cut out the effects of gas pressure.

 

 

I think I'm trying to speculate about "why" gravity happens as observed:

- Curvature causes space in one direction to appear be more open or "roomy" in one direction vs the opposite direction.

- Random motion, or oscillation, would favor the open direction over the more closed direction.

- When you move in the open direction, space warps around you, so that it continues to open more in the direction toward the gravitational mass. What this means is that once you move in the direction of open space, that space closes up a bit around you so that it is not as easy to move back as it was to move forward. So I speculate that it is not static spacial curvature that explains gravitational acceleration, but it is that the geometry of curved space appears to CHANGE as you move through it, that explains it.

- Random movement or oscillation would continue in all directions, but the slight favoring of one direction would continually add momentum in that direction.

 

Yes... I think that's the way to fix this. It's not the difference between inside and outside the cylinder that is important, but the changing shape of space from one location to the next that's important. The cylinder might only be useful to illustrate the meaning of "open space being more roomy".

 

 

Note: I don't know if I'm getting "open" and "closed" right here. I'm assuming that space opens toward a gravitational mass, because an object falling into that mass will see space appear to "open up and get roomier". Am I backwards?

Edited April 26, 2011 by md65536

[imatfaal]

IS the cylinder a good example for your thought experiment - is it really curved in the intrinsic/extrinsic curvature sense. Surely a triangle on a cylinder is the same as a triangle on a flat plane and Euclidian geometry applies. I am struggling to remember this from half understood lectures.

 

I second Spymans suggestion that you get the thread moved to GR/SR in Physics

 

edit

 

From further recollection jolted by a recent post by ajb that I cannot locate now - you need to consider a vector moving on a closed path around the surface which only changes direction with the path - on a curved surface on completion of the path it is at an angle to its starting position, on a flat surface it goes back to the same angle.

 

further edit

http://math.ucr.edu/home/baez/gr/parallel.transport.html

 

found the post - think I have this right and not a-about-f

http://www.scienceforums.net/topic/56707-trying-to-make-sense-of-gravity/page__view__findpost__p__603451

Edited April 26, 2011 by imatfaal

[md65536]

I think I'm trying to speculate about "why" gravity happens as observed:

- Curvature causes space in one direction to appear be more open or "roomy" in one direction vs the opposite direction.

- Random motion, or oscillation, would favor the open direction over the more closed direction.

- When you move in the open direction, space warps around you, so that it continues to open more in the direction toward the gravitational mass. What this means is that once you move in the direction of open space, that space closes up a bit around you so that it is not as easy to move back as it was to move forward. So I speculate that it is not static spacial curvature that explains gravitational acceleration, but it is that the geometry of curved space appears to CHANGE as you move through it, that explains it.

- Random movement or oscillation would continue in all directions, but the slight favoring of one direction would continually add momentum in that direction.

 

 

I suspect that I got it all backwards so I'll try restating it... backwards...

 

- Curvature means that space will be more closed towards a gravitational mass, however it appears flat locally.

- Local flatness means that random or oscillatory motion in any direction is equally probable as any other direction.

- When moving toward a gravitational mass, the closed space flattens as you move into it (ie. closed space seems to open up as you move into it). Again this means that space warps around you and changes the probability or "ease" of moving back to the location you just came from. -- I can't describe this easily. I might have to draw a diagram.

- Continually flattening space in one direction makes random movement in that direction more likely, which continually adds momentum in that direction.

[md65536]

IS the cylinder a good example for your thought experiment - is it really curved in the intrinsic/extrinsic curvature sense. Surely a triangle on a cylinder is the same as a triangle on a flat plane and Euclidian geometry applies. I am struggling to remember this from half understood lectures.

 

I second Spymans suggestion that you get the thread moved to GR/SR in Physics

I dunno how to get it moved. Either place is good for me. It seems to be half trying to understand GR, and half speculating based on that.

 

I think the cylinder example is bad or misleading for several reasons. I don't think the shape of the volume actually matters, only the relative volumes of specific spaces, and how they change when viewed from different perspectives (most important of which are the changing viewpoints of a gravitationally accelerated test particle).

 

I'll have to check out those links and try to understand them.

 

 

From an external perspective, my example doesn't show anything, because we don't see space curvature significantly changing as a test particle (with low enough mass) moves through it.

From the perspective of the moving particle, space changes a lot as it moves through highly curved space.

 

From an external perspective, geodesics are good for describing what's going on, because they show the differing curvature of space that the test particle moves through.

From an internal perspective, geodesics that I'm moving on are not so good for describing anything, because all geodesics that I'm on appear straight. The changing curvature of perpendicular geodesics that I approach and leave behind, might demonstrate it.

 

 

 

 

[Spyman]

Note: I don't know if I'm getting "open" and "closed" right here. I'm assuming that space opens toward a gravitational mass, because an object falling into that mass will see space appear to "open up and get roomier". Am I backwards?

Maybe I mixed it up too, lets try again:

 

Since the speed of light is constant for all observers, when local time ticks slower deep down in the gravity well, for the local observer than for a distant observer, then local space should seem equally larger for a distant observer than for the local observer.

 

So if an object that is 1 lightsecond long in flat space, is placed on Earth's surface it would locally take 1 second for light to go from one end to the other, but for the distant observer 1 Earth second is ~0.6963 nanoseconds longer.

 

[math] t_{local}=t_{distant}*\sqrt{1-\frac{2Gm}{rc^2}} [/math]

 

G = 6.67428×10^-11 m³kg^-1s^-2

m = 5.9736×10²⁴ kg for Earth

r = 6 371 000 m mean for Earth

c = 299 792 458 m/s

 

Thus for the distant observer the clock ticks slightly more while the same lightsignal travel across the object which means that the distant observer also measures this object to be ~209 millimeters longer down on Earth than up in flat space.

 

Which means that a straight cylinder in flat space moved down to Earth would appear to get streched slightly conical with its base diameter a little larger than its top diameter.

 

 

On point 1: One of my assumptions is that "curved space" must always appear to be a different shape from some different perspectives, because locally it appears flat (which is the same as saying that all geodesics that pass through a point appear straight when viewed from that point???). If curved space looked the same (ie flat) from everywhere, then it couldn't be called curved.

So I tend to talk about curved space looking different from "inside" (flat?) and "outside".

HOWEVER, I think that if the space in the cylinder has significant curvature from one end to the other, then a viewpoint that moves around inside the cylinder would see the cylinder change shape as the viewpoint moved. So... would the cylinder always appear uniform, but as you move from one side to the other it appears to change size? (It still seems intuitive that one side would always be bigger than the other.)

In FLAT space the interior angles of a triangle always add up to 180 degrees and the circumference of a circle is always the diameter multiplied with pi. CURVED space is when those requirements are not fullfilled. As such flat space is where there is no influence of gravity at all and from there the curving increases with the strength of gravity.

 

Locally you can't see the curving because it is a very tiny effect and since there is nothing to compare against when everything curves with space. If you bring your straight ruler there to check it will bend exactly as much as space making space seem straight. So curved space looks flat locally, it's when observers compare their local space and time with a different environment they see time dilation or length contraction.

 

If an observer moves up and down in the cylinder placed upright on a gravity source then he and all equipment he brings with him will also be subjected to the warping of spacetime, so he will not be able to measure any different size locally of the cylinder. However if he measures the top and bottom diameter he would be able to see that the bottom is greater.

 

 

On point 2: Yes, I think that this is a better way to put it. Okay so if we instead say the cylinder IS uniform, and appears uniform in flat space, then from outside it appears narrower on the side closer to a gravitational mass. From inside, particles look uniformly distributed in a uniform(ish) volume, and from outside they appear slightly more dense in the narrower part.

It seems certain that with significant enough curvature, the cylinder would look distorted from any perspective.

Ok, it seems like I messed up, sorry if I turned things upside down for you. As said above the cylinder should be wider closer to the source of gravity and more narrow on the side towards empty space. Which means that from an outside view the media inside the cylinder would appear less dense closer to the source of gravity, before gravity compresses the media towards the bottom.

 

 

On point 3: Yes, there's something I'm missing there. I think I'm failing to consider the effect of acceleration. Curvature causes acceleration -- I think I'm showing that in a hand-wavy way -- but the equilibrium state of gas in a volume depends on its acceleration, not just on the shape (as affected by curvature) of the volume. The particles would *accelerate* due to a random drift through changing space, but the momentum they gain would cause a non-random motion. In equilibrium, it is only gas pressure that counteracts gravitational force and allows the particle's motion to appear to be a "random drift". Perhaps using a single particle in an essentially endless tube would be a better example, and then only its inertial motion would be considered, but accelerated relative motion would have to be accounted for.

I might have misinterpreted you but I think you are also getting into a hypothetical side effect of space warping instead of normal gravity and not only a confusion of pressure/equilibrium effects.

 

 

So I'm wrong that the particles would appear uniformly distributed from inside the volume. They would still "feel" that the cylinder is accelerating around them, and would still experience the pressure difference within the cylinder.

Yes, there is a valid reason for submarines to have a maximum depth, however if the cylinder would be in freefall orbiting Earth the media can't feel any acceleration from the cylinder, but are still inside the range of Earth's gravity. The effect you seem to be targeting should make a difference here, very slightly increasing pressure towards Earth, se below.

 

 

I think I'm trying to speculate about "why" gravity happens as observed:

- Curvature causes space in one direction to appear be more open or "roomy" in one direction vs the opposite direction.

- Random motion, or oscillation, would favor the open direction over the more closed direction.

- When you move in the open direction, space warps around you, so that it continues to open more in the direction toward the gravitational mass. What this means is that once you move in the direction of open space, that space closes up a bit around you so that it is not as easy to move back as it was to move forward. So I speculate that it is not static spacial curvature that explains gravitational acceleration, but it is that the geometry of curved space appears to CHANGE as you move through it, that explains it.

- Random movement or oscillation would continue in all directions, but the slight favoring of one direction would continually add momentum in that direction.

 

Yes... I think that's the way to fix this. It's not the difference between inside and outside the cylinder that is important, but the changing shape of space from one location to the next that's important. The cylinder might only be useful to illustrate the meaning of "open space being more roomy".

Well, I think we start to dip our toes in the speculation area now. My understanding of gravity in relativity is that objects follow straight paths through spacetime due to inertia and that the curving of space causes these straight lines to bend. I am not able to explain it better than that.

 

The geometry of space does not change as you move through it, locally you can not measure any change and from the outside view the only change done to the field of gravity is caused by your mass, which is equal in both directions.

 

However the geometry of spacetime is different closer to a gravity source, so the speculation part that random movement might slightly favoring wider space is beyond my knowledge, but I think if it exists it would be a very tiny side effect and not what we normally mean with the force of gravity.

 

As said above, if this effect is real then it should be possible to verify it by experiment in orbit.

 

 

I suspect that I got it all backwards so I'll try restating it... backwards...

 

- Curvature means that space will be more closed towards a gravitational mass, however it appears flat locally.

- Local flatness means that random or oscillatory motion in any direction is equally probable as any other direction.

- When moving toward a gravitational mass, the closed space flattens as you move into it (ie. closed space seems to open up as you move into it). Again this means that space warps around you and changes the probability or "ease" of moving back to the location you just came from. -- I can't describe this easily. I might have to draw a diagram.

- Continually flattening space in one direction makes random movement in that direction more likely, which continually adds momentum in that direction.

Repeating/restating:

- Curvature means that distances seems greater closer towards a gravitational mass from a distant observer's view.

- Even if it space appears flat for the local observer, it's only flat without influence from gravtity.

- When moving deeper into a gravity well, time passes more slowly and distanses increases from an outside view.

- Local observers are not able to measure any difference since they don't have anything to compare against inside their frame of reference and in small enough areas space is approximately flat.

 

 

I think the cylinder example is bad or misleading for several reasons. I don't think the shape of the volume actually matters, only the relative volumes of specific spaces, and how they change when viewed from different perspectives (most important of which are the changing viewpoints of a gravitationally accelerated test particle).

The apparent change from different perspectives is unable to affect particles, the difference in geometry on the up and down side of the particle is independent of observers.

 

 

From an external perspective, my example doesn't show anything, because we don't see space curvature significantly changing as a test particle (with low enough mass) moves through it.

From the perspective of the moving particle, space changes a lot as it moves through highly curved space.

From an external perspective we do see space curvature change significantly for the test particle moving through it.

In the perspective of a moving particle space is not changing, everywhere along the cylinder the diameter will locally be of the same size but if space is highly curved then the diameter will appear to be greater in the down direction and smaller in the up direction from the current location.

[md65536]

Since the speed of light is constant for all observers, when local time ticks slower deep down in the gravity well, for the local observer than for a distant observer, then local space should seem equally larger for a distant observer than for the local observer.

Please tell me you have that backwards! If I get this wrong yet again that will make it just about every time that I got it backwards, and I'd have to conclude that I have some kind of neurological disorder.

 

Local time ticks at a "normal" rate for all observers. Time ticks slower in a gravity well according to distant observers.

 

If I got that wrong it wouldn't be the first, hundredth, or last time.

 

Thus for the distant observer the clock ticks slightly more while the same lightsignal travel across the object which means that the distant observer also measures this object to be ~209 millimeters longer down on Earth than up in flat space.

 

Which means that a straight cylinder in flat space moved down to Earth would appear to get streched slightly conical with its base diameter a little larger than its top diameter.

I think that's backwards???

 

Consider geodesics to show this. Light appears to follow a curved path around a strong gravitational mass, because that is the shortest path through curved space. If you moved onto the geodesic, space would flatten, and the geodesic would appear straight. If you imagine straightening a curve, you do it by expanding space on the "inside" of the curve. If you moved from a distant location to a geodesic near a large mass, space would appear to expand more on side of the mass, until the geodesic is straight.

 

It is length-contracted for distant observers... lengths are shorter according to distant observers. -- Suddenly I have this feeling that I've said this before only to find out I was wrong.

 

It is not moving the cylinder that makes it expand; it is moving the observer.

 

If an observer moves up and down in the cylinder placed upright on a gravity source then he and all equipment he brings with him will also be subjected to the warping of spacetime, so he will not be able to measure any different size locally of the cylinder. However if he measures the top and bottom diameter he would be able to see that the bottom is greater.

I see... so my original example, described from the "inside" perspective, is not right and not very useful.

 

However... with a long enough cylinder and enough curvature, the farther end of the cylinder can't be considered to be in "local spacetime". So the local area of the cylinder can always be the same size, no matter where you go in the cylinder, but the farther parts of it can still be warped.

 

Ok, it seems like I messed up, sorry if I turned things upside down for you. As said above the cylinder should be wider closer to the source of gravity and more narrow on the side towards empty space. Which means that from an outside view the media inside the cylinder would appear less dense closer to the source of gravity, before gravity compresses the media towards the bottom.

 

I might have misinterpreted you but I think you are also getting into a hypothetical side effect of space warping instead of normal gravity and not only a confusion of pressure/equilibrium effects.

I think you had it right the first time, and only I was backwards!

 

Yes... I'm suggesting that a hypothetical side effect of space warping can completely account for gravitational acceleration (would require math to show this). I want to abandon the "cylinder and gas pressure" example because it is complicated and misleading... or really just wrong.

 

 

Well, I think we start to dip our toes in the speculation area now. My understanding of gravity in relativity is that objects follow straight paths through spacetime due to inertia and that the curving of space causes these straight lines to bend. I am not able to explain it better than that.

Yes, I agree.

And I think that gravity can be completely explained by the differing curvature of geodesics, as a test particle shifts from one geodesic to another as it moves in multiple directions (due to random motion or due to oscillatory motion in 2 or more dimensions). As a demonstration of this, we could draw a bunch of curved geodesics around an imaginary massive object, then repeatedly trace distances of some fixed unit, along the current geodesic, in random directions*, and we would end up moving toward the gravitational mass.

 

Granted I haven't done this...

 

* moving along a curved geodesic would change your direction relative to an external perspective such as that of the person who is tracing these paths. That would have to be accounted for when choosing random directions. The random directions would have to be evenly distributed according to the test particle, not according to the person tracing. This would better be simulated with a computer.

 

Space curvature is very slight, and the distances particles move when oscillating is very small (so the different geodesics they travel on are only VERY slightly different). However, gravity is very weak, and if you oscillate a few billion times in a short period of time, the effect would add up.

 

SO, this is all a way that FOR ME satisfactorily explains gravity from the perspective of a distant observer.

I think that the same thing can be explained, from the perspective of a test particle, in terms of the way that space appears to warp as you move in multiple directions onto different geodesics.

 

As said above, if this effect is real then it should be possible to verify it by experiment in orbit.

The idea is a hypothetical explanation of how GR works. I would say that if it predicts a difference from GR then it is wrong.

 

 

In the perspective of a moving particle space is not changing, everywhere along the cylinder the diameter will locally be of the same size but if space is highly curved then the diameter will appear to be greater in the down direction and smaller in the up direction from the current location.

Exactly! Far away parts of the cylinder will appear a different size, but when you move to them, they appear to be the "normal" size. That means that these parts of the cylinder (and the space they occupy) are distorting as you move through it. Local space appears flat; distant space appears distorted; moving into that distant space makes it local which makes it flat which can only be done if it changes from distorted to flat.

[Spyman]

Please tell me you have that backwards!

Hmmm, ok lets see now, I said: "local time ticks slower deep down in the gravity well, for the local observer than for a distant observer".

 

You said: "Local time ticks at a "normal" rate for all observers. Time ticks slower in a gravity well according to distant observers."

 

Wikipedia says: "the lower the gravitational potential, the more slowly time passes."

 

 

I don't understand, we all seem to be in agreement with each other, or???

 

 

I think that's backwards???

Well I might be, maybe imatfaal or someone else have this knowledge and can confirm which way it is.

 

Until then lets review my logic in a thought experiment:

 

If you have a bar that is 299 792 458 meters long and place a blinking lightsource in one end and a mirror in the other, such that any observer will be able to see the lightsource flash and then the reflection in the mirror. Then they can measure the time it takes for the lightsignal to propagate across the bar.

 

The bar is attached with wire to a winch under a spacecraft at very high altitude. One astronaut is placed locally on the bar with one clock and one ruler and a second astronaut with a clock and a telescope that stays on the spacecraft is given the task to observe the bar from distance.

 

At a given signal the bar is lowered down into Earth's gravity field, as the bar gets closer to Earth it gets deeper down into the gravity well and therefore time ticks slower for the local bar riding observer according to the observer at the spacecraft.

 

For the local observer on the bar 1 second takes 1 second and since the bar is 299 792 458 meters long according to his ruler he will always measure a 1 second long duration with his clock between the lightpulses from the lightsource and the mirror.

 

The observer at the spacecraft looks through the telescope and confirms that according to the local clock, down on the bar, it takes 1 second between the lightpulses from the lightsource and the mirror, however according to his own clock it takes slightly more time, since the clock on the bar ticks slower.

 

Since lightspeed is constant for all observers, the observer at the spacecraft concludes that the bar is expanding during the descent.

 

 

It is not moving the cylinder that makes it expand; it is moving the observer.

I don't see any difference whether the bar and the observer originates at Earth and then the observer travels up to a distant location, or of they originates at the distant location and the bar travels down to Earth. In both cases the observer would measure a difference in the length of the bar.

 

 

However... with a long enough cylinder and enough curvature, the farther end of the cylinder can't be considered to be in "local spacetime". So the local area of the cylinder can always be the same size, no matter where you go in the cylinder, but the farther parts of it can still be warped.

Yes, the notion of local is approximate and with precise enough equipment you would be able to notice the warping very close, so technically one side of a local thin line is warped towards a gravity source and the other side away from the source.

 

 

SO, this is all a way that FOR ME satisfactorily explains gravity from the perspective of a distant observer.

As I understand your description I am unable to view it as an explanation of normal gravity interpreted from GR. Testparticles slightly favoring one direction due to random or oscillatory motion in curved space seems to me to be something else.

 

 

The idea is a hypothetical explanation of how GR works. I would say that if it predicts a difference from GR then it is wrong.

I don't think you are predicting an difference from GR though and if there is an effect like this then it would be a different sideeffect of GR which I think should be much smaller than normal gravity, maybe to small to measure with current technological level of equipment.

 

 

...and if you oscillate a few billion times in a short period of time, the effect would add up.

If the frequency of oscillations have influence would laserlights of different colors get separated by gravity?

 

 

Far away parts of the cylinder will appear a different size, but when you move to them, they appear to be the "normal" size. That means that these parts of the cylinder (and the space they occupy) are distorting as you move through it. Local space appears flat; distant space appears distorted; moving into that distant space makes it local which makes it flat which can only be done if it changes from distorted to flat.

Maybe I am only nitpicking here but space is distorted by gravity and that doesn't change when observers move through those parts, it is the observer and his equipment that gets distorted equally so space appears flat.

Edited April 28, 2011 by Spyman

[md65536]

At a given signal the bar is lowered down into Earth's gravity field, as the bar gets closer to Earth it gets deeper down into the gravity well and therefore time ticks slower for the local bar riding observer according to the observer at the spacecraft.

I think this is confusing. What does it mean for time to tick slower for an observer according to another observer? Local time ticks at a normal rate, according to all observers.

 

 

I would restate it as such: As the bar gets closer to Earth it gets deeper down into the gravity well and therefore a clock in the locale of the bar ticks slower according to the observer at the spacecraft.

 

The observer at the spacecraft looks through the telescope and confirms that according to the local clock, down on the bar, it takes 1 second between the lightpulses from the lightsource and the mirror, however according to his own clock it takes slightly more time, since the clock on the bar ticks slower.

 

Since lightspeed is constant for all observers, the observer at the spacecraft concludes that the bar is expanding during the descent.

This is a tricky puzzle! Your logic seems okay but I think the trick to this is that for a distant observer, the path of light will be curved. If the path of light appears to follow the length of the bar, then the bar will also be curved.

 

Then, since the bar (or at least the path of light) is curved, it appears to us that there is a "short-cut" from end to end, which is a line from bar end to bar end that appears straight to us. The length of the straight line would necessarily be shorter than the (curved) length of the bar.

 

Could it be that we're both right? According to the distant observer, the bar curves but does it also stretch in length, while at the same time the ends get closer together?

 

Let's say space is warped in a way that the local observer on the bar sees the entire thing in flat space, and it remains 1 lightsecond long.

According to the distant observer, the bar's clock ticks slower. So I agree with your logic, the bar must appear longer to the distant observer.

I think the bar would curve for the distant observer, which means that the distance between bar ends is less than the length of the bar.

But what does that mean???

Do we say the distant observer sees length expansion, because the bar length expands?

Do we say the distant observer sees length contraction, because the distance between bar ends is shorter than the length of the bar, even if the distance between ends remains at 1 lightsecond long?

Or do we say the distant observer sees length contraction, because the distance between the bar ends actually becomes shorter than 1 lightsecond?

 

I should have split this thread into 2. If it remains a puzzle I'll post this in the relativity forum.

 

Yes, the notion of local is approximate and with precise enough equipment you would be able to notice the warping very close, so technically one side of a local thin line is warped towards a gravity source and the other side away from the source.

But... I think that no matter how curved space is, you can find a small enough volume around yourself in which space is flat, to any degree of flatness you need.

If we have a 1m bar and our equipment can detect a warping, then we can use a 1 micron bar, and it will be flatter.

Certainly the limit of spacial curvature is flat, as distance from the observer approaches 0. I don't know if there's more to it than that. Is that alone enough to say that local space is flat, or does the way it approaches flatness matter?

 

As I understand your description I am unable to view it as an explanation of normal gravity interpreted from GR. Testparticles slightly favoring one direction due to random or oscillatory motion in curved space seems to me to be something else.

 

 

I don't think you are predicting an difference from GR though and if there is an effect like this then it would be a different sideeffect of GR which I think should be much smaller than normal gravity, maybe to small to measure with current technological level of equipment.

Yes... I can't call it an explanation of gravity without the math to show that it matches Newtonian gravity.

One problem with this is that it seems as if the "oscillation distance" would affect rate of acceleration; if a random walk with small steps moves you in a certain direction, a random walk with large steps should move you there quicker? So wouldn't an electron moving around a nucleus have greater mass??? And wouldn't that also imply that greater masses would accelerate faster than smaller masses??? Or would the constant speed of light fix everything, implying that all "random or oscillatory steps" are done at the same speed, regardless of the size of the steps?

 

Too many unanswered questions to say it explains gravity.

 

HOWEVER, I still think it is possible to use this reasoning to show that spatial curvature and 2d oscillatory motion will cause acceleration, and that's a start. It is a leap to think it would account fully for gravity, but it is an imaginable leap. Needs work.

 

If the frequency of oscillations have influence would laserlights of different colors get separated by gravity?

 

 

No, this only affects energy that is oscillating in 2 or more dimensions.

This idea assumes that everything (light or objects) consist of energy that is always moving (at c). In order for something to remain stationary, it would vibrate, so there must be an ability to "move backward" (otherwise it would shoot off in one direction at c). Light would never be accelerated this way, because it never moves back.

 

Also, if something was moving in 1 dimension along a single geodesic, that uhh... that thing about closed space being "harder or less likely" to move into does not apply. If you squish a line, it is still a line. So a one-dimensional oscillation is not enough.

 

Maybe I am only nitpicking here but space is distorted by gravity and that doesn't change when observers move through those parts, it is the observer and his equipment that gets distorted equally so space appears flat.

 

What I mean is that space appears different for different observational viewpoints,

so if you move from one viewpoint to another, you will see space "morph" from how it looks at one place, to how it looks at another.

 

You won't just see this: Space and distances actually change as you move.

 

I suppose one clarification might be that it is only distant objects and space that appear to change shape and/or size... however you can make distant space into local space by moving into it, so somewhere along the line that space has to warp.

 

 

[Spyman]

This is a tricky puzzle! ... If it remains a puzzle I'll post this in the relativity forum.

Short of time, but I found the thread I was looking for: time dilation in gravitational potential what about distance?

[md65536]

Short of time, but I found the thread I was looking for: time dilation in gravitational potential what about distance?

That's a good thread. I think that I've got some of it right, and some of it -- still/again -- backwards.

 

Okay I propose a new law of relativity:

 

The Law of Inverse Reasonability: If anything seems to make sense with relativity, then the opposite is true.

 

Outline of proof: Relativity implies that different observers see different things. Since the universe is consistent, that requires that for any observed effect of relativity, a converse effect must be observable from somewhere else. Therefore anything that makes sense from one perspective, will be completely wrong when misapplied to another frame or perspective. Due to the statistical certainty of mixing up some detail of different viewpoints or frames or clocks or measuring sticks or light paths or etc etc etc, the law holds.

 

 

This allows us to iteratively build an explanation for any aspect of relativity. The "Convergent Oscillating Explanation Algorithm":

1. Explain something any way you want so that it makes sense to you.

2. By the law of inverse reasonability, the opposite will be true.

3. Discuss the opposite until that too makes sense. Again by the law of inverse reasonability, the opposite of what you made sense of is true.

4. Repeat until the explanations become convergent.

 

This of course requires that the explanations get a little bit closer to the truth when discussed. I think this may be the method I always use in these forums, to explain things with relativity... I just don't know how many iterations it takes to acquire convergence!

 

 

 

Okay that was a bit of a digression.

 

Back to the example:

As a distant observer nears a bar that is "submerged" in a gravity field, she will see the curvature of the bar straighten, and see the length of the bar shorten (opposite of what I'd been suggesting). I'm still not sure what will happen to the end points of the bar (I don't think they will converge; they may stay the same distance apart?; I still think they will diverge, so that the space between the ends of the bar increases as you move into it and it flattens -- but they certainly wouldn't diverge as much as I'd originally assumed).

 

 

 

 

Another digression: Suppose you had a closed loop of spacetime inside a blackhole horizon, by which I mean a geodesic in a loop: If light leaves a point in a certain direction, it will return to that point via another direction. I don't know if this is possible. If you moved into the black hole, the loop would flatten into a straight line and appear infinitely long. That seems to describe closed space inside a strong gravitational field, rather than open space as I was originally calling it.

 

if there is an effect like this then it would be a different sideeffect of GR which I think should be much smaller than normal gravity, maybe to small to measure with current technological level of equipment.

The effect would have to be on the order of 1 part in 3x10^7, if that statement makes sense.

If a particle accelerates from rest to 10m/s after 1 s in Earth's gravity, it will travel about 5 m. But if the particle is made up of oscillating energy traveling at c, that energy will "travel" 3x10^8 m in the same time.

 

The "smallness" of the effect I'm predicting is similar to the smallness of gravity, like say we were to compare the difference in freefall speed over 1 second compared to c. It makes sense that it would be small, because it would predict small changes in speed.

Edited April 28, 2011 by md65536

[Spyman]

You won't just see this: Space and distances actually change as you move.

Well, I don't know if it's relevant or not but I want to return to my point here and try to explain it further.

 

I do see your point, if we continue to use your cylinder then an observer in freefall straight down through its center will measure the diameter in front of him to be wider than the local diameter and the diameter behind to be narrower. As he falls down he will notice that when he reach a new location that he previously measured to be wider, that it now has shrinked to the value his last location had and when he looks back up he can see that the previously location now looks narrower than what it was when he was there.

 

Yes, space and distances actually change for the moving observer.

 

However and this is what I tried to say, for an distant observer looking at the freefaller in the cylinder, he views that space is not changing, instead it is the freefaller that changes. For the distant observer the curvature of space is already mapped out around the source of gravity and it is the freefaller that "morph" to fit in his current local space.

 

Maybe I am nitpicking and arguing semantics, both observers are entitled to claim equal value of their views and for both of them the world follows the laws of nature as normal. But if you want to explain gravity from a distant perspective then you need to consider that from the distant view, space is more warped towards the source of gravity and the freefaller is subjected to the local warping when he gets submerged into it.

 

A moving object actually change from the view of the distant observer.

 

When you make distant space into local space by moving into it, would it make any difference if it's space that is changing or the objects?

[Spyman]

If a particle accelerates from rest to 10m/s after 1 s in Earth's gravity, it will travel about 5 m. But if the particle is made up of oscillating energy traveling at c, that energy will "travel" 3x10^8 m in the same time.

"Although the bending of light can also be derived by extending the universality of free fall to light, the angle of deflection resulting from such calculations is only half the value given by general relativity."

http://en.wikipedia.org/wiki/General_relativity#Light_deflection_and_gravitational_time_delay

[md65536]

However and this is what I tried to say, for an distant observer looking at the freefaller in the cylinder, he views that space is not changing, instead it is the freefaller that changes. For the distant observer the curvature of space is already mapped out around the source of gravity and it is the freefaller that "morph" to fit in his current local space.

 

Maybe I am nitpicking and arguing semantics, both observers are entitled to claim equal value of their views and for both of them the world follows the laws of nature as normal. But if you want to explain gravity from a distant perspective then you need to consider that from the distant view, space is more warped towards the source of gravity and the freefaller is subjected to the local warping when he gets submerged into it.

I agree and I think that's the problem. It's not a problem with GR, but a problem people have with understanding it. We naturally want to describe things from the perspective of a distant observer, because that is the way we have always understood our world. But describing what is happening from this perspective is not always intuitive. I think the view from the perspective of affected particles better illustrates "how gravity works". Since both views are valid, what a particle might "see" corresponds to what a distant observer sees, and SR and GR fully explain that correspondence.

 

 

Perhaps a different way to sum up my thoughts in this thread (while removing all of the incorrect stuff) is as such:

- If a test particle moves from point A to point B toward a gravitational mass, it moves into relatively more-closed space which flattens. The space it came from opens. The distance from A to B according to an observer at A, is less than the distance from point B to point A according to an observer at point B. Therefore, the trip toward a gravitational mass is easier than the trip away from the mass, in terms of effort required, from the viewpoint of the traveler. (From a distant viewpoint, the equivalent might be that the trip from A to B is easier than the converse, due to the greater mass a particle has at B vs A.)

- If energy oscillates, and in doing so physically moves toward and away from a gravitational mass, then it will move toward it more easily. This slight bias will build momentum and produce a noticeable effect after millions or trillions of oscillations.

- Therefore, oscillatory movement and spacial curvature directly imply acceleration.

 

 

 

What I don't know is: Is this even correct? Does this account for gravity, or is there a lot more to it, that would make this effect negligible?

 

 

With this modified version, it seems like 1d oscillation is enough (as long as the movement is properly aligned). I'm not sure if this modification is essentially the same, or (if not) whether either version is correct.

 

Note: There is still contention I think on the first point about which distance would appear greater???

 

I do concede that this might be a negligible effect. For example, in my original gas pressure example, I might have tried to claim that the pressure in the center of a sun is somehow only "an illusion of spacetime curvature", which is false. If you take a balloon filled with air to the bottom of a pool, it will compress, and the balloon's size may change slightly (maybe immeasurably so) due to a change in spacetime curvature between the top and bottom of the pool, but that doesn't explain at all why the balloon compresses.

 

"Although the bending of light can also be derived by extending the universality of free fall to light, the angle of deflection resulting from such calculations is only half the value given by general relativity."

I'm not trying to account for the bending of light paths. I think spacetime curvature intuitively explains that.

 

Light travels in a straight line in one direction. I don't think it's valid to describe energy actually traveling along the oscillating path one draws when illustrating light as a wave. ??? Even if it is, it does not apply here. When we speak of distances, we're talking about the straight line distances that light travels, not the length of a wavy line. Light of a higher frequency (thus more oscillations per unit of straight-line distance) does not travel a greater distance than light of a lower frequency.

 

The oscillations I'm talking about with massive particles involve actually physically moving back and forth across a distance, which photons do not do. Thus, my conjecture would not explain any acceleration of photons, but that is good, because GR predicts no acceleration.

Edited April 30, 2011 by md65536

[md65536]

I've made a mistake and want to correct it.

 

If space is continuous, the travel distance from A to B should be the same as from B to A. You should be able to keep splitting up that distance into smaller steps until you're only moving a distance of epsilon each step, where epsilon is small enough such that it is completely within local "flat" space, such that the forward distance from X to X+epsilon is the same as the backward distance. Therefore continuous movement from A to B, through continuously changing space, would be the same distance as traveling through continuously changing space in the opposite direction.

 

One way to fix it is to assume continuous movement is impossible, ie. that there is a quantum distance across which energy "leaps" rather than travels through. This quantum distance would place a minimum bound on epsilon, and if it's large enough so that the spacial curvature across that distance is not absolutely negligible, then the theory might still have hope.

 

Version 3 of the conjecture:

- Assume energy leaps across some distance of epsilon. Let point B be separated from A by a distance of epsilon, in a direction toward a gravitational mass. Again, the distance from A to B (viewed from A) will be smaller than the distance from B to A (viewed from B), and so it is slightly easier to leap toward a gravitational mass than away from it.

- Assume energy continuously oscillates at some fixed rate, and in doing so physically moves toward and away from a gravitational mass, then it will move toward it more easily. This slight bias will build momentum and produce a noticeable acceleration after millions or trillions of oscillations.

- Therefore: Spacetime curvature, plus constant motion evenly distributed in all directions, plus energy leaping across a quantum distance, implies gravitation.

 

 

 

 

When a distant observer sees a particle traveling along a curved geodesic, they will see the particle rotate to follow the curve, while the particle itself does not experience any rotation (it is following a straight geodesic). Therefore, if the particle is moving in all directions with equal probability from the particle's perspective, it will not be moving in all directions with equal probability according to a distant observer. It will appear to favor moving toward the gravitational mass, as the distribution of directions seems "pinched" into higher density in that direction. This requires moving along a curving geodesic, which means not just moving toward and away from the mass, but also perpendicular to that, which seems to restore the requirement that the oscillations must be in 2 or more dimensions.

 

But then again... you wouldn't need to see the particle rotate at all. If the particle is leaping back and forth across a fixed distance (from the particle's perspective), then from the distant perspective the particle is leaping a different distance toward gravitational masses than away from it, on each oscillation. So again only 1 dimensional oscillation may suffice.

 

I don't know how the changing mass of the particle would come into the picture, and whether or not it would only affect the distant observer's take on it. Perhaps the mass times leap distance remains invariant, so each leap seems to take a fixed amount of energy, regardless of observer location.

 

I think that the 2 different explanations for a distant observer (1 for traveling toward and away from the mass, and one for traveling perpendicular to that) might mean that there will be a gravitational effect regardless of the direction of oscillation. We might drop the requirement that oscillation is evenly distributed in all directions, and instead only require that for any given direction, forward and backward motion occurs on average with equal probability or rate.

Edited May 1, 2011 by md65536

[md65536]

We might drop the requirement that oscillation is evenly distributed in all directions, and instead only require that for any given direction, forward and backward motion occurs on average with equal probability or rate.

Nope, I change my mind again. If energy is oscillating in one dimension I assume it will be along a geodesic?, in which case it will never leave that geodesic. Then if you could "polarize" energy so that it's oscillating in only one direction, it wouldn't be accelerated due to gravity in a direction perpendicular to its oscillation, yet it could still be accelerated back or forth along the geodesic. That means 2 equal masses might accelerate you differently depending on your orientation relative to them. This does not match reality.

 

1D oscillation might produce specific acceleration. Universality of gravity would require that the oscillation happens in all directions (including all spatial dimensions) with a uniform probability distribution, at least on average.

 

 

[Spyman]

The oscillations I'm talking about with massive particles involve actually physically moving back and forth across a distance, which photons do not do.

Sorry but I don't have time to read through all that has been posted lately, but this part seems wrong.

 

There is an simple experiment that most peoples can do to measure the speed of light, which relies on the fact that light actually does move back and fourth physically at its frequency like this:

The picture is taken from an article here: http://www.null-hypothesis.co.uk/science//item/measure_speed_light_microwave_chocolate

 

The article explains how to measure the speed of light by melting a chocolate bar in a microwave owen with a known frequency and measuring the distance between the melting spots.

 

Mid Morning Experiment: Speed of Light

By Mark Steer

 

Einstein realised that the speed of light was one of the defining measures of the Universe. Remember E=mc2, the beautiful little formula that wrapped up the theory of relativity? Well ‘c’ is the speed of light. And you can measure it. All you need is a microwave, a ruler and a bar of chocolate (and maybe a calculator).

[md65536]

There is an simple experiment that most peoples can do to measure the speed of light, which relies on the fact that light actually does move back and fourth physically at its frequency like this:

Obviously there is a difference between the distance a photon travels, in a straight line, at a velocity of c, versus the oscillation of the photon at a given frequency.

 

Do you agree that the red line in your diagram does not show the distance d that a photon travels, when we speak of the velocity of light being v = d/t?

 

Do you agree that light travels in a straight line and is not constantly changing directions?

d is a straight-line distance when we speak of the speed of light. Photons travel in a straight line.

THIS is the movement that applies to my conjectures.

 

If some part of light physically moves back and forth, it is something else. Clearly, there can be different types of oscillation that behave differently in spacetime, and would need to be described differently.

 

My conjecture would apply to photons themselves, and probably massive particles as a whole, and possibly any components of a particle that make up its mass. It would not apply to whatever part of a photon could be said to be oscillating. It would not apply to oscillations that do not contribute to a particle's mass.

 

---

 

Granted, my conjecture does really on oscillation to explain gravity.

If all oscillations are physically equivalent, and there is no way to explain how it applies to some but not all oscillations, then the conjecture is probably wrong.

As is, I have no way to explain a difference, or why the oscillation that you're talking about does not apply.

 

(I could always guess, and say that the oscillation of light is continuous, and does not involve any random or quantum "leaping back and forth" of the wave-like aspects of light.)

Edited May 3, 2011 by md65536

[Spyman]

An electromagnetic wave travels in a straight line at the velocity of c but it is a WAVE and the tiny packets called photons have both the properties of waves and particles. Something is going through the chocolate bar and causing the melting spots, if it is not photons then what is it?

[md65536]

An electromagnetic wave travels in a straight line at the velocity of c but it is a WAVE and the tiny packets called photons have both the properties of waves and particles. Something is going through the chocolate bar and causing the melting spots, if it is not photons then what is it?

I've never been able to reason about the wave-like nature of light, but I accept the wave-particle duality without understanding it.

My conjecture is based only on reasoning involving a particle-like nature.

I think that's valid... you can discuss particle-like properties of light that are valid despite the duality.

 

Either way, I still think it's valid to say "light never jumps backward in any direction while propagating".

 

If this conjecture can't stand on its own without addressing the wave-like nature of light, I am unable to provide that.

 

 

[md65536]

I'm thinking I should try simulating this on the computer, unless anyone knows why this might be a waste of time.

 

 

What I would simulate is this:

On a one-dimensional line, have a mass at x=0 with some familiar value (such as Earth's mass) and a test particle at some arbitrary x₀.

Calculate distances on this line using some kind of spacetime equations (which hopefully I can find on wikipedia!), set up with curvature based on the mass.

"Leap" the test particle back and forth along this line by some fixed distance in the particle's frame, at a speed of c.

Edit: Somehow accumulate and apply the test particle's velocity.

Plot the location of the test particle over time.

 

Questions to answer:

How does the leap distance matter?

Does a random speed < c and/or random leap distance give better results? If so how quickly does the result converge?

 

A successful outcome would involve fully accounting for gravity when using real-world constants, within some reasonable error.

A failed outcome would involve no definitive acceleration in the test particle.

A possible outcome according to replies to this thread would be that any simulated acceleration is negligible compared to g₀.

 

 

If such an experiment would be valueless, I might avoid trying to do it. Would the results of a simulation matter?

Edited May 24, 2011 by md65536

[Spyman]

The main value of the simulation is not the results themselves but what you might learn by doing it.

[md65536]

The main value of the simulation is not the results themselves but what you might learn by doing it.

So far I've learned that I have no idea about the things I've been talking about! I have mental ideas of what curvature means, but then if I try to do calculations based on that, I'm stuck. Which means my "understanding" is not usable, which means I don't really understand curvature. I really need to learn the math.

 

 

 

 

But of course that didn't stop me. I found a post on another site which said that the curvature caused by Earth is in the range of 1mm.

As a guess I figured the effects of this curvature might be inversely proportional to r (I also tried 1/r^2 but 1/r seems to work better... not perfect though).

I was going to describe the calculations I'm using but I realize it must be wrong because it doesn't have a property of local flatness. So I think I must be overestimating the effects of curvature. Which is unfortunate because to get results close to Newtonian gravity, I had to use a curvature of 17.6mm instead of 1.

 

But when I tweak values, I can get results that are close to Newtonian. It is tempting to try to get them to match (eg. I wonder if I can make it match by including the effects of time dilation etc), but that's a red herring if I'm not even using curvature even remotely right.

 

Another problem to address is numerical error. I'm adding tiny values (the difference in x after leaping back and forth a very small distance) to very large ones (x is around the radius of the earth). I'm surprised I got values at all. I may be able to mitigate this by separating big variables from small, and making sure no variable includes both big and small. For example, I would have 2 variables for x... one for its position ("big") and another for the distance it traveled since the beginning of the simulation (small, at least to start with). Also I'm using mainly Euler method. Also I'm using a huge leap distance of 0.1m (otherwise I'd have to simulate billions of oscillations without getting anywhere)... I have to find a way to approximate a few billion tiny oscillations in a single calculation, while making sure the errors are kept small.

 

 

 

 

Implementing this forced me to figure out a way that acceleration plays a part. The best I have so far, is this:

Suppose a particle leaps 1 unit in one direction, and then 1 unit in the opposite direction, but due to changing measurements of distance, say it is 1 trillionth of a unit away from its initial position.

The time that it has taken to make these leaps at a speed of c is t = d/v = 2/c. It has moved 1 trillionth of a unit in a time of 2units/c, which I'm treating as a tiny change in velocity that the particle gets to "keep"... if it leaps once, it acquires a tiny additional velocity and keeps moving with that velocity while continuing to oscillate. That would mean that after a trillion (an arbitrary example number here) oscillations, it would approach a significant fraction of c (I'd have to include time dilation and length contraction if the velocity gets large).

 

 

So in summary: Don't know what I'm doing yet, but I can fudge this in several ways to force some results. At this point I don't expect it to work if I make some reasonable corrections, but I'll try to figure out some corrections.

Edited May 27, 2011 by md65536

[DrRocket]

 

Now imagine that the same volume consists of highly curved spacetime.

For this thought experiment, let us suppose that the volume looks to us, as outside observers, as a long uniform cylinder, ...

 

I hate to burst your bubble, but in terms of differential geometry a cylinder is flat. You can roll the surface isometrically onto a plane.

 

Maybe you should regroup and learn enough mathematics to understand curvature a it. There is no simple explanation of Riemannian curvature.