Can Negative Energy Theoretically Enable Faster-Than-Light Travel in Sci-Fi?

[Longstreet]

I know crazy unsupported theories aren't supposed to be posted, but I have been working on some fiction which requires faster than light travel. I have read many other guides to ftl in SciFi, and I had origionally fitted in a form of hyperdrive, which would simply avoid the issue by ignoring the relevant dimensions of space. But, I don't like it either. Anyway, if it's ok here is what I made up so far. Admin I don't mind to delete if it really shouldn't be here:

The speed of light is the relative propagation of disturbances to the background “medium”. All energy propagates at the speed of light, either in the form of a photon or a fermion. Negative energy can be used to oppose this.

When normal energy propagates it warps space and time and causes the phenomenon known as gravity. Everything in the immediate vicinity is effected by this and everything slows down, like it got stuck in a muddy space-time rut. The negative energy acts in the opposite way, altering space and time so that everything speeds up. A single notohp (get it, negative photon), or small number of notohps, will still traverse at the speed of light. But as these particles build the cumulative effect of the negative energy causes the relative propagation to exceed the speed of light.

The amount of negative energy needed to complete a hyperspace jump depends on the mass of the ship, how far the destination is, and how quickly the arrival is desired. The mass of the ship and the distance traveled both increase the demand in a linear fashion; in other words twice the distance means twice the demand. This may be larger or smaller depending on how much in-jump navigation is required. The amount of negative energy used through the time dimension has a minimum at the speed of light which is considered the optimal. Going too slow actually increases the demand in a linear fashion. However, as the traversal time goes to zero, the needed negative energy approaches infinity.

Of course, causality is a concern, but I think I can say that if you change reference frames such that causality would be violated that the negative energy wouldn't necessarily look as negative, lessening the exceedence of the speed of light and allowing causality to be restored. I'm wondering if that even makes sense in the concepts of the paradox, even though negative energy can't exist in the equations.

Also it seems that the ship would have to be isolated from the effects of negative energy for the ftl to be usefull for a number of reasons; the ship falling apart and the crew aging and dieing from the opposite of the twin paradox.

So, I guess my question now is does this make any more sense than the other SciFi forms of ftl, or is it just as "magical"? What really would happen if you had negative energy, assuming of course it could exist appart from normal energy.

 

[dicerandom]

Fermions (i.e. half-spin particles: electrons and the like) don't propogate at c. Photons are actually bosons, they're spin 1, although not all bosons propogate at c either.

One thing that's always bothered me about FTL in Sci-Fi is that there's an implicit assumption that you can define simultaneaty across large distances in a curved space, which I do not think is true.

 

[franznietzsche]

One thing that's always bothered me about FTL in Sci-Fi is that there's an implicit assumption that you can define simultaneaty across large distances in a curved space, which I do not think is true.

In general, its not. Simultaneity is not universal, not even in flat minkowksi space.

 

[Robine]

Your approach is of course "magical" because it is not currently believed anything can go faster than light. But when some way is found to circumvent that all ideas about FTL will come under new scrutiny.

 

[Longstreet]

I don't know why I'm so concerned over the details. I think that even if ftl was plausable theoretically it wouldn't be very practical. After I posted I though, you know, I wish I didn't post that since there really isn't any way to qualify anything. I have read in other forum about Heim theory, and of course there was an article proposing that alternate geometries would allow alternate speeds of light; using dark energy particles I think, which is similar to the concept of negative energy.

 

[selfAdjoint]

But be really aware of what dicerandom said:

One thing that's always bothered me about FTL in Sci-Fi is that there's an implicit assumption that you can define simultaneaty across large distances in a curved space, which I do not think is true.

SF authors from the golden age onward have used this or that bafflegab to achieve FTL in their stories but how many have indicated they understood the consequences? FTL means time-travel, at least to some observers, simultaneity is gone from the galaxy, and negative energy matter would enable constant costless acceleration.

When I read that Honor Harrington zips hither and yon and when she gets back her elapsed time exactly adds up to that of the people who stayed home, I can only smile, and concentrate on the adventures and personalities.

 

[Longstreet]

Here is a question. Say there is a cylinder planet where the diameter is << than the length. Bob and Alice live millions of miles apart. There are also satelites in orbit as in the picture I attached. If they send a laser beam direclty from one house to the other, and also a beam that is relayed through the salelites, which signal arives at the other house first?

Edit: I guess the question should be, can the laser relayed through the satelites ever get there before the direct one.

 

[JesseM]

Here is a question. Say there is a cylinder planet where the diameter is << than the length. Bob and Alice live millions of miles apart. There are also satelites in orbit as in the picture I attached. If they send a laser beam direclty from one house to the other, and also a beam that is relayed through the salelites, which signal arives at the other house first?

In SR, the fastest way for a light signal to get between two points should always be a straight line through space('straight' in an inertial coordinate system), so a one-to-one signal should be the fastest. If you ask the question in GR figuring out the answer would be more complicated, but if the gravity is less than or equal to earth-gravity I doubt the answer would change.

 

[Longstreet]

Yes. Clues are planet, millions of miles, and the general construct is a question reguarding general relativity. You don't even need SR to answer the question in flat space.

 

[dicerandom]

Note: this is very much off the cuff, basically I'm assuming that the metric for your cylinder world will have roughly the same characteristics as the Kerr metric for a spherical rotating mass.

With the spinning cylinder you're going to end up with two major curvature effects (I'm thinking of Christoffel symbols here):

1) There's the standard radial 'pull' $\Gamma^r_{tt}$ due to the mass of the cylinder.
2) There's also a contribution like $\Gamma^\phi_{tt}$ due to the rotation of the cylinder.

Now your light from the laser will follow a geodesic, so you should be looking at the geodesic equation:

$$\frac{d^2 x^\alpha}{d\tau^2} + \Gamma^\alpha_{\mu \nu} \frac{d x^\mu}{d\tau} \frac{d x^\nu}{d\tau} = 0$$

So initially your photon will have velocities in t and z, the t component of the velocity will result in "accelerations" in the r and $\phi$ directions, so basically your laser will end up dropping towards the surface of the cylinder and picking up rotation along with the cylinder as well.

Now, unless the cylinder is rotating extremely quickly I can't imagine this effect (namely the spin) being nearly large enough to justify taking the light beam out to a larger radius. I think that the magnitude of the rotation will also be limited like it is with the case of the Kerr black hole, so you'll have a definite upper limit imposed on how quickly you can spin this thing.

 

[Longstreet]

I'm more interested in time and distance differences between the two houses, and between the "satelites". Will the light beam traveling from one house to the other take any longer than from one satellite to the other.

 

[pervect]

While you could work out the geodesic equation in the curved coordinates, the easiest solution is to use Minkowski coordinates (i.e. non-rotating coordinates), and then you know that light follows a straight-line path.

Your diagram does not show the curvature of the cylinder. the ideal path will be tilted slightly "up" because the cylinder is curved.

 

[Longstreet]

Maybe I need to be more explicit. Time passes more slowly in the houses than in the satelites, right? This is my only assumption so far from what I have gathered from other websites, sense I don't know the math to find out myself.

Say you have a light clock manufactured by AB corp. One clock is in Alice's house on the surface and one is in the satellite orbiting above her house several thousand miles away (this is a really big planet btw.). If alice looks up at the satelite will she see the clock running faster than her own? I'm guess the answer is yes, but that the clock also looks smaller so the speed of light is not violated here?

Now, the two houses (Alice's and Bob's) can act like one giant light clock, and the two satellites act like a giant light clock, and let's say that AB corp. built them too (ie they are idential in their reference frame).

If my previous assumptions are true, then Alice and Bob will, from the surface of the planet, see the satellite light clock run faster than their house light clock. But that the satellites are somehow closer together than the houses preserving speed of light. But now how can Alice and Bob still independently verify that they see their satellite orbiting directly above their house?

Ok, I'll eagerly await replies. :)

 

[pervect]

Maybe I need to be more explicit. Time passes more slowly in the houses than in the satelites, right? This is my only assumption so far from what I have gathered from other websites, sense I don't know the math to find out myself.

Yes, this is correct, and time will pass the fastest at the center of the cylinder.

Say you have a light clock manufactured by AB corp. One clock is in Alice's house on the surface and one is in the satellite orbiting above her house several thousand miles away (this is a really big planet btw.). If alice looks up at the satelite will she see the clock running faster than her own? I'm guess the answer is yes, but that the clock also looks smaller so the speed of light is not violated here?

Alice will see the clock running faster than her own. Assuming that by speed you mean the rate of change of the distance coordinate with respect to the time coordinate, Alice will not see a constant speed of light from her location. However, if Alice visited the light clock, she would measure the speed of light as 'c'.

Now, the two houses (Alice's and Bob's) can act like one giant light clock, and the two satellites act like a giant light clock, and let's say that AB corp. built them too (ie they are idential in their reference frame).

If my previous assumptions are true, then Alice and Bob will, from the surface of the planet, see the satellite light clock run faster than their house light clock. But that the satellites are somehow closer together than the houses preserving speed of light.

OK, this is the tricky point you are missing.

The "speed" of light, as definied by the rate of change of the distance coordinate with respect to the time coordinate, is not constant.

What is constant is the speed of light as measured by a local observer, using local clocks and rulers.

An example might help. Let us suppose you are in a spaceship accelerating at 1g, using acceleration to create "artificial gravity" rather than rotation as in your example. Using standard coordinates, clocks "above" you will tend to tick faster. The metric of space will, however, not change at all. You can find the "speed" of light to be as fast as you like by looking at a light beam sufficiently far "above" you.

At 1g of acceleration, roughly 1 light year above you light will be traveling twice as fast (in coordinate terms) than it will be at your location.

It's not directly relevant, perhaps, but it's interesting to note that a person 1 light year above you will accelerate at only 1/2 g to "hold station" a constant distance above you (while you accelerate at 1g). This is related to the "Bell spaceship pardox".

 

[Longstreet]

It may be a minor point, but the planet is solid, and Alice and Bob live on the surface like on earth. But the matter has been stretched into a cylinder (instead of a sphere). The cylinder isn't necessarily rotating.

Ok, so what I am hearing is that Alice can indeed observe what looks like faster than light travel between the two satellites, as long as she doesn't try to measure it herself. By that I mean that Alice knows Bob's house is 3 million km away by surface travel. Ignoring the fact that you can't have a prefectly straight laser beam over that distance because of gravity (ie they have some kind of relay system that make practically a straight path operating at c), they should be able to send emails directly between the houses in 10 seconds (3x10^9m/3x10^8m.s = 10s).

The two satellites are directly above each house, and they can fire a signal directly up to hit the satellite. The satellite can then send a signal to the other and then back down to the other house. I don't know how to calculate the time difference (so I leave out exact satellite distances), but let's just say for argument sake it takes 9 seconds to relay the emails.

pt1: From the point of view of Alice and Bob is this not a form of ftl communication? In other words, the fastest they can signal each other on the surface is in 10 seconds with a beam traveling at c. But an alternate path, shorter I guess, only takes 9 seconds. Is this causing any "time travel" problems?

pt2: If it is possible to have negative energy, would the argument not be reversed that a cylinder planet made of negative energy could relay a signal faster than the satellites, instead of slower?

pt3: Is this a form of ftl communication, and do "time travel" problems occur?

variations: Since the length of the cylinder planet is still undefined, it could stretch from here to alpha centari. Also, it needen't really be a planet, but simply a small path of energy/negative energy. The planet analogy is simply to relate to what we already know from GR.

 

[JesseM]

It may be a minor point, but the planet is solid, and Alice and Bob live on the surface like on earth. But the matter has been stretched into a cylinder (instead of a sphere). The cylinder isn't necessarily rotating.

Ok, so what I am hearing is that Alice can indeed observe what looks like faster than light travel between the two satellites, as long as she doesn't try to measure it herself.

Wait, where do you think you heard pervect saying that the signal could travel between them in less time if it was beamed up through the satellites than if it was sent directly from one to the other? I didn't read him as saying that in any of his responses.

 

[pervect]

Why make your planet a cylinder? If you want to make life simple, make it a very large "Alderson disk".

[add]
I assume you are familiar with that idea from your SF background, if not the wikipedia has an article on it at

http://en.wikipedia.org/wiki/Alderson_Disc

I discussed the metric for an infinite Alderson disk (i.e. a 2-d plane) in another thread recently - if we have the plane being the x-y plane, the metric is just

ds^2 = (1+g|z|)dt^2 - dx^2 - dy^2 - dz^2

Light in such a metric does not follow a straight line path. Furthermore, a lightbeam following the optimum path will cover more distance than it will if constrained to move on the z=0 plane, if that's what your eariler remarks were about.

This is not really going "faster than light", though, it's just an observation that the optimum path for light is not a path that's restricted to the z=0 plane. In coordinate terms it might look like FTL, but in local terms it's just that light, following the optimum path, gets there before light following a non-optimum path.

There is a very simple way to compute the optimum path. Imagine that, instead of being on a planet with gravity, you are on a big accelerating spaceship with the apparent gravity being due to your acceleration.

Then you can adopt an inertial coordinate system, and light travels in straight lines in the inertial coordinate system, while the spaceship accelerates (and thus does not travel in a straight line).

The light that will get between two points the fastest is light that travels a straight line in an inertial coordinate system. From the POV of the passenger on the rocket, the light will follow a curved trajectory, a lot like a thrown softaball, as if it were being "bent" by gravity.

Light that is forced to co-move with the rocket (z=same z as rocket) will travel a longer distance, and hence take a longer time. You can see this from the fact that this path is "curved" in the inertial frame.

Thus it is possible to cover more than 1 light-year of distance on the z=0 plane in 1 year by shooting light upwards at the right angle and letting it follow its "natural" path. All this means is that the real distance (the Lorentz interval) between two points is shorter than the coordinates would make you think.

[add]
Here's an anology.

If you are on the Earth, and you want to take the shortest distance between two points at the same lattitude, you do not follow a curve of constant lattitude. Instead, you follow a "great circle" path.

I believe that this problem (trajectory of light in an accelerating metric) has been worked out in detail by another poster BTW - if you're interested, I'll try and look up the details (when I have more time, right now I have to get going).

 

[Longstreet]

That's kind of my point. If there ever is an ftl system it will not be violating speed of light within it's reference. However, the effect can be characterised as providing a "more optimal" path of transmission. In my example it happens to be that signals exhanged on the planet is more optimal to go through space.

However, the premise of my orgional post is saying that instead of the planet being made of normal energy/matter, that it would be made of negative energy. The assumption now is that a signal's optimal path would be to dip *toward* the surface, instead of away. In the reference of the rest of the universe it would seem that this has the effect of ftl transmission, even though detectors on the negative planet would still detect it as a max of c...

Wait, where do you think you heard pervect saying that the signal could travel between them in less time if it was beamed up through the satellites than if it was sent directly from one to the other? I didn't read him as saying that in any of his responses.

Assumtion 1: the two satellites together are considered a light clock by how quickly they can exchange signals.

Assumption 2: The two houses together are considered a light clock by how quickly they can exchange signals.

Assumption 3: They tick at identicle rates if they are in the same reference.

Assumption 4: When the house clock is on the surface and the satellite clock is in orbit, however, the house clock will tick slower than the satellite clock.

Conclusion: A signal relayed by the satellites will travel faster than directly between the houses.

 

[JesseM]

Assumtion 1: the two satellites together are considered a light clock by how quickly they can exchange signals.

Assumption 2: The two houses together are considered a light clock by how quickly they can exchange signals.

Assumption 3: They tick at identicle rates if they are in the same reference.

Assumption 4: When the house clock is on the surface and the satellite clock is in orbit, however, the house clock will tick slower than the satellite clock.

Well, the rate that separated clocks tick always depends on your coordinate system. It may be true that the most practical coordinate system for dealing with a cylindrical mass would have this property, but it might not be true in all coordinate systems.

Conclusion: A signal relayed by the satellites will travel faster than directly between the houses.

Again, I think this might depend on your coordinate system. But if you have Alice send one signal directly to Bob, and at the same moment send a signal to the satellite above her which relays it to satellite above Bob which in turn relays it down to Bob, in this case there will be an objective, coordinate-independent truth about which signal reaches Bob first. But in this case you also have to take into account the time for the signal to go up from one satellite and down from the other satellite, in whatever coordinate system you're using.

edit: I guess if you make the distance between Alice and Bob arbitrarily large, you can make the time for the signals to go up and down be an arbitrarily small fraction of the time for the signal to go from one satellite to another--was this the idea behind putting them on a cylinder?

 

[Longstreet]

Yes. If I assume that time is dilated on the surface as compared to the satellites, I wouldn't think my coordinate system would matter.

 

[pervect]

That's kind of my point. If there ever is an ftl system it will not be violating speed of light within it's reference. However, the effect can be characterised as providing a "more optimal" path of transmission. In my example it happens to be that signals exhanged on the planet is more optimal to go through space.

However, the premise of my orgional post is saying that instead of the planet being made of normal energy/matter, that it would be made of negative energy. The assumption now is that a signal's optimal path would be to dip *toward* the surface, instead of away. In the reference of the rest of the universe it would seem that this has the effect of ftl transmission, even though detectors on the negative planet would still detect it as a max of c...

I would not really describe the situation as FTL. There is a minimum time required to cover a certain amount of distance, and that minimum time path is covered by light. The only somewhat unusual thing is that this minimum time path is not one which holds the height coordinate of the lightbeam constant.

Assumtion 1: the two satellites together are considered a light clock by how quickly they can exchange signals.

Assumption 2: The two houses together are considered a light clock by how quickly they can exchange signals.

Assumption 3: They tick at identicle rates if they are in the same reference.

Assumption 4: When the house clock is on the surface and the satellite clock is in orbit, however, the house clock will tick slower than the satellite clock.

Conclusion: A signal relayed by the satellites will travel faster than directly between the houses.

Actually, you can't quite say that. You gain an advantage from the height of the satellites (we are back to normal gravity), but you have to use the metric to evalaute $d\tau$ and integrate it over the path to compare the total elapsed time. The path to and from the satellite will contribute to this intergal, adding extra delay. So you have to check to see if added delay offsets the gain...

Another point is that the path "along the ground" is not a path that light would take unless forced (by putting it through a fiber optic cable, for instance). Light, on its own, would naturally follow the bent path I described earlier. If you shot a light beam out horizontally, it would "dip" in the gravity, and never reach the second observer.

 

[Longstreet]

What if the houses where at the center of the planet. There is no net gravitational acceleration and so light could travel directly from one house to the other in a straight line without being forced. I don't know if GR still says that clocks at the center of a symetric mass are still dilated compared to clocks a significant distance away from the planet.

 

[JesseM]

Another point is that the path "along the ground" is not a path that light would take unless forced (by putting it through a fiber optic cable, for instance). Light, on its own, would naturally follow the bent path I described earlier. If you shot a light beam out horizontally, it would "dip" in the gravity, and never reach the second observer.

In the case of the infinite plane you mentioned earlier, if light were shot out along the surface of the plane, would it continue along the surface instead of curving in one direction or another? It seems like it would have to, just by symmetry. So is that a situation where you can have multiple different null geodesics that go between two points on the surface A and B, with signals arriving faster if they take one path instead of another?

 

[pervect]

In the case of the infinite plane you mentioned earlier, if light were shot out along the surface of the plane, would it continue along the surface instead of curving in one direction or another? It seems like it would have to, just by symmetry. So is that a situation where you can have multiple different null geodesics that go between two points on the surface A and B, with signals arriving faster if they take one path instead of another?

No, the case I was considering was a flat disk / infinite plane heavy enough that it has significant surface gravity.

Light would still be deflected downwards by the gravity of the disk/plane. You can find the exact path messily by solving the geodesic equation, or easily by pulling a trick and replacing the gravity of the disk with an equivalent accelerating spaceship. In the later case, light follows straight lines in the inertial (non-accelerating) frame.

Either way you analyze it, the light is bent downwards by the gravity, so to get from A to B via a "straight line" you have to angle the light up slightly. Very slightly - the light will drop 16 feet in 1 second (the spaceship approach is the easiest way to see why this is true), but it will travel 186,000 miles in that second.

There is actually something I have to correct in my earlier remark, now that I think about it. For a non-light object, it makes sense to integrate dtau over the path to find the time.

For light, dtau = 0 everywhere along the path. There's still a way of determining the "length" of the path, but it's different than I described. (Mathematically it's called an affine length, physically it would be equivalent to the process of counting the number of wavelengths along the path).

 

[JesseM]

No, the case I was considering was a flat disk / infinite plane heavy enough that it has significant surface gravity.

Right, that's what I understood you to be saying.

Light would still be deflected downwards by the gravity of the disk/plane.

I was asking what would happen to light that was in the same 2D plane as the physical plane, assuming the plane has infinitesimal thickness--ie the light is emitted at 0 height from the plane, right at the surface. If it were emitted either above or below the plane it would be deflected towards the plane, so by symmetry doesn't that mean if its height is zero it'll go in a straight line?

 

[pervect]

OK - while I don't generally think of light as traveling well through an infinitely dense plane, I think the answer to your question if the infinitely dense plane was also transparent would be yes.

You can easily set up other situations where different light beams travel paths of different lengths (affine lengths) to reach the same point.

Gravitational lensing can cause the same sort of effect - one light beam whizzes by a massive object and is bent to meet another light beam which does not travel close to the object, for instance.

 

[Longstreet]

Is time dilated in the "middle" of the plane?

 

[pervect]

Is time dilated in the "middle" of the plane?

The plane case is somewhat ill-behaved because it is of zero thickness (it is convenient to use mathematically outside the plane, though).

If we make the plane a slab of finite thickness, so that is is better behaved, there will be some time dilation between the surface of the slab and the middle.

The dilation effect will be (in the weak field limit) proportional to the potential energy. This will in turn be proportional to the surface gravity * thickness. (There's an additional factor of 1/2 if we do the intergal).

If we take the limit of a very thin slab (the limit as the thicknes approaches zero) the potential energy tends towards zero, so there will be little time dilation between the surface and the middle of the slab for a thin slab (holding the surface gravity constant).

Without hyper-dense materials, you'll probably need to make the slab somewhere around the Earth's thickness to make it have the proper surface gravity. You would then have about as much time dilation between the surface of the slab and the interior as there is between the surface of the Earth and the interior of the Earth (that's a rough order-of-magnitude argument, not a rigorous analysis). This amount of time dilation would be easily measurable with todays atomic clocks, but not particularly significant otherwise.

 

[Longstreet]

So, just with the regular old earth. To be clear, you're saying that using a clock in free space (far away from the earth) as reference, that a clock on the surface will be dilated, but a clock in the exact geometric center is not dilated; and a transition period of varing dilation between the center and the surface, and again between the surface and the reference clock.

I would have thought the center of the Earth would have time dilation too since it's surround by matter/energy.

 

[dicerandom]

I would have thought the center of the Earth would have time dilation too since it's surround by matter/energy.

But the distribution of that matter and energy is spherically symmertrical to a good degree of accuracy. Classically, all the bits of matter in the Earth's crust are pulling you in every direction, but all of those little pulls exactly cancel out and result in zero net pull.

 

[pervect]

So, just with the regular old earth. To be clear, you're saying that using a clock in free space (far away from the earth) as reference, that a clock on the surface will be dilated, but a clock in the exact geometric center is not dilated; and a transition period of varing dilation between the center and the surface, and again between the surface and the reference clock.

I would have thought the center of the Earth would have time dilation too since it's surround by matter/energy.

I guess I wasn't clear at all :-(.

I'm saying that the clock at the surface will run at approximately the same rate as the "buried" clock, because they are at the same potential. The thinner the slab, the closer the clock at the surface is to having the same potential as the buried clock.

I was using the clock on the surface of the slab as the reference clock, not a clock at infinity.

Wth the metric I posted earlier for an infinite slab (which is also the metric for the metric for an accelerating spaceship) the metric coefficient g_00 which controls time dilation increases without limit as height increases. This means that clocks just keep ticking faster and faster as one goes higher and higher. This state of affairs does not allow one to normalize clocks on the ground to clocks at infinity, so one choses to normalize clocks relative to some other point, such as the surface of the "Earth" (which is what I did).

If you think strictly in terms of slabs, and ignore the spaceship anology, the infinite metric coefficients do not actually hapen with a finite sized slab. Far enough away from the slab, the slab appears like a point mass rather than a slab.

The infinitely increasing metric coefficients do occur for accelerating spaceships, though.

And in all cases, it turns out to be convenient to use the surface of the slab as a reference point, so that's what I did - unfortunately I didn't get around to actually mentioning that that's what I was doing :-(.

Hopefully this clears up what I was trying to say...

 

[Longstreet]

But if you have a sphere that's not the case, because the potential is zero in the center?

 

[pervect]

But if you have a sphere that's not the case, because the potential is zero in the center?

Huh?

It's very straightforwards - the higher you are, the faster your clocks tick. Also, the higher you are, the greater your potential energy.

Something at the center of the Earth is as low as you can get. This also means its as slow as you can get.

Potential as applied to GR really works only in the weak field approximation anyway - if you're not familiar with it, don't worry about it too much. (But I thought you already knew how it worked which is why I didn't go into a lot of detail).

The approach of setting the potential zero at infinity, and making potential a negative number is very standard, and is an assumption used in most formulas for time dilation. Unfortunately it is not possible to use this standard assumption with the "infinite plane" scenario, or with the equivalent "accelerating spaceship" scenario, as I tried to explain.

However, "higher=faster" still works just fine with both of the above scenarios.

This information is _all_ contained in the metric coefficient g_00. Proper time, the time read by a local clock, will always be $\sqrt{|g_{00}\Delta t|}$, where $\Delta t$ is the coordinate time interval. Thus when $|g_{00}|$ is large in magnitude, clocks tick fast, relative to the reference clock.

This comes straight from the defintion of the metric, which gives the square of the proper time (dtau) as a function of coordinate time intervals (dt) and coordinate displacements (dx,dy,dz). If the an object is standing still, there is no coordinate displacement, and proper time is given by the above formula as a function of coordinate time.

Take a look at the metric I posted earlier again:

$$g_{00} = (1 + g|z|)^2$$

Thus $$\sqrt{|g_{00}|} = 1+ g|z|$$

This means that clocks on the surface (z=0) tick normally, clocks keep ticking faster the higher they go (as z increases, so does the rate at which clocks tick) - and (with this particular metric) there is no limit as to how fast a clock can tick - the higher it goes, the faster it ticks.

When I (or anyone) says a clock ticks "fast" or "slow", they obviously have to compare the clock in question to some reference clock. (A clock always ticks at 1 second per second if you compare it to itself).

The reference clock is at whatever location has g_00 = 1 - for the simple reason that this is the only place where coordiante time is equal to proper time via the above formula.

For the metric I quoted, this reference clock is at z=0, the origin of the coordinate system.

This is different than the location of the refernce clock for the Schwarzschild metric, which is at r=infinity.

It is necessary to have a different reference location for the two metrics, it may be confusing, but there is no way around it.

 

[Longstreet]

Doh! Never mind. I guess that was my problem. My picture is upside down.

So back to the cyliner example, if the cylinder was very massive. There is a relatively small hollow shaft through the center length of the cylinder. Now the houses are down there, which are relatively weightless. Now they can beam a laser directly between the houses in a streight line, no?

Oh, also, is there any length contraction for things along the cylinder compared to things "high up"?

 

[pervect]

Right, your second potential diagram is the correct one - the one labelled "correct potential".

The houses can beam directly between them, and it will be a "straight" line. But it won't be the shortest straight line between the two points - that privilege will still go to a signal that starts going up, gets bent by gravity, and then comes back down at the right location.

(It will take a very long cylinder before the light beam to curve appreciably, of course).

The shortest distance between any two points in a curved geometry is always a straight line (because a straight line has the property that any "nearby" path is always longer) - but a straight line connecting two points is not necessarily the one with the shortest distance! There can be multiple straight lines connecting two points in curved geometries, and they do not necessarily have the same length.

A geometrical example might help. Unfortunately I can't draw in ascii that well easily here on the board.

A straight line between A and B goes over a big mountain.

A (^) B

^ being the "big" mountain.

This is a "straight line" path, but it doesn't mean that it's not better to go around the mountain rather than over it! There can be some other straight-line path that goes "around" the mountain that will be the shortest distance between A and B