Now, remember how many molecules we're dealing with here. For example, a mol of iron typically weighs about 56g, and that's 6.02 * 10^23 atoms. A 1-mol sphere of iron would have a radius of a bit over 1cm (thus diameter about 2cm, or roughly the size of a golfball, for a neat bit of symmetry). Passing through a 2cm-block of iron requires passing a stupendous number of 1-in-6-billion chances. Eventually the odds will get you.
I don't think it works like that. You are talking like the entire mol of iron is in the path of the neutron. It isn't. Only a very small percentage of the atoms are anywhere near its path.
Another factor is that, unlike neutrinos, neutrons are not electromagnetically neutral -- since neutrons is composed of quarks, not only do they have a magnetic dipole moment (meaning they generate a magnetic field, couple to magnetic fields and to dynamic electric fields), but they also have an electric dipole moment.
http://en.wikipedia.org/wiki/Neutron_electric_dipole_moment
The neutron electric dipole moment (nEDM) is a measure for the distribution of positive and negative charge inside the neutron. A finite electric dipole moment can only exist if the centers of the negative and positive charge distribution inside the particle do not coincide. So far, no neutron EDM has been found. The current best upper limit amounts to |dn| < 2.9×10^-26e·cm.[1]
That being said, the Standard Model predicts |dn| to be on the order of 10^-31, but it's not a proven fact yet. And if it exists, it's incredibly tiny.
Build a man a fire, warm him for a day,
Set a man on fire, warm him for the rest of his life.
General Relativity and the metric expansion of the universe has two very unintuitive consequences:
1) Even though the maximum speed at which anything can move is c, it's still possible for two objects to recede from each other faster than c. 99% of people (a small percentage of which are scientists themselves) immediately reject this notion, even though it's a direct prediction of GR.
2) Even more unintuitively, and even if you accept the above, if two objects are receding from each other faster than c (due to an expanding universe), light can still travel from one object to the other, even though at no point does it travel (locally) faster than c. (This is easiest understood by using the snail-on-a-stretching-rubberband analogy.)
I have two questions about this:
1) If two objects are receding from each other slightly faster than c (due to the metric expansion of space) and the first object sends light to the other, what's the color of the light when it arrives? (I'm thinking about redshift here.)
2) I think there's a cutoff point for how fast the two objects can recede from each other before light can reach one from the other. In other words, if the two objects recede faster than a certain speed, light will never be able to travel between them. What is this speed?
1) If two objects are receding from each other slightly faster than c (due to the metric expansion of space) and the first object sends light to the other, what's the color of the light when it arrives? (I'm thinking about redshift here.)
If the two objects are receding from each other faster than the speed of light, they cannot send light to each other. However, they might still be able to see each other if they were closer together in the past. Imagine a pair of galaxies that are pulled further and further apart by the expansion of the universe. At time t1, they are finally separated far enough that they are receding from each other faster than the speed of light. Any photons sent out after this by galaxy A will never be received by galaxy B (unless the expansion speed of the universe changes). But light sent out before t1 will be received, but after a large delay. This delay grows longer and longer as we approach t1 (because the distance the light has to travel grows while it is travelling), at which point it becomes infinite. This means that from one of the galaxies' point of view, the other one will always be visible, but it will be a frozen, fading, ever more red-shifted past image of the galaxy, which never updates to reflect what happened after t1.
Warp wrote:
2) I think there's a cutoff point for how fast the two objects can recede from each other before light can reach one from the other. In other words, if the two objects recede faster than a certain speed, light will never be able to travel between them. What is this speed?
If the two objects are receding from each other faster than the speed of light, they cannot send light to each other.
That's not what I have understood, and it's precisely a big source of confusion and misunderstanding.
To understand why one object can send light to the other even though the latter is receding (slightly) faster then c, consider the classical "snail on a stretching rubber band" analogy:
Imagine that there's a hypothetical, infinitely stretchable rubber band, initially 1 m in length. A snail on one end starts moving towards the other end at 1 m/h. However, the rubber band is stretched at 1.1 m/h. Will the snail ever reach the other end?
Perhaps a bit surprisingly, the answer is yes. Did the snail at any point have a local speed larger than 1 m/h? No. Yet it still reaches the other end.
And perhaps a bit more surprisingly, light in an expanding universe behaves like the hypothetical snail on the hypothetical stretching rubber band. As the universe expands, it "drags" light with it, in the same way as the rubber band drags the snail. At no point is the local speed of the light different from 1c, but it still reaches its destination.
This means that if the universe were expanding at such a rate that two planets are receding from each other at eg. 1.1c, it means that one planet can send light to the other (and the light will eventually reach it.)
I have two questions about this:
1) If two objects are receding from each other slightly faster than c (due to the metric expansion of space) and the first object sends light to the other, what's the color of the light when it arrives? (I'm thinking about redshift here.)
2) I think there's a cutoff point for how fast the two objects can recede from each other before light can reach one from the other. In other words, if the two objects recede faster than a certain speed, light will never be able to travel between them. What is this speed?
Good questions. I'll post a link to the paper I used when answering something similar: http://arxiv.org/pdf/astro-ph/0310808v2.pdf
Equations (1) and (2) relate the redshift z to the recession speed according to GR and SR, respectively. What's important to know is that while SR's equation is straight-forward, GR's depends largely on the model and the time elapsed since the beginning of the universe.
The paper also has a graph plotting those equations using convenient values, it looks like this one: http://en.wikipedia.org/wiki/File:Velocity-redshift.JPG
Infinite redshift means you're unreachable by particles at that recession speed. The graph shows that SR is wrong, predicting z=infty for v=c. The red blur for GR is in fact a variety of possible models, all of them approximate fairly Hubble's law (linear) and SR for small recession speeds, but diverge on the velocity for higher redshifts.
So, by looking at the graph, since z is approx 1 when v=c, you would observe light at half its original frequency. And the largest recession speed observable could be anything between 2c and 4c.
That made me wonder why one would assume GR's superluminal speeds instead of SR's subluminal ones if redshifts are what we can effectively measure. But it seems the authors of that paper already took care of that and showed experimental data that rules out SR's predictions by 23 standard deviations. I didn't read this part, but it looks good enough for me :)
If the two objects are receding from each other faster than the speed of light, they cannot send light to each other.
That's not what I have understood, and it's precisely a big source of confusion and misunderstanding.
Let's do this properly, then.
The scale factor a(t) indicates the ratio of the size of the universe at time t to the size now, at the current time (t0). Using this, we can define a set of co-moving coordinates X = x/a, where x is the normal distance. The nice property of these coordinates is that they compensate for the expansion of the universe - that is, galaxies that are moving away from us only due to the expansion of the universe will have a time-varying x, but a constant X.
Let us calculate how light moves in terms of these co-moving coordinates, assuming light is emitted at the present time (t0). We then get X(t) = int(dX/dx * dx/dt * dt,t=t0..t) = int(1/a c dt, t=t0..t). So how far light has gotten by time t depends on exactly how the universe expands (a(t)). Right now, the universe seems to be transitioning from a phase where a = (t/t0)^(2/3) to one where a = exp(H(t-t0)). In the former, the integral evaluates to 3ct0([t/t0]^(1/3)-1), and in the latter, it becomes c/H*(1-exp(-H(t-t0))).
What we are interested in is how far light can get, since that determines whether light emitted from a far away galaxy can ever reach us (remember that we are calculating co-moving distances here, so expansion is already taken into account). We see that in the first case, we can get arbitrary large numbers for X by inserting large numbers for t. This means that for a matter dominated universe (the first case), there is no limit to how far light can get if we wait long enough - a galaxy will never become invisible. This is not that surprising, since a universe in this phase is slowing down, and hence, if we wait long enough, expansion will no longer be a problem. However, in the second case, a vacuum energy dominated universe, there is a limit to how far light can get. If we let t -> infinity in that expression, we find X = c/H. This is the most realistic case, actually, since the universe is becoming more and more vacuum energy dominated.
To answer Warp's question, then, we need to relate this to the expansional velocity. An object a (normal) distance x away is exapanding away from us with velocity Hx = HaX. If we are considering light emitted right now, then a(t0) = 1, so we simply get v = HX. And inserting the expression for the distance for the furthest light that can ever reach us in a vacuum-energy dominated era, X = c/H, we find v = c. That is, galaxies that are expanding away from us faster than c cannot communicate with us.
So the conclusion is still the same as previously.
Physically, the mechanism is that in a given time period dt, light moves c dt towards us, but space between us and it is expanding by Hx dt. If Hx dt > c dt, then the light will not have made any progress, and will get further and further away. If H is constant (the vacuum energy case), then it will never arrive at all. The only reason why light eventually arrives in the matter domination case is that H decreases with time, and eventually falls low enough that Hx < c.
Right now, the universe seems to be transitioning from a phase where a = (t/t0)^(2/3) to one where a = exp(H(t-t0)). In the former, the integral evaluates to 3ct0([t/t0]^(1/3)-1), and in the latter, it becomes c/H*(1-exp(-H(t-t0))).
Davis-Lineweaver wrote:
The myth that superluminally receding galaxies are beyond our view, may have propagated through some historical preconceptions. Firstly, objects on our event horizon do have infinite redshift, tempting us to apply our SR knowledge that infinite redshift corresponds to a velocity of c. Secondly, the once popular steady state theory predicts exponential expansion, for which the Hubble sphere and event horizon are coincident.
Right now, the universe seems to be transitioning from a phase where a = (t/t0)^(2/3) to one where a = exp(H(t-t0)). In the former, the integral evaluates to 3ct0([t/t0]^(1/3)-1), and in the latter, it becomes c/H*(1-exp(-H(t-t0))).
Davis-Lineweaver wrote:
The myth that superluminally receding galaxies are beyond our view, may have propagated through some historical preconceptions. Firstly, objects on our event horizon do have infinite redshift, tempting us to apply our SR knowledge that infinite redshift corresponds to a velocity of c. Secondly, the once popular steady state theory predicts exponential expansion, for which the Hubble sphere and event horizon are coincident.
That's a nice, pedagogical paper you found p4wn3r. I haven't read all of it yet, but it appears to agree with what I said above: as long as the H is decreasing we can see recessional velocities higher than c, but if H is constant, c is the limit. The steady state universe is one (obsolete) example of a case with constant H. Inflation is another. And a dark-energy-dominated universe is a third. The latter is the case we seem to be approaching right now, and if our current understanding of the universe (LCDM) is correct, we will be thoroughly dark-energy-dominated in a few tens of billions of years.
So, by looking at the graph, since z is approx 1 when v=c, you would observe light at half its original frequency. And the largest recession speed observable could be anything between 2c and 4c.
Thanks. But now I'm puzzled why the answer to the second question depends on things that intuitively seem irrelevant, and why it isn't an exact number. (After all, speed c isn't dependent on anything, and the metric expansion of the universe is just about changing geometry. Why would any of this depend eg. on the age of the universe?)
So, by looking at the graph, since z is approx 1 when v=c, you would observe light at half its original frequency. And the largest recession speed observable could be anything between 2c and 4c.
Thanks. But now I'm puzzled why the answer to the second question depends on things that intuitively seem irrelevant, and why it isn't an exact number. (After all, speed c isn't dependent on anything, and the metric expansion of the universe is just about changing geometry. Why would any of this depend eg. on the age of the universe?)
Didn't you read my explanation, Warp? The question you are asking is if light emitted at some point will *ever* reach us at some point in the future, so of course it depends on how the universe is going to expand in the future. If you imagine your snail on a rubber band, it has a huge effect on the progress the snail makes if you stop pulling on the band, or if you start letting it contract, for example.
Didn't you read my explanation, Warp? The question you are asking is if light emitted at some point will *ever* reach us at some point in the future, so of course it depends on how the universe is going to expand in the future. If you imagine your snail on a rubber band, it has a huge effect on the progress the snail makes if you stop pulling on the band, or if you start letting it contract, for example.
Of course the rate of expansion affects whether light reaches the other object or not. That's not what I was asking. I was asking about "the largest recession speed observable could be anything between 2c and 4c." I don't understand why that depends on anything else then the recession speed itself. Why isn't the maximum observable recession speed a certain fixed value?
Also note that in my original question I'm talking about a hypothetical situation where the expansion of the universe is such that the two objects are receding from each other at a constant speed (that's larger than c). (I suppose this means that the expansion of the universe would have to be asymptotically decelerating for that to happen.)
If you don't like the expansion of the universe for this because it's too hypothetical (the universe does not expand like that), then take the ergosphere of a rotating black hole instead, if you like. (AFAIK the same phenomenon can be arranged there.)
That's a nice, pedagogical paper you found p4wn3r. I haven't read all of it yet, but it appears to agree with what I said above: as long as the H is decreasing we can see recessional velocities higher than c, but if H is constant, c is the limit. The steady state universe is one (obsolete) example of a case with constant H. Inflation is another. And a dark-energy-dominated universe is a third. The latter is the case we seem to be approaching right now, and if our current understanding of the universe (LCDM) is correct, we will be thoroughly dark-energy-dominated in a few tens of billions of years.
I put that citation to say that if you assume exponential expansion, you'll conclude correctly that no observable galaxies can recede faster than light. Working with any of these three theories indeed doesn't matter.
However, the authors of the article are emphatic saying that we have always observed galaxies receding faster than light. The problem is that when you set a(t) = exp(Ht) you're extending the expansion of the dark-dominated era to the end of time, this is a good approximation for short time intervals, but like in the snail in the rubberband analogy, it can take enormous amounts of time for the encounter to happen, so this approximation breaks apart. In the context of the Lambda-CDM model, at no moment the Hubble sphere is an event horizon, it's irrelevant if we're passing through a dark-dominated era, since galaxies outside the sphere can reenter it when this era ends.
EDIT:
Warp wrote:
Of course the rate of expansion affects whether light reaches the other object or not. That's not what I was asking. I was asking about "the largest recession speed observable could be anything between 2c and 4c." I don't understand why that depends on anything else then the recession speed itself. Why isn't the maximum observable recession speed a certain fixed value?
You got it wrong. The graph I showed you doesn't contain only one line for General Relativity, but a red blur that represents a lot of models that make good predictions. Some of them put the cutoff at 2c, others at 4c, we don't know which is correct, because the evidence isn't strong enough to do that. The values of 2c and 4c aren't precise anyway, I just looked at the graph and guessed close values. If you really want only one value, I think the red line is the one with most acceptance, so look at where that line gets flat, just don't take it as a sure answer though.
You got it wrong. The graph I showed you doesn't contain only one line for General Relativity, but a red blur that represents a lot of models that make good predictions. Some of them put the cutoff at 2c, others at 4c, we don't know which is correct, because the evidence isn't strong enough to do that. The values of 2c and 4c aren't precise anyway, I just looked at the graph and guessed close values. If you really want only one value, I think the red line is the one with most acceptance, so look at where that line gets flat, just don't take it as a sure answer though.
I'm still not quite sure I understand. Wouldn't it be enough to plug the numbers in the GR equations and calculate the result? (I know that when talking about GR, it's not always a simple task. I myself would even dare to try, as the math involved goes well beyond my understanding. However, it sounds like there should be one exact solution, and not a range of possibilities. A bit like there's only one exact solution to the Schwarzschild radius for a given mass, for instance.)
If you know anything about differential equations, you know that solutions to such equations depend on the boundary conditions imposed upon them. GR has a set of coupled nonlinear differential equations at its core; different boundary conditions change the solutions to these equations. Since GR gives solutions not only to the dynamics of all energy, but also for the dynamics of space-time itself, space-time itself is dependent on the boundary conditions you impose on the equations.
So it is unrealistic to expect be able to "plug the numbers in the GR equations" and expect a single result -- each set of numbers gives a different (but related) space-time, with different dynamics.
Boundary conditions are restrictions placed on the solutions of partial differential equations; they are similar to initial values for ordinary differential equations: they limit the set of all possible solutions of a given equation (or set of equations) to those that match the specified problem/physical system.
Example of initial values: in Newton's equation of motion (which are ordinary differential equations), you have to supply the starting position and velocity of all particles in order to determine their motion.
Example of boundary conditions: in GR, one possible set of boundary conditions (that is often used in numerical integrators) is this: first, you pick a space-like slice of space-time (a good approximation for the concept is to think "constant time"). On this slice, you then need to specify (a) the distribution of stress-energy-momentum, (b) the intrinsic geometry of this slice and (c) the rate of change of these two, evaluated on the slice (there is a problem in what I said -- (a) and (b) are tightly coupled, and cannot be specified truly independently; but it is irrelevant here). The equations can then be integrated forwards or backwards in "time".
The need to specify how energy/geometry is at one slice, plus how they are changing, is what prevents you from getting a single answer from your original question -- it depends on too many things.
The need to specify how energy/geometry is at one slice, plus how they are changing, is what prevents you from getting a single answer from your original question -- it depends on too many things.
So, if I understand correctly, a proper answer to the question would depend not only on the distance between the two planets and the rate of expansion of space, but also other details such as the geometry of spacetime between the two planets (which might eg. be curved) and the exact manner in which spacetime is expanding (among other things)?
Put simply, yeah, that's pretty much it. You probably were confused because you thought pure GR (the Einstein Field Equations) uniquely determines the evolution of a mechanical system. It doesn't, and things get worse if you take into account stuff like charge and temperature.
Different people will explain this in different ways. The Einstein equations are an overly complicated mathematical way to relate the geometry of spacetime with the matter distribution. Do we know the exact spacetime geometry? No. Do we know the exact matter distribution? Neither. So, if we don't impose conditions that are physically observable for certain phenomena, GR becomes a useless theory.
In fact, if someone taught you all the math right now and asked immediately after: "give me a solution", you could ṕick any Lorentzian metric, and through some simple math operations find out what the other side of the equation is, and you'd have a solution with little effort.
The challenge is to impose symmetries and boundary conditions that describe a physical problem well and play with the algebra until you get a solution that can actually be tested empirically.
However, the authors of the article are emphatic saying that we have always observed galaxies receding faster than light. The problem is that when you set a(t) = exp(Ht) you're extending the expansion of the dark-dominated era to the end of time, this is a good approximation for short time intervals, but like in the snail in the rubberband analogy, it can take enormous amounts of time for the encounter to happen, so this approximation breaks apart. In the context of the Lambda-CDM model, at no moment the Hubble sphere is an event horizon, it's irrelevant if we're passing through a dark-dominated era, since galaxies outside the sphere can reenter it when this era ends.
Yes, that's true, but in the Lambda-CDM model you can't break out of a dark energy era, so if the concordance model is correct, the universe will become ever more dark energy dominated. As you say, light from previous eras will still be reaching us during this time, and so we will still in theory be able to observe objects receding superluminally (albeit enormously redshifted). But the question I thought Warp was asking was whether any object receding at speed x *today* will be possible to see in the furture, not whether we currently can see any objects that were receding at speed x when their light was sent out. And the answer to that is that as the universe becomes increasingly dark matter dominated, the speed x approaches c from above.
Warp wrote:
Of course the rate of expansion affects whether light reaches the other object or not. That's not what I was asking. I was asking about "the largest recession speed observable could be anything between 2c and 4c." I don't understand why that depends on anything else then the recession speed itself. Why isn't the maximum observable recession speed a certain fixed value?
Imagine a universe which first expands ridiculously fast, so that some object is
moving away from you at 100000c. At this point the object emits light towards you. If the expansion continues at the same speed, this light will not be able to reach you, but if the universe later stops expanding, or starts contracting, then the light would have no problem reaching you. So as you can see, in theory there is no limit to what recessional velocities could be observed. In practice, though, the universe doesn't expand like that, and the article p4awn3r linked to looks at realistic scenarios for the expansion, getting 2-4c for the maximal recessional speed.
Warp wrote:
Also note that in my original question I'm talking about a hypothetical situation where the expansion of the universe is such that the two objects are receding from each other at a constant speed (that's larger than c). (I suppose this means that the expansion of the universe would have to be asymptotically decelerating for that to happen.)
To have two objects receding from each other at constant speed requires decelerating expansion, as you say. v = HaX' = const => da/dt = v/X' => a(t) = 1 + v/X'*(t-t0) when we require a(t0) = 1, and where both v (recession speed) and X' (comoving coordinate of object) are constant. The propagation of light is then: X(t) = int(c/(1+v/X'*(t-t0) dt, t=t0..t) = cX'/v * log(v/X'*(t-t0)+1). This expression is not bounded, so no matter which v you choose, you will always be able to observe it (highly redshifted) if you wait long enough in this case. Again, physically this is because the universe is slowing down, letting the light eventually start to make progress and reach us, even if it was initially carried away from us.
If a photon were to orbit a rotating black hole at the outer limit of its ergosphere in the opposite direction of the rotation, it would look stationary from the perspective of the rest of the universe.
(I'm assuming that this would be a completely hypothetical situation that probably can't happen in practice due to a myriad things, eg. because of quantum fluctuations and whatnot, but in theory it could be possible.)
But what would this photon "look like" from the outside (assuming there was a way to measure it)? What would its properties appear like (such as wavelength)?
What would the rest of the universe look like from the perspective of the photon itself?
(I don't think the obvious answer "it would look normal" is correct because space is bent quite a lot that close to a rotating black hole. Not only is light curving down due to the gravity of the black hole, but it's also curving due to frame dragging.)
Another question: A gravitational singularity doesn't need to be a point. However, can it be infinite? (If yes, would it be an infinite line or surface? Can it be a 4-dimensional surface?)
But what would this photon "look like" from the outside (assuming there was a way to measure it)? What would its properties appear like (such as wavelength)?
Another question: A gravitational singularity doesn't need to be a point. However, can it be infinite? (If yes, would it be an infinite line or surface? Can it be a 4-dimensional surface?)
I don't think there is anything that would fundamentally forbid that, it just couldn't happen in the universe we live in.
Just as with the alcubierre drive, not everything that can exist can be created. A collapsing star or galaxy or anything of finite mass and size will not end up any larger than that. And since our universe seems to be finite, it couldn't even be artificially constructed. Remember, the biggest singularity known to men filled out the whole universe, and even that was way too small to be infinite.
If you wish to stop constraining yourself by reality and assume an infinite universe with enough dimensions, then you could probably construct a space-time with an infinite, 4-dimensional, kitten-shaped singularity. No guarantees whether it'll be stable.
Also, gravitational singularities may or may not actually exist inside black holes; see Fuzzball theories. I don't know how much acceptance there is in the scientific community, but I found it to be an interesting approach to the problem. Does one of the actual physicists in here know more about it?
