This is called the Lorentz-FitzGerald contraction. When you observe a moving object, it appears shorter, or thinner, *along the axis of its motion* relative to you. Likewise, you appear thinner relative to the moving object (who does not feel thin at all). There’s a formula for this, but it’s irrelevant for the discussion below.

You also appear to gain mass (using a similar formula). This is also irrelevant for the following discussion, but I thought I’d toss it out there.

So here’s the rub. You will often hear someone say or write, “When an object nears the speed of light, the universe flattens to a thin sheet from the viewpoint of an observer on that object.”

Just to clarify, this is bullshit. If you are cruising along at near-light speed, then all matter, relative to your frame of reference, is moving in the opposite direction at near-lightspeed. That’s okay so far. Except, the universe is expanding. And the farther out you go, the faster it’s expanding, such that there are regions of space expanding away faster than the speed of light (the expansion of “space” is apparently able to ignore the whole “speed of light” limit thing; go figure).

So when you attain a certain velocity, you become stationary relative to another part of the universe that is moving away from Earth at the same speed. There is no shortening of length or thickness for that object, that part of the universe.

Take the Andromeda Galaxy for example, moving toward us at 110 kilometers per second. When we measure the galaxy in the direction of its travel, along its axis of motion, it’s foreshortened in that direction. Now, fire up your rockets so you’re traveling at 100 kilometers per second in the same direction, and Shazam! The entire galaxy poofs back out to its real shape in its own frame of reference that happens to coincide with your own. Relative to you, the Andromeda Galaxy is no longer moving.

So, back to the expanding universe; as your spaceship speeds up more and more, there’s always a part of the universe that’s moving at the same speed at which you are traveling (a comoving reference frame). It won’t look compressed or thinner or foreshortened at all. In fact, if we take the viewpoint that all parts of the universe are essentially equal, (that is, there is no “center” of the universe) then the universe doesn’t compress into a pancake at all as you near the speed of light; it’s just that the non-foreshortened part of it, the part that matches your current velocity, is farther and farther away from you. But the overall volume will appear unchanged.

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I think none of these is right. The idea between the “escape velocity being faster than the speed of light” is relative to the delta between the gravitation potential of the observer and the potential at the event horizon. From an infinite distance, we observe an event horizon at a certain radius. Should the event horizon suddenly disappear if we are in an inertial frame starting our fall into the black hole? Starting at what distance? A thousand miles? A light year?

The more likely result is that the event horizon moves inward as you approach it. You are in a deeper gravity well as you approach the black hole, thus the difference between your local gravity potential and that of the event horizon, to maintain a high enough value for the escape velocity to equal the speed of light, requires that the event horizon continuously move away from you (toward the singularity) as you move toward the singularity. You never quite catch up with it. There’s a Wikipedia article that says this explicitly, but then, it’s a Wikipedia reference (Event Horizon). Sometimes they’re wrong, but usually they’re dead-on.

An interesting consequence of this is that if you maintain a certain orbit near the event horizon, and your version of the event horizon is closer to the singularity than that of a more distant observer, then a photon just outside your observed event horizon could reach you just fine, even though it cannot reach the more distant observer. Having received that photon, you could transmit the data from it outward, (boosting the frequency) as the distance from your gravity well to the distant observer requires an escape velocity somewhat less than the speed of light. Is this a loophole?

Why, then, do we think that a photon below the event horizon (for the observer at infinity) can’t escape the confines of the black hole? Is it only because it would be red-shifted to a zero frequency? Or is that false?

Escape velocity is merely a calculation of the velocity required to go from one gravity potential to another. If you are already in a gravitational well (like the outer edge of the Milky-Way galaxy) with an escape velocity of 300km/s, this has no effect on the escape velocity from Earth (11km/s), or the velocity needed to orbit Earth (7.5km/s). Likewise, consider a photon trapped just beyond the event horizon as viewed from an observer at infinity. To the guy in orbit around the black hole, the difference in potential is much smaller, and his relative event horizon is closer to the singularity. Won’t he see that photon? Can’t he receive it from the domain outside his apparent event horizon, but inside the event horizon of the observer-at-infinity? And then capture the photon and retransmit it?

So, even though a photon by itself can’t escape the event horizon of the observer-at-infinity, an intermediate process (natural or human) could conceivably pass a photon up through overlapping light cones, even though the light-cones at either end don’t overlap. This might eliminate the question of whether information can escape a black hole or not. The infalling observer can see what’s happening beyond the outer event horizon, and pass the information on, since his own event horizon is even closer to the event horizon.

]]>Let’s start with some fundamental concepts about photons. It’s generally believed that photons are their own antiparticles, and also that the speed of light is the ultimate speed past which nothing can travel. Also, in a photon’s frame of reference, the distance from source to destination appears to be zero, and it takes zero time to travel that distance. This led me to speculate that light might, in fact, travel at an infinite speed, and that somewhere out there, there is a geometry in 4D space-time where that makes sense, where, when we try to measure it, we see light ambling along at a tedious 300,000 km/s. It would also explain why you can’t travel faster than the speed of light; the speed is, in fact, infinite. It’s really hard to go faster than that. The difficulty lies in finding that geometry. My second supposition regarding photons is that they are always emitted perpendicular to the path of travel (through time) of the originating particle. In a standard space-time diagram, assuming a velocity of c, this leads to the light cone diagram. In the new geometry, assuming an infinite photon speed, the picture is a little different, but still leads to the well-known equations we are used to.

Looking at Figure 1, the vertical line A represents the source of photons a and b, which travel instantaneously to the observer on line B. Line B observes the two events a and b separated by time ct_{2}, and from B’s perspective, the object has moved away a distance x, which equals a-b. In A’s proper time, the time between the two events is merely c time t, and the distance is zero, so the interval is s=ct_{1}. From B’s proper time, the duration is ct_{2} and the distance A has traveled away from him is x, so the measured interval between the two events is s=√(ct_{2}^{2}-x^{2}), which should be familiar to everyone.

In Figure 2, we see what happens as B gets closer to the X axis. But it still produces the common formula for the interval. What the diagram does not explain is why the speed of light appears to be roughly 300,000 km/s.

However, what Figure 1 can do is allow us to derive the standard time dilation formula:

∆t_{1}=∆t_{2}/√(1-v^{2}/c^{2})

How do we get there?

Note the velocity of B away from A is

v=x/t_{2}

_{ }So x^{2}=(vt_{2})^{2}

From before, we had s=ct_{1} in A’s reference frame and s=√(ct_{2}^{2}-x^{2}) from B’s perspective. Set the two equations equal, and we get

(ct_{1})^{2}=(ct_{2})^{2}-x^{2}

Substituting for x^{2} we get

(ct_{1})^{2}=(ct_{2})^{2}-(vt_{2})^{2}

^{ }Divide it all by c^{2} and pull the t_{2} out of the two terms on the right gives us

(t_{1})^{2}=(t_{2})^{2}•(1-v^{2}/c^{2})

Or, ∆t_{1}=∆t_{2}•√(1-v^{2}/c^{2}), which should be familiar to a lot of you out there. It’s the standard formula for time dilation due to relative velocity.

So Figure 1 works out that if the line is at a 45 degree angle, then v=c/√2, which shouldn’t be a surprise. And as v gets closer and closer to c, then the graph gets flatter and flatter. But this graph is based on the idea that c is infinite. Why is it that when we measure it, it’s always 300,000 km/s? Clearly, if you set up an experiment that bounces light back and forth between mountain tops, and c=∞, then your test should show that light moves instantaneously. Not 300,000 km/s. And yet, we always measure c at the same boring 300,000 km/s (yeah, I know this isn’t exact value-it’s 299792.458 km/s. Get over it-this is easier to type).

So what is it that makes an infinitely fast photon measure as a finite number in all frames of reference? Is it the expansion of spacetime? Is it the curvature of spacetime due to gravity affecting the value of c that we measure? Is it that this theory is just wrong? I don’t know yet – this would obviously have to be resolved before this model made any sense, and may or may not appear in a later entry. Any speculations about this are welcome.

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In this log of ramblings, I’ll offer up a bunch of foolish ideas on physical reality. I’m a big fan of determinism, so be forewarned. I also think of time as an actual, physical dimension. If you happen to join me on this warped (!) journey of speculation, I’d love for you to tear my arguments apart, tell me what’s wrong, and perhaps help shape the speculations into something that makes a coherent sense of reality, or assure their demise.

]]>First, if you aren’t familiar with the Grandfather Paradox, you can read up on the subject at Wikipedia at https://en.wikipedia.org/wiki/Grandfather_paradox.

Start with the idea that we are always travelling at the speed of light. You, the Sun, the Earth, your brother Bob, everything is travelling at the speed of light through the time dimension, all going the same direction so your relative speed is zero. This is a pretty common concept in modern physics, so I’m not going to expand on that here. Just more physics weirdness.

So, let’s say I get the dubious urge to go murder my grandfather at some time in the past, before I was born. Using some fantastic time machine, I go back in time to when Grandpa was just a young fella and shoot him. What happens then?

Well, imagine that space and time are a 4-dimensional matrix, but that changes made in the matrix can only propagate forward at the speed of light. Remember, that’s how fast we move through time. Eventually, the change (where I no longer exist) reaches the point where I would have gone back in time, but the particles that would have made up my body go shooting forward past that point and never go back. Well, they aren’t “shooting forward” so much as redirecting the world line of their old path at the speed of light. Now, instead of making a U-turn and heading toward their fate with my grandfather, the bullet-magnet, they continue forward in time. The old worldline going backwards collapses/disappears at the speed of light and eventually catches up to where Granddad is, and lets him live. I’m born again! And I foolishly decide, again, to go back in time and kill my granddad.

What does this result in? An oscillation. The world line shifts back and forth between the two realities, carrying the data from both possible realities, like a sine wave on a current. Just as a single electrical sine-wave can contain positive and negative values as it propagates through a wire, so can events toggle on a worldline as that worldline propagates through time. Even past the point where I made my fateful decision, the world line is toggling back and forth; both realities are true, taking their turn as the decision I made causes both of them to be real. The duration of this toggling or oscillation would be twice the duration of the time from when I went back in time to when I snuffed Gramps; the duration of the whole loop.

It’s my belief (not necessarily shared by many others) that we live in a four-dimensional space time that exists perpetually as a 4D matrix, and that what we perceive as our consciousness exists at each point along that worldline. There is the version of you that you perceive now, and the version that existed when you kissed your wife for the first time, savoring the moment you’ve forgotten. Kind of a repetitive immortality.

But what I’m suggesting above is that multiple realities can exist on a single worldline; you don’t need multiple universes dividing every time a critical decision is made or a quantum observation collapses a wave function, or a bit of antimatter goes back in time and changes an existing chemical configuration. Both events occur and exist on a changing, fluctuating, dynamic 4D worldline. There’s the version of you that remembers killing your Grandfather, and the version that never existed, propagating through time, one behind the other forever on the oscillating world line.

The normal view of a 4D worldline is of a static deterministic universe, bound by the future and past configuration of an unchanging worldline. Another view is that every decision, human or quantum, splits the universe into a multiverse, a crowded infinity of infinities. This version allows us to stick with one universe, but to modulate our worldline to allow multiple realities to exist along a single timeline.

Possibly, an outside observer could interface with either version of your reality, based on where he encounters your worldline from his own worldline. Could that be the “collapse of the wavefunction” we talk about? Good grief, that would make the Schroedinger’s Cat conundrum actually possible. Dead and alive! I always thought of it as complete nonsense.

One issue with this model is that each worldline, as it moves from one reality to another, may have to move instantaneously from the collapsed worldline to the new worldline. I think. No real way to test it, that I can imagine. Mmm…maybe pick a subatomic decay process that can have multiple durations, then have someone record the decay time data, then take off (with that data) somewhere at high-speed so your worldline is no longer in sync with the experiment’s original timeline, fast enough that the time separation is greater than the decay time variance. Then, come back and see if the recorded time is the same as it was before; you’ll have two sets of readings of the duration of a single decay, and they might not agree. Wouldn’t that be something?

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There are no strange ideas in this entry. In fact, I might call this a boring entry. If you want to read the weird stuff, read one of my other entries.

I was sitting around reading a primer on particle physics (L.B. Okun) today, and got thinking about the conservation of linear and angular momentum.

Conservation of linear momentum means that, when you chuck something out the rear end of your spaceship, then your spaceship moves the opposite direction, so m_{1}v_{1} = m_{2}v_{2 . } So, if you toss out a small mass of propellant from one end at a really high velocity, then you move the much larger mass the opposite direction at a much slower velocity (along with your remaining propellant). This is an exponential relationship, but that’s not what this blog is about today, so you can forget learning about that useful tidbit of knowledge.

Anyway, to increase your velocity, you have to chuck part of your mass the opposite direction. Pretty basic. If you just move stuff around inside your ship, the ship won’t move at all (except incrementally for the duration that you move around in the ship, but you won’t acquire a continuous velocity). Anyone who’s been on one of those playground spinners and tried to throw your body one way or the other knows how that works. You throw your body forward a foot, and the disk rotates a foot and stops.

So you can’t change the momentum of an object by moving stuff around inside. Not even if you have the rocket inside an enclosed sphere. The sphere won’t move.

I was recently (foolishly) wondering if that was true of angular momentum, too, if you had a rotating planet or moon, is there some way you can get energy out of the rotation by diddling around with the insides, somehow tapping the angular momentum of the planetoid for energy. Ultimately I realized you cannot in a closed system, but it should have been obvious to me all along. However, as with a rocket, you can change the angular momentum by ejecting part of the object. You can even speed up the spin a lot or slow it down.

Satellites do this sort of thing all the time. Usually they have spin they want to get rid of, and they call the technique “momentum dumping”. Two methods known to me involve extending tethers (like ballerina arms) to slow down the satellite’s spin, then releasing the tethers, or spinning up a high-speed gyro in the opposite direction of your spin (potentially dumping the core of the gyro, though I’m not certain any spacecraft does that – usually they use the gyros to turn the spacecraft both ways, hoping the overall effect will cancel out, and when the spin in one direction gets to be too much, they finally use propellant to dump the angular momentum). These are called Control Moment Gyros, or CMGs, and they usually have a minimum of three on board to cover the 3 axes.

Carrying this concept one step further, since you can eject propellant from a ship to make it go faster in a straight line, you can similarly spin up a chunk of mass from your planetoid in the opposite direction of your planetoid’s spin to make the planetoid spin faster, then eject that spinning mass into space. What amused me about the idea is that it’s essentially the same as a rocket ejecting propellant linearly to increase linear momentum, but here you are ejecting an object with accelerated angular momentum to increase the angular momentum of your planetoid. The difference being, you don’t ever have to eject the mass; it’s rotating in place, like a CMG.

That’s it. Not really that interesting, I guess. The equivalency of the two systems and the idea of “rotational rocketry” just struck me as amusing.

]]>A number of times in physics books and articles, I’ve come across people stating that “as an object approaches the speed of light, its relativistic mass increases, so it’s harder and harder to accelerate the steadily increasing mass, thus, you can’t speed up.”

I’d like to call bullshit on this argument. As your ship’s mass increases, so does the mass of your propellant in equal parts. If your mass doubles, so does the propellant mass being ejected out of your rockets, and you’re still obeying the law that for every action there’s an opposite and equal reaction. It’s just as easy to accelerate as it was before. You still can’t measurably *reach* the speed of light, but that’s for another entry I’m working on.

See? Easy peasy.

]]>One of the discussions centered on whether there is a “background field” of gravity or not, or whether it’s even sensible to discuss such a thing since, in an infinite field of equally distributed mass (or gas, or 1 atom per cubic light year, whatever), all the forces around you seem to cancel out. The mass of the universe to your left is equal in size to the mass of the universe on your right; you feel a net acceleration of zero. I argued that even though the field was “flat”, there was still a field there. He argued that a field implied a gradient; there is always a force.

We did not come to a satisfactory conclusion. It might merely have been the fact that we were defining the same terms in different ways in our heads. I’m not sure. I thought my argument was rock-solid.

So, here is my side of it.

Some of you are probably familiar with Newton’s Shell Theorem. It’s in his Principia Mathematica, and if I remember right, he solved it without using calculus. Basically, what it says is this; if you are inside a spherically symmetric shell of mass, then you feel no gravity pulling you any direction. It’s a bit non-intuitive. Let’s say the Earth is hollow, and the entire planet’s mass has been compressed into a thin spherical shell a few centimeters (or meters, it doesn’t matter) thick. If you are floating around in your Nike Space Suit inside this shell, you will not be pulled toward the center, or the inner surface of the shell, or anywhere else inside the shell. Wherever you are put, you will remain.

Personally, I think this is one of the coolest theorems ever.

It’s also true that if you are outside the surface of a spherically symmetric planet, then it doesn’t matter how dense it is, at a given radius you will feel the pull of a certain amount of gravity. If you are in orbit above the Earth, and the Earth suddenly becomes a black hole of the exact same mass, you will remain in orbit, totally unaffected by that change. That’s pretty cool, too. Given that the gravitational force is based on F=GMm/r^{2}, this should be kind of obvious. Neither your mass (m) or the mass of the Earth (M) has changed, your orbital radius is the same, and G is the gravitational constant. Ergo, the density of the object you are orbiting at a radius “r” from the center of the object is irrelevant.

So, that was me drifting from the actual subject. Shell Theorem—let’s get back to that.

As you might know, the clock of anyone in a gravity field runs slower than that of a clock outside of that gravity field. This is called gravitational time dilation (and is equal to ∆t’ = ∆t √(1-2MG/rc^{2}) for a non-rotating sphere). A person on Earth actually ages slower than a person in deep space, according to relativity. This was verified with clocks flying around the Earth in the Hafele-Keating experiment. Before you ask, yes, they took into account Earth’s rotational speed, the speed of the airplane (in both directions relative to Earth’s rotation) with regard to Special Relativity’s time dilation due to velocity. It was a nice experiment.

Let’s say we’re using a hollow Earth from Newton’s shell theorem. As you get closer to the Earth, you are in a deeper gravity well, and the outside observer sees your clock slow down. There’s a small hole in the planet, and you pass into the planet, where everything is pulling you in opposite directions equally, so you seem to feel no force. And yet, your time dilation effect does not suffer a discontinuity, jumping suddenly to that of the outside observer. You are in a denser gravity field, but a flat gravity field. [to the physics majors out there, for god’s sake, if my terminology sucks, please correct me]. Your time dilation will be just the same as if you were standing on the surface of the planet.

So now you have a flat gravitational field (no “force” pulling you in one direction, that is, all forces pulling you equally in all directions). And yet, even in this apparent lack of gravity, where you can’t actually tell that you’re in a gravity well, your time runs slower than the time of someone far from the planet.

I extended this argument to the rest of the universe. If mass was distributed equally around you, even though you felt no force one way or the other, there would still be a background gravitational field. Gravitational time dilation implies a gradient; for time dilation to be relevant, you need someone in a weaker gravitational field measuring your time. However, both the measured and the measurer can be in locally flat gravitational fields.

Does a flat gravitational field curve spacetime by itself? Or is it only the gradient between two different gravitational fields that curves spacetime? My general opinion is that you don’t need the gradient for the curvature of spacetime. If you have an infinite universe with equally distributed mass, then from some arbitrary center, it will appear to curve spacetime until it closes the dimensions of universe into closed loops, like the inside surface of an event horizon (though other arbitrary centers will have different, yet overlapping event horizons – a subject I will touch upon another day). Likewise, if you have a large, thin flat sheet of soapy water in the air, fluctuations are going to cause it to form bubbles, closing up the edges. In spacetime, there may be a similar tension (gravity?) that closes the edges together into a 4D hypersphere.

How would you test the curvature of space inside a shell? I’m not entirely sure. I think the universe we have is a good test case, however.

]]>Some reactions result in the release of more than two photons. A particle and antiparticle meet, three photons are emitted. The photons are lower energy, but the reverse reaction, 3 photons meeting, is a much, much lower probability than 2 photons (gamma rays) meeting. Still, on rare occasions, it might happen.

In fact, it’s my belief that if you have enough photons, even low-energy photons, passing through the same bit of space at the same time, you can also have pair-production, spitting out particles and antiparticles. One calculation for photons from the cosmic microwave background radiation (CMB) estimates 400 photons per cubic centimeter, average, plus whatever higher-energy visible light and gamma rays pass through from billions of stars. And there are a lot of cubic centimeters in a light-year (about 4.9 x 10^{50}). Even if the probability of pair production is very, very low, I still imagine that it would happen on occasion.

As a side-note, the probability of a positron and electron meeting in deep space is very high, since they attract one another, while the probability of two gamma rays meeting at just the right time in just the right way is fairly low. The reaction looks symmetric, but the probability of it happening in a certain direction is much higher one way than the other. Ditto for any two-particle reaction that creates three particles. This contributes to the increased entropy of the universe and the “arrow of time”; there’s a preferred direction for these subatomic reactions to occur.

]]>Due to the nature of how light moves through space, when you look back 14 billion years to the farthest reaches of the universe, you are actually looking at a very small volume. But the image, warped as it is, is spread out and fills the farthest regions of the sky, like a view through a concave lens. If you were able to look all the way back to the tiny point of the Big Bang, the image would be smeared and spread out across the 14 billion light-year shell, any detail of the event washed out by turning the fine detail of a tiny event into a picture the size of the universe.

So, when you look up at the night sky, everywhere you look in the blackness of deep space, 14 billion light years away, is actually the same small point.

Does this make sense? I’m trying to think of a good analogy for this, but it just isn’t coming to me. Maybe like starting with a tiny drawing on the surface of a tiny sheet of rubber, then stretching it out so that the sheet of rubber stretches all around you in a sphere, like the inside of a balloon, and then trying to figure out what the original picture looked like.

This, of course, begs the question of what physicists are calculating when they measure the accelerating expansion of the universe. If the universe was physically smaller 14 billion years ago, and now the remaining image of it is spread out over a sphere with a radius of 14 billion light years, that’s going to come off as an acceleration; the farther you look, the smaller the original volume and the more the image is spread out over the apparent warped view of the current volume. And, of course, 14 billion years ago, the universe actually *was* expanding a lot faster than it is now. It’s a double-whammy of accelerations. Most physicists are a hell of a lot smarter than me, so I’m guessing that both these accelerations have been calculated into the “accelerating expansion of the universe” equation. I can only speculate that there is a third element. I wish I could find out without wading through a lot of really obscure math.