Scenario in an attempt to understand relativity

[CosmicVoyager]

What we think of as motion in 3D is really rotation in 4D. So different speeds in three dimensions correspond to different angles in 4D.

You are saying things are rotating in time? Go forward and back? I do not get that.

 

[lugita15]

You are saying things are rotating in time? Go forward and back? I do not get that.

No, you misunderstood me. Let's say two people Alice and Bob both have xy coordinate systems with the origins being the same but the x-axis of Bob's system is rotated by 10 degrees with respect to Alice's. Then if you have some weird-shaped object, the "shadows" or projections it casts on Alice's x-axis and y-axis, will be very different from the projections on Bob's axes. But some properties, like the area of an object and the distance between two points will be invariant in all coordinate systems.

What does this have to do with relativity? We have four mutually perpendicular, xyzt. And now Bob is moving at say 500 meters per second with respect to Alice. What does that look like geometrically? It turns out it corresponds to rotating Bob's x-axis and t-axis by some angle, just like in the previous case his x and y axes were rotated by some angles. It's a bit counterintuitive, but looking at and drawing spacetime diagrams might help. I suggest the book Spacetime Physics.

 

[Rap]

Yes. I am having difficulty picturing it. I don't think I am going to understand what you are trying to explain in just words without an actual picture or animation. I would expect such animations to exist already.

I don't know the directions of the lines, and I don't see how the light appears to be moving at the same speeds to both observers. I need to see observers and light in 4D as a series of 3D grids, or in 3D series of 2d grids.

Don't worry about how they see the light moving at the same speed right now. Just understand that this picture is unvarying, and it is the bottom line. Any physics, any geometry, any description of the universe, of what happens in the universe, how things interact, how all the lines connect up, all happen in this unvarying spacetime, independent of any observer.

For example, if an electron and a positron collide and emit a gamma ray, there are two straight lines that come together and meet. At the point that they meet, another line goes away from that vertex, and it is the photon. These three lines and the point at which they meet are absolutely fixed in spacetime. The angles between the lines are absolutely determined and fixed in spacetime.

You say you don't know the directions of the lines. How should I describe the directions of the lines? One way to describe the direction of a line is to establish a coordinate system. If we are in a 3D spacetime, we can choose three orthogonal axes, whose directions we know, and then we can describe the direction of the line. Otherwise you cannot talk about the direction of a line, you can only talk about the angle it makes with another line, and the plane of that angle.

Lets say an observer is a point in space, and in an "inertial frame". This means they exist as a straight line in spacetime. That line is their "world line". You can think of it as the observer traveling along this line. The 2D space that is orthogonal to that persons world line and passes through them is what they experience as their 2D space at that time. Everything in that 2D space happens simultaneously. Anything "above it" happens later, everything below it happens "before". Other observers that are moving at a constant velocity with respect to you have world lines that are separate from you and at an angle with respect to your world line. The 2D space orthogonal to their world line, passing through them, is what they experience as their 2D space "now". Everything on that plane is simultaneous to them. Obviously you and that person will disagree on what is simultaneous. If you see two events (like two firecrackers going off) and you say they happened in the same place, the other observer will say no, they happened in different places. But if you both understand relativity, you will both agree on what happened in spacetime, as you must.

This is all spacetime geometry. You have to learn spacetime geometry, just like you learned plane or solid geometry. Its the same kind of thing, only a bit more complicated. Once you learn spacetime geometry, you will find that all the paradoxes are just the result of people not understanding spacetime geometry, treating special relativity like a bunch of recipes in a cookbook, which may or may not have any relationship to each other, all kinds of strange voodoo going on with no rhyme or reason. Once you understand spacetime geometry, its like solving a problem in plane or solid geometry. Do it step by step, and you are done.

 

[ghwellsjr]

I'm assuming that when you say "accelerate them together" you mean that they are connected at some point, either at their common edge or at the other end of the shorter one or at the center of the shorter one or at any other point. Then this is no different than if you had marked off lines on the longer one at 100 unit intervals and asked about how those markings are any different than the ends of the shorter object next to it. Maybe I didn't understand your scenario but I think if you analyze it carefully you will see that you cannot learn anything about how connected things contract during acceleration.

No, I don't mean connected. It would work if they were connected :-) By accelerate together, I mean at the same rate, starting at the same time.

So read again.

If you have two objects (They would each have two rockets attached on their sides at their midpoints.) of different lengths side by side with either their forward edges or backward edges aligned, and accelerate them at the same rate starting at the same time, then edges should get out of alignment. Because they are both contracting toward their midpoints the same *percentage*, which is a different length for each object. For example, if one is 1000 units long and the other is 100 units long, and the midpoint of the shorter object is 50 units from the edge of the longer object, and they both contract 50%, the edge of the longer object will move in 250 units while the edge of the shorter object only moves in 25 units.

I wish I could draw pictures on the screen with a stylus.

*edit* Added illustration.

Yes, you are right, if you have more than one rocket and each one is accelerating a different rigid structure with the same acceleration, then you will be able to see differences in their positions after the acceleration, unless the rockets are adjacent to each other.

But I thought we covered this issue earlier when discussing the accelerations of the individual mirrors in my animation. Why are we going over this again?

Is it that you are suggesting that this would be a good test of length contraction? If so, then, yes, except for the very practical matter that we cannot perform such a test because of the difficulting of individually accelerating massive objects and the problem of getting volunteers to go on these trips because we probably couldn't get them back, but even if we could, they'd probably be dead for a host of reasons (g-forces, impacts with debris in the way, so much energy required we'd have to make it a one-way trip, etc.).

Maybe you could think of a more practical experiment.

 

[ghwellsjr]

@ghwellsjr

Uh oh. I think I thought of another problem. What happens if the light source is moving with me? Would the animation me the same, or does the line emitted perpendicular to my direction of motion move along forward with me?

I addressed that issue way back on this post:

We have two observers, I, George, will be the first one and we'll assume that I'm stationary and I have a flash bulb that I have arranged to be energized when you, the CosmicVoyager traveling toward me in a straight line at half the speed of light arrive at my location. You carry a stop watch, as do I, which we both start at the moment of the flash. You continue on without stopping or slowing down. The very bright flash of light will expand outward from its starting point in a perfect sphere getting bigger at the speed of light. According to Special Relativity, I will measure myself to be in the exact center of the expanding sphere of light. That makes sense, doesn't it, since I set off the light and I'm not moving? But according to Special Relativity, you will also measure yourself to be in the exact center of the expanding sphere of light and that doesn't seem right, does it, because you are moving with respect to the source of the light? But would it make sense if you were the one carrying the light source and to have set it off when you arrived at my location? Maybe, but in this case, I, too, would measure myself to be in the center just like you would. And that doesn't seem right but it really doesn't matter what the speed of the source of the light is or the speed of the observers, they all will think they are in the center of the expanding sphere of light. This can only work if the speed of light is the same for all observers, do you agree with that?

And it wouldn't change the animation at all.

 

[LeeJeffries]

That is awesome! Thanks :-)

I will think about it some more and see if it raises any questions. I took a conceptual basic physics class which focused on understanding. I wish there was a conceptual relativity class, and a conceptual quantum physics class.

We know that time dilation is more than a technique to make things work out. We have actually measured it with pairs of high precision clocks. Is there a way to measure length contraction to know objects are actually getting thinner? Since density is increasing, it's gravity should be affected. You could pass in front or behind it closer to it's center since it is narrower and experience stronger gravity.

Sort of off-topic but addresses your first wish

If you search for TTC Quantum Mechanics, TTC Particle Physics and TTC Relativity you'll get some somewhat lengthy but very informative lectures from the teacher training company (I think that's what it stands for)