Relativity Wheel

[A-wal]

Good afternoon, or whatever. I came up with this about eight years ago so I think it's about time I actually checked if it works.

 

There's only one spacial dimension (the one the object is moving in) to worry about in relativity and time is no different to space so you can derive time dilation and length contraction using two spacial dimensions. The four dimensions are at right angles to each other so if you draw a horizontal line and then draw another line the same length at an angle to it with 90 degrees representing the speed of light (so if you want to compare objects moving at half the speed of light relative to each other draw the second line at a 45 degrees to the first) then you just need to trace vertically down from the tip of the second line to see how much time dilation and length contraction there is by simply measuring how much shorter the second line is to the first one in the horizontal dimension.

 

You can see that at the speed of light the second line is infinitely time dilated and length contracted because it goes straight up. At low relative velocities there's very little time dilation and length contraction because if you trace down from the tip of the second line at low angles it's almost at the tip of the horizontal line but the same change in angle (relative velocity) makes more of a difference the higher the relative velocity. If you want to view the second line as the object at rest and the first one as moving then just turn it so that the second line is horizontal. In theory this should work perfectly. That's all of the relationships in special relativity expressed in detail and without a single equation. Actually all the equations are there but they're hidden behind geometry so simple it could be taught in primary schools.

 

If this does work then it should be able to handle acceleration as well be simply using a curved line. As the velocity increases, the angle changes to create an acceleration curve. As long as both lines are the same length it should work and the length of the curved line (it's length as measured by a straight line) should give you the amount of time dilation the accelerator experiences relative to the straight horizontal line of the inertial object, I think.

[A-wal]

So does this work? I can't find a simple table for time dilation and length contraction at various speeds. An object at rest relative to the observer would be a following a path parallel to the the observers line and so would be moving at the speed of light in the horizon time axis and not moving at all in the vertical space axis and an object moving at the speed of light relative to the observer wouldn't be moving at all on the time axis and so would be infinitely time dilated and length contracted, so the two ends of the spectrum are right and the length contraction increases at a faster rate with the same amount of acceleration at higher relative velocities, so that's a good start. An object moving at half the speed of light relative to the observer would be a line at a 45 degree angle, this:Obviously this is very rough, I did it in paint but it's good enough for now just to see if it could work. That looks like too much length contraction to me for half the speed of light. I'm sure this can work because the dimensions are at right angles to each other and there's only two dimensions that are shortened due to relative velocity. Maybe this gives the overall amount of time dilation and length contraction together so that it's the square root of the difference between the two lines that gives length contraction?

 

Edit: 0.25c and 0.75c.

Edited June 9, 2015 by A-wal

[Pmb]

re - time is no different than space

 

Quite wrong. There is nothing in relativity which says or implies such a thing. This is a common misconception. Time is a dimension in spacetime. That doesn't mean that time and space are the same thing. Time is measured with a clock and space is measured with a rod. If time and space were actually the same thing then you could rotate a clock into a clock. Since we know such things are meaningful we can easily see that time is not the same as space. Einstein made this quite clear in an article he wrote in nature.

 

Also the four dimensions are not at right angles to each other. That's not a physically meaningful statement. The only thing that can be at right angles are spatial axes. On a spacetime diagram you can draw the time axis at a right angle to a spatial axis if you use Cartesian coordinates. Only in a Cartesian coordinate system are the axes at right angles to each other.

 

To draw a spacetime diagram you draw a horizontal axis with the +x direction to the right. Then you draw the time axis vertically, i.e. at a right angle to the x-axis. The worldline of a photon moving in the +x direction is a straight line making an angle of 45 degrees with the x-axis in the counter clockwise sense. This is only if you scale the time axis by plotting it as x⁰ = ct where t is time and c is the speed of light.

 

Being able to see time dilation and Lorentz contraction on such diagrams requires much more work. You first have to superimpose the new axes over the old ones. The new x' axis will make an angle theta with the x-axis in the counter clockwise direction. The new time axis will make an angle theta with the old one in the clockwise direction. Events are then plotted in the new coordinate system by drawing lines parallel to the new axis. These lines won't be perpendiculars to the new axis though as they are when you change Cartesian coordinates from one frame to a new one rotated with respect to the first. Then you have to calibrate the axes which can be done using the invariant (ct)² - x² - y² - z² = (ct')² - x'² - y'² - z'²

[A-wal]

re - time is no different than space

 

Quite wrong. There is nothing in relativity which says or implies such a thing. This is a common misconception. Time is a dimension in spacetime. That doesn't mean that time and space are the same thing. Time is measured with a clock and space is measured with a rod.

You don't say. :)

 

I think you mean you could rotate a clock into a ROD and that since we know such things areN'T meaningful. That's a complete misconception of the principle that time is physically equivalent to space. Both devices measure length. Both can be transformed into the other. You can express time as a spatial distance and a spatial distance as a duration. They're completely interchangeable and on paper this makes them physically identical.

 

No, it's not simply that you can draw an axis is at 90 degrees. The three spatial dimensions are obviously at right angles to each other and special relativity shows that time is no different by treating coordinate time as simply another spatial dimension. In relativity there's no distinction and you don't need to specify which dimension is undergoing length contraction and which is undergoing time dilation because they're physically identical. You can swap them over and it makes no difference what so ever because all four dimensions are part of a four dimensional lattice with each dimension at right angles to the other three.

Edited June 13, 2015 by A-wal

[A-wal]

A quarter the speed of light:

 

Three quarters the speed of light:

[Pmb]

You didn't grasp what I meant when you responded saying "That's a complete misconception..."  I was explaining that physically time is not equivalent to space. What you lack the understanding of is that it only applies to the equations of physics and not to physical reality. It's only treated as being equivalent mathematically. Every expert in relativity knows this and I know some of the best experts. Einstein made this very clear in his article A Brief Outline of the Development of the Theory of Relativity in Nature Fe. 17, 1921 page 783

 

From this it follows that, in respect of its role in the equations of physics, though not with regard to its physical significance, time is equivalent to the space co-ordinates (apart from its relation to reality).

If someone were to look at a manifestly relativistic covariant equation then they'd be unable to tell which was the space coordinate and which was the time coordinate unless they looked at the metric or how the coordinates are defined. When I explained that one cannot rotate a clock into a rod it means that time is not physically the same thing as space. I didn't create that analogy. It was created by Richard C. Tolman and is found in his relativity text.

 

All that stuff you wrote about clocks and rods being transformed into another is just a sign of your ignorance of special relativity. I know what I'm talking about since I've been a relativist for over two decades and I run into ignorant people such as yourself on a daily basis posting nonsense like this. And I've been trained by some of the best relativists and physicists that there are.

[A-wal]

You can also use this method to show the path of an accelerating object by simply curving the line so that it moves to a higher relative velocity. If the acceleration remains constant then the angle will gradually decrease as the object's velocity increases relative to the inertial observer on the horizontal axis. If someone with a genuine understanding of relativity could confirm that this way of expressing relative velocities gives accurate measurements of the amount of time dilation and length contraction that occur or post a formula for comparison so it can be properly checked I'd be grateful. 

 

You didn't grasp what I meant when you responded saying "That's a complete misconception..."

You made an error and I corrected it. I understood exactly what you meant despite you saying that: "If time and space were actually the same thing then you could rotate a clock into a clock. Since we know such things are meaningful we can easily see that time is not the same as space." When in fact you meant: If time and space were actually the same thing then you could rotate a clock into a rod. Since we know such things aren't meaningful we can easily see that time is not the same as space. With regard to the physical relationships described by special relativity, time and space are equivalent and interchangeable and that was the point.

 

All that stuff you wrote about clocks and rods being transformed into another is just a sign of your ignorance of special relativity.

What exactly is it about "all that stuff I wrote about clocks and transformed into another" that you believe demonstrates an ignorance of special relativity? Please be specific.

 

I'm curious, how do you believe somebody with that much ignorance of special relativity would be able to brake it down to its simplest form and come up with a way to express all physical relationships described by the theory and more besides using just a piece of paper and angles? Blind luck?

 

I know what I'm talking about since I've been a relativist for over two decades and I run into ignorant people such as yourself on a daily basis posting nonsense like this. And I've been trained by some of the best relativists and physicists that there are.

I find that extremely difficult to believe, because of statements such as this:

Also the four dimensions are not at right angles to each other. That's not a physically meaningful statement. The only thing that can be at right angles are spatial axes.

General relativity uses four dimensional spacetime as framework and each dimension is at right angles to the other three. It's what the whole theory is based on. If what you claim was true then you really should know that so I hope for your sake that it's not true.

 

And this:

What you lack the understanding of is that it only applies to the equations of physics and not to physical reality. It's only treated as being equivalent mathematically. Every expert in relativity knows this and I know some of the best experts.

I would expect somebody with a genuine understanding of physics to know that the equations of physics are a description of physical reality so it makes no sense to claim that it only applies to the equations and not physical reality itself. The point is that in regards to the physics of special relativity, time and space are interchangeable. That's the point I was making and that's what's important for this topic.

 

The fact that they're interchangeable allows you to substitute time with a second spacial dimension and show the amount of time dilation and length contraction that occurs at different relative velocities. This isn't just a way to show the relationship between relative velocity and time dilation/length contraction, it's literally why it happens. The only difference is that it uses two spatial dimensions instead of one spatial dimension and time. It works because there's no distinction in relativity.

Edited July 6, 2015 by A-wal

[A-wal]

Now for the fun part. I don't think you'll find this in special relativity but accelerations add together in exactly the same way that relative velocities do. Acceleration is to energy as relative velocity is to mass. An accelerating object can slow the speed of light from their frame of reference by increasing their acceleration but they can never slow it to the same velocity as themselves. The way accelerations have less effect of the velocity of light relative to an accelerator is identical to the way increases in velocity have less effect on an objects velocity relative to a second object the higher the velocity gets.

 

This means that the diagrams I posted can just as easily be used to see the effects of acceleration simply by substituting acceleration for relative velocity. Now the horizontal axis represents zero acceleration and the angle is the amount of acceleration, so a constant rate acceleration will be represented by a straight line rather than a curved one. The amount of shortening you get when you trace down from the tip of the accelerator's line is the amount of time dilation and length contraction the accelerator experiences to keep their velocity below the speed of light relative to the speed of light itself.

[CraigD]

So does this work? I can't find a simple table for time dilation and length contraction at various speeds.

You can make your own table easily using the reciprocal of Lorentz factor, [math]\alpha = \frac1{\lambda} = \sqrt{1-v^2}[/math], which you can write in this simple form if the unit of the relative velocity [math]v[/math] is the speed of light [math]c[/math]. From this, you can get a table for either time dilation or length contraction, such as this one.

[math]v \,\,\,\,\,\, \alpha[/math]

0.0 1.00000000

0.1 0.99498744

0.2 0.97979590

0.3 0.95393920

0.4 0.91651514

0.5 0.86602540

0.6 0.8

0.7 0.71414284

0.8 0.6

0.9 0.43588989

1.0 0.0

 

Rel.GIFObviously this is very rough, I did it in paint but it's good enough for now just to see if it could work.

It’s useful and mathematically OK to visualize [math]\alpha[/math] like this, because the equation for a unit circle, [math]x^2 +y^2 = 1[/math], can be algebraically transformed into it:

[math]x^2 +y^2 = 1[/math]

[math]x^2 = 1 -y^2[/math]

[math]x = \sqrt{1 -y^2}[/math]

 

When using a circle like this

to visualize the Lorentz factor, though, it’s important not to confuse the sketch with a Minkowski spacetime diagram, like this one

(image from Richard Baker’s wonderful Sharp Blue pages)

 

A 45° line in the unit circle visualization of the Lorentz factor for [math]v=0.5 \,\mbox{c}[/math] is not the same as the one in the spacetime diagram for [math]v=\mbox{c}[/math].

[A-wal]

Cheers. I'm not seeing the equations in the right format. I'm seeing [ math ]...[ /math ] without the spaces.

 

I can't get this to work anyway. I've compared it to the time dilation and length contraction at various relative velocities and it doesn't seem to match. It works at zero relative velocity and at the speed of light but it's the small bit between I'm having trouble with. :)

 

I'm sure this can work if it's done right. I'm going to try again when I get round to it.

 

Edit:

Ah, I can see the diagrams now. Still not seeing the equations though. So this does work. Damn, I thought I'd found a cool new way of describing it.

Edited July 18, 2015 by A-wal

[pgrmdave]

The three spatial dimensions are obviously at right angles to each other and special relativity shows that time is no different by treating coordinate time as simply another spatial dimension. In relativity there's no distinction and you don't need to specify which dimension is undergoing length contraction and which is undergoing time dilation because they're physically identical. 

Time and space are the same except for when they're distinctly different.  There are distinct differences between a spatial dimension and a time dimension, mostly having to do with causation.  Your left and my left can be different, your up and my up can be different, but your future and my future are always in the exact same direction.

[Pmb]

A-wal claims that - ...special relativity shows that time is no different by treating coordinate time as simply another spatial dimension. In relativity there's no distinction and you don't need to specify which dimension is undergoing length contraction and which is undergoing time dilation because they're physically identical.

 

Clearly wrong. E.g. time dilates whereas space contracts.  By merely looking at the metric one can see which coordinate is the temporal one and which are the spatial ones. And while particles travel in any direction in space they never travel backwards in time. By looking at the Lorentz transformation which relates coordinates between inertial frames its obvious that the temporal and spatial coordinates transform differently. No relativist/physicist worth his salt would make such a clearly wrong statement.

 

pgrmdave - A-wal has all of this quite wrong. His grasp of relativity and physics in general is quite poor since he makes a lot of mistakes like this. Your response is quite correct. I've provide other opinions from Einstein and his colleagues on the subject of the difference between time and space in spacetime in what follows.

 

As Einstein himself said, this is wrong. From Nature, Feb. 17, 1921, page 783

 

From this it follows that, in respect to its role in the equations of physics, though not with regard to its physical significance, time is equivalent to space co-ordinates (apart from their relations of reality).

Of course this point of view is universal for those who have a correct understanding of relativity, which A-wal does not. This means that all of Einstein's followers agree with what Einstein said here. For example; from Relativity; Thermodynamics and Cosmology by Richard C. Tolman, Dover Pub, page 29. In the section 14 entitled The three plus one dimensions of space-time

 

In using this language it is important to guard against the fallacy of assuming that all directions in the hyper-space are equivalent, and of assuming that extension in time is of the same nature as extension in space merely because it may be convenient to think of them as plotted along perpendicular axes. A similar fallacy would be to assume that pressure and volume are the same kind of quantity because they are plotted at right angles in the diagram on a pv indicator card. That there must be a difference between spatial and temporal axes in our hyper-space is made evident, by contrasting the physical possibility of rotating a meter stick from an orientation where it measures distances in the x-direction to one where it measures distances in the y-direction, with the impossibility of rotating it into a direction where it would measure time intervals - in other words the impossibility of rotating a meter stick into a clock.

 

From a much more modern text Introduction to Special Relativity by Wolfgang Rindler, page 51

 

Care must be taken, however, not to regard spacetime as a straightforward generalization of ordinary Euclidean three-space to four dimensions, with time as just one more dimension. Owing to the distribution of signs in the metric, the time coordinate x^0 is not on the same footing as the three space-coordinates, and spacetime consequently has non-isotropic properties quite unlike Euclidean space.

 

Note: It's important to note that one of the major differences between Euclidean space and spacetime is that they have very different metrics.

Edited August 23, 2015 by Pmb

[A-wal]

A-wal claims that - ...special relativity shows that time is no different by treating coordinate time as simply another spatial dimension. In relativity there's no distinction and you don't need to specify which dimension is undergoing length contraction and which is undergoing time dilation because they're physically identical.

 

Clearly wrong. E.g. time dilates whereas space contracts.  By merely looking at the metric one can see which coordinate is the temporal one and which are the spatial ones. And while particles travel in any direction in space they never travel backwards in time. By looking at the Lorentz transformation which relates coordinates between inertial frames its obvious that the temporal and spatial coordinates transform differently. No real relativist/physicist would make such a clearly wrong statement.

You are so funny!

 

The way that time dilates and length contracts are physically identicle! The fact that they have different names doesn't mean a thing. :) They're both distance shortening and they both happen in unsion which is why this works.

[CraigD]

 

A-wal claims that - ...special relativity shows that time is no different by treating coordinate time as simply another spatial dimension. In relativity there's no distinction and you don't need to specify which dimension is undergoing length contraction and which is undergoing time dilation because they're physically identical.

 

Clearly wrong. E.g. time dilates whereas space contracts. By merely looking at the metric one can see which coordinate is the temporal one and which are the spatial ones. And while particles travel in any direction in space they never travel backwards in time. By looking at the Lorentz transformation which relates coordinates between inertial frames its obvious that the temporal and spatial coordinates transform differently. No real relativist/physicist would make such a clearly wrong statement.

You are so funny!

 

The way that time dilates and length contracts are physically identicle! The fact that they have different names doesn't mean a thing. :) They're both distance shortening and they both happen in unsion which is why this works.

 

<moderator hat>

I think we’d be more aligned with Hypography’s of “science for everyone” if we didn’t call one another “no real relativist/physicist” and “so funny!” Let’s be more polite and respectful, please!

</moderator hat>

 

I don’t have the physics education that Pmb has. I had 8 hours of undergraduate Modern Physics on my way to a BS in Mathematics and a long, ongoing, essentially non-scientific professional career in health care software engineering. But I think I can explain his response to A-wal enough to bridge the gap between them.

 

One of the most useful tools for understanding Special Relativity is the concept of Minkowski spacetime (M or R^{3,1}). Essentially, it allows us to take the familiar idea of 3-dimensional Euclidean space (E³ or R³), in which any point can be described by a ordered collection of 3 real numbers, or “3-vector”, and add to its 3 spatial (which can also be called length) dimensions a 4th dimension representing time (duration).

 

Here’s an example of how this can be used. I want to represent a simple body (or geometric figure), a cube 1-meter on its side, in E³, using the simple Cartesian coordinate scheme. I could do this with 3 (x,y,z) 3-vector pairs, (0,0,0)-(1,0,0), (0,0,0)-(0,1,0), (0,0,0)-(0,0,1), understanding that they are intended to define a cube in units of 1 meter.

 

This body can be transformed by rotating it.

 

For example, rotating it 45% on its z then y axis can be done by multiplying each vector by the rotation matrixes

[math]

\begin{bmatrix}

\cos 45^\circ & -\sin 45^\circ & 0 \\

\sin 45^\circ & \cos 45^\circ & 0 \\

0 & 0 & 1 \\

\end{bmatrix}

\begin{bmatrix}

\cos 45^\circ & 0 & \sin 45^\circ \\

0 & 1 & 0 \\

-\sin 45^\circ & 0 & \cos 45^\circ \\

\end{bmatrix}

[/math]

giving (0,0,0)-(0.5,0.5,2^0.5), (0,0,0)-(-2^0.5,2^0.5,0), (0,0,0)-(0.5,0.5,-2^0.5), which is approximately (0,0,0)-(0.5,0.5,0.7071), (0,0,0)-(-0.7071,0.7071,0), (0,0,0)-(0.5,0.5,-0.7071)

 

 

In E³, this body is “timeless” (and thus can’t have velocities), bounded only in space. To be bounded in M it must have defined time coordinate. So I could represent my cube existing for 1 second with this ordered collection of (ct,x,y,z) 4-vectors:

((0,0,0,0)-(0,1,0,0))-((299792458,0,0,0)-(299792458,1,0,0)),

((0,0,0,0)-(0,0,1,0))-((299792458,0,0,0)-(299792458,0,1,0)),

((0,0,0,0)-(0,0,0,1))-((299792458,0,0,0)-(299792458,0,0,1)),

 

Because M has time, bodies represented in it have velocities (velocity = [math]\Delta[/math]length/[math]\Delta[/math]time) and speeds. Thus, these bodies can not only be transformed by rotating them, but by giving them non-zero velocities.

 

For example, changing the 1 meter x 1 second cube in the previous example, which has speed 0, to have a velocity of 179875474.8 m/s (0.6c) in the x direction can be done by multiplying each vector by the Lorentz transformation matrix

[math]

\begin{bmatrix}

1.25 & -0.75 & 0 & 0 \\

-0.75 & 1.25 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1 \\

\end{bmatrix}

[/math]

giving

((0,0,0,0)-(-0.75,1.25,0,0))-((374740572.5,-224844343.5,0,0)-(374740571.75,-224844342.25,0,0)),

((0,0,0,0)-(0,0,1,0))-((374740572.5,-224844343.5,0,0)-(374740572.5,-224844343.5,1,0)),

((0,0,0,0)-(0,0,0,1))-((374740572.5,-224844343.5,0,0)-(374740572.5,-224844343.5,0,1))

 

Here I need some help :Exclamati I think my naive assumptions about simultaneity in setting up my initial R^3,1 figure so simply has made their transformation arithmetically correct, but confusing!

[Pmb]

re - the way that time dilates and length contracts are physically identical !

 

Far from it. There were too many errors in your claims for me to want to bother correcting so I'll just correct this one to make a point, i.e. you don't know what you're talking about in relativity,

 

In what follows let there be an inertial frame S and let S' be an inertial frame in standard configuration with S.

 

Let the time interval between two events A and B which occur at the same location in S be dt (dt = proper time). Then it's value in S' will be dt' = dt/sqrt[1 - v^2/c^2]. I.e. dt' > dt  - Time dilation.

 

Let there be a rod of proper length L be at rest in S. Then the rod's length in S' will be L' = L*sqrt[1 - v^2/c^2]. I.e. L' < L - Length contraction.

 

This clearly demonstrates that you claim is wrong, as usual.

[Pmb]

CraigD - The gap between A-wal and I is that I thoroughly understand SR and he has a very poor grasp of it, close to that of a layman. And when I point out his mistakes his understanding is so poor he can't understand the correction to it. Not to mention that his personality is not of the type that allows one to admit their mistakes. And he kept it up in this thread posting rude responses when I corrected him. Why do you allow him to use that kind of tone with people?

 

If you'd like to see a nice application of spacetime see my page at http://home.comcast.net/~peter.m.brown/sr/invariant_mass.htm

[A-wal]

re - the way that time dilates and length contracts are physically identical !

 

Far from it. There were too many errors in your claims for me to want to bother correcting so I'll just correct this one to make a point, i.e. you don't know what you're talking about in relativity,

If an object is moving away at a high relative velocity then its speed (distance over time) is slowed by the same amount in both dimensions or the two would cancel each out. I can't believe I even have to explain this. You even posted an Einstein quote saying that they're mathematically identical! The rate that time dilates and the way that length contracts are physically identical!

 

You said I made too many wrong claims to correct but the one you chose is a massive error on your part. You are just so full of crap. Name one genuine error I've made.

 

This is basic fundamental stuff that you have again shown that you don't understand, like not knowing that gr describes gravitation as the process of massive objects following straight paths through curved spacetime or thinking that it's not possible to prove a scientific model wrong. I'll add this one to the list. I would expect someone with no understanding of relativity to be able to pick up a book and within a few hours make less mistakes and have a deeper understanding than you, and yet you claim to be an expert?

 

In what follows let there be an inertial frame S and let S' be an inertial frame in standard configuration with S.

 

Let the time interval between two events A and B which occur at the same location in S be dt (dt = proper time). Then it's value in S' will be dt' = dt/sqrt[1 - v^2/c^2]. I.e. dt' > dt  - Time dilation.

 

Let there be a rod of proper length L be at rest in S. Then the rod's length in S' will be L' = L*sqrt[1 - v^2/c^2]. I.e. L' < L - Length contraction.

 

This clearly demonstrates that you claim is wrong, as usual.

That shows that it happens at the same rate! What the hell is wrong with you?

 

CraigD - The gap between A-wal and I is that I thoroughly understand SR and he has a very poor grasp of it, close to that of a layman. And when I point out his mistakes his understanding is so poor he can't understand the correction to it. Not to mention that his personality is not of the type that allows one to admit their mistakes. And he kept it up in this thread posting rude responses when I corrected him. Why do you allow him to use that kind of tone with people?

:cry: ing again because you're not getting your own way. The only one who's making mistakes is you and the only one who should be warned about their conduct because of the way they talk to people is also you. You do it to others as well in other topics. What would motivate a person to do that? I'm guessing it comes from of lack self-esteem stemming from failing at everything you try so you feel the need to pretend to be some kind of condescending expert (which would be annoying even if you did have the first clue of what you're talking about) on something that you can't grasp yourself.

 

The only person I don't show respect to here is you because you clearly don't deserve any.

Edited August 29, 2015 by A-wal