Is Time Dilation Misunderstood in Popular Science Videos?

[m4r35n357]

Sorry but I can't restrain myself . . ;)

The ship's pilot is a trained, intelligent person, who understands relativity. He calculated the route before traveling, and knew full well that at 0.8c he would age 7.5 years over the course of the trip, even without checking his clock at the destination.

Describing this situation in terms of length contraction is merely inviting the division that seb7 cannot resist the urge to perform (if he knows to divide the numerator by gamma first then he also knows the spacetime interval).

Space does not magically contract at the whim of the traveler. It is merely an illusion. Here's how I would describe the situation:

The Doppler factor is 3, so over the whole journey the traveler sees the home clock advance by 2.5 years (1/3 of his own clock rate), and the destination clock advance by 22.5 years (3 times his own rate).

Before the journey the home clock reads (say) 0 years and the destination reads -10 years (they are synchronized). After the journey the home clock reads 0 years and the destination clock reads 10 years.

So, at the start of the journey, the destination clock jumps (in reality there would be finite acceleration so it wouldn't really be instantaneous) back by 2.5 years (from -10 to -12.5), and at the end of the journey the home clock jumps back 2.5 years (from 2.5 to 0).

Clear as mud? Maybe, but no sleight of hand with space "compressing" (I just hope I've got the maths right).

 

[m4r35n357]

Mr Tompkins in Paperback by Gamow (first 4 chapters)

But that is just wrong. Nobody would ever see (with their eyes) what is in those illustrations. The "real" situation is far more varied and interesting.

In particular see http://www.spacetimetravel.org/tuebingen/tuebingen.html

 

[FactChecker]

But that is just wrong. Nobody would ever see (with their eyes) what is in those illustrations. The "real" situation is far more varied and interesting.

It is a dream where the speed of light is vastly reduced so that the relativity effects are evident in everyday life. Gamow was a famous scientist.

 

[m4r35n357]

It is a dream where the speed of light is vastly reduced so that the relativity effects are evident in everyday life.

I know that.

Please read my edited post, in particular the second link. Here is a quote:

"However, this picture is entirely wrong. It is wong because Gamov did not take the consequences of the finite speed of light into account. In everyday life we may to all intents and purposes pretend that the speed of light is infinite, since we are always concerned with much smaller velocities. But when a velocity close to the speed of light is involved, this is not justified any more. On the contrary, what we should see in this case would be substantially influenced by the fact that light propagates at a finite speed."

 

[FactChecker]

I know that.

Please read my edited post, in particular the second link. Here is a quote:

"However, this picture is entirely wrong. It is wong because Gamov did not take the consequences of the finite speed of light into account. In everyday life we may to all intents and purposes pretend that the speed of light is infinite, since we are always concerned with much smaller velocities. But when a velocity close to the speed of light is involved, this is not justified any more. On the contrary, what we should see in this case would be substantially influenced by the fact that light propagates at a finite speed."

Yes. There is also a rotation. It is important in explaining some of the paradoxes. I am sure that Gamow was aware of that. So what? It takes about an hour to read the first 30 pages of the book. It is fun to read and educational, if not complete. The criticism that Gamow did not take into account that the speed of light is finite puzzles me. This book describes a dream where the speed of light is not only finite, it is small. Other than the rotation, I don't know what the complaint is.

 

[m4r35n357]

Yes. There is also a rotation. It is important in explaining some of the paradoxes. I am sure that Gamow was aware of that. So what? It takes about an hour to read the first 30 pages of the book. It is fun to read and educational, if not complete.

More quotes:

"But it was not until 1924 that Anton Lampa [4] described the visual effects for the first time."

"And this notion was in fact illustrated by the physicist George Gamov in his 1940 book “Mr. Tompkins in Wonderland” "

So he must have known (or been ignorant of the facts, which I doubt), and yet he still wrote the book, and people are still being taught that this is what we "see" more than 90 years after it was proved false . . .

 

[FactChecker]

More quotes:

"But it was not until 1924 that Anton Lampa [4] described the visual effects for the first time."

"And this notion was in fact illustrated by the physicist George Gamov in his 1940 book “Mr. Tompkins in Wonderland” "

So he must have known (or been ignorant of the facts, which I doubt), and yet he still wrote the book, and people are still being taught that this is what we "see" more than 90 years after it was proved false . . .

The rotation is the only effect I see mentioned in the link. Obviously Gamow took the finite speed of light into account. I can't see how the link could possibly imply otherwise.

 

[m4r35n357]

The rotation is the only effect I see mentioned in the link. Obviously Gamow took the finite speed of light into account. I can't see how the link could possibly imply otherwise.

Gamow's illustrations are based on a literal coordinate description of the scene. The simultaneity that underpins the coordinate description is not observable, since it is spacelike. We observe down light cones; we do not observe simultaneity. Terrell rotation is just one very specific example of several effects that Gamow omits (aberration, doppler, beaming).

Anyway, I think this thread is in danger of going off-topic, so I'll stop now as I am in danger of hiding post #31, which was my on-topic point.

 

[FactChecker]

Gamow's illustrations are based on a literal coordinate description of the scene. The simultaneity that underpins the coordinate description is not observable, since it is spacelike. We observe down light cones; we do not observe simultaneity. Terrell rotation is just one very specific example of several effects that Gamow omits (aberration, doppler, beaming).

Yes. The illustrations in Mr Tompkins are not 100% complete and accurate. I don't think they were meant to be. But the OP video was so bad that alternatives were called for. In the first 30 seconds of that video, it gave rotating, orbiting planets as examples of inertial reference frames. Certainly that misconception would cause problems in the most elementary physics classes.

 

[Herman Trivilino]

The Doppler factor is 3, so over the whole journey the traveler sees the home clock advance by 2.5 years (1/3 of his own clock rate), and the destination clock advance by 22.5 years (3 times his own rate).

You criticized Gamow for using "sees" in the same way that you are using it. By the time the traveler arrives at the destination he will not have received all the signals yet, so he won't have yet "seen" the clock advance by that much. He'll have to wait for some time while the signal completes its journey. Also, when the ratio of frequencies is 3 the ratio you are using for clock rates is not $\frac{1}{3}$, it's $\frac{3}{5}$.

That ratio describes what people call time dilation. You may prefer to not call it that, but it is nevertheless not an illusion. Likewise for length contraction. If you like, you can think of those terms as describing a ratio that's not equal to one. The fact that people are using different terms than you to describe the same thing you are describing does not make what those terms are describing an illusion.

 

[m4r35n357]

You criticized Gamow for using "sees" in the same way that you are using it. By the time the traveler arrives at the destination he will not have received all the signals yet, so he won't have yet "seen" the clock advance by that much. He'll have to wait for some time while the signal completes its journey. Also, when the ratio of frequencies is 3 the ratio you are using for clock rates is not $\frac{1}{3}$, it's $\frac{3}{5}$.

That ratio describes what people call time dilation. You may prefer to not call it that, but it is nevertheless not an illusion. Likewise for length contraction. If you like, you can think of those terms as describing a ratio that's not equal to one. The fact that people are using different terms than you to describe the same thing you are describing does not make what those terms are describing an illusion.

Thanks for taking the time to reply, but I don't think you are getting my point. The Doppler factor is 3 or 1/3 so that is the rate the the traveler sees (compared to his own clock). I prefer not to call it time dilation because it is not time dilation, it is the Doppler factor (you could say it "includes" time dilation, gamma (which is 3/5), or whatever). He sees all this happen in real time so he does not have to wait for anything.

I am using the word "sees" to mean precisely what it says. I might have made an error somewhere, but not where you are pointing to.

I think you might be suffering from Mr Gamow's "education" ;)

Try looking at this if you don't see my point, and the link inside it too. Drawing an accurately scaled spacetime diagram will help too.

 

[Herman Trivilino]

If errors were made in your analysis, please tell us what you think they were so that we can get them fixed. Then I might be able to understand what it is you're saying.

The ship's pilot is a trained, intelligent person, who understands relativity. He calculated the route before traveling, and knew full well that at 0.8c he would age 7.5 years over the course of the trip, even without checking his clock at the destination.

Can you show us how he did that? And explain what the calculation means. Like, for example, if using the Lorentz transformation, explain what the transformed and untransformed values mean, and what a comparison of them means.

The Doppler factor is 3, so over the whole journey the traveler sees the home clock advance by 2.5 years (1/3 of his own clock rate), and the destination clock advance by 22.5 years (3 times his own rate).

How does he "see" that the clock has advanced 2.5 years? Does he look at it with a telescope or does he "see" it some other way?

 

[m4r35n357]

If errors were made in your analysis, please tell us what you think they were so that we can get them fixed. Then I might be able to understand what it is you're saying.
Can you show us how he did that? And explain what the calculation means. Like, for example, if using the Lorentz transformation, explain what the transformed and untransformed values mean, and what a comparison of them means.
How does he "see" that the clock has advanced 2.5 years? Does he look at it with a telescope or does he "see" it some other way?

1. I don't know of any, that's why I'm asking.
2. $\tau^2 = t^2 - x^2 = x^2 / v^2 - x^2 = 100 * (1/0.64 - 1) = 56.25 = 7.5^2$
3. Yes, with a telescope
4. Goodnight, it's 11pm here and I have to work tomorrow . . . enjoy ;)

 

[m4r35n357]

OK I found my error, and had to get up early to correct it! I shall start afresh:

The ship's pilot is a trained individual who understands relativity. He planned the route before traveling, and knew full well (by calculating the spacetime interval as in #43) that at 0.8c he would age 7.5 years over the course of the trip, even without checking his clock at the destination.

The Doppler factor is 3, therefore he also knows in advance that over the whole journey the home clock will advance by 2.5 years (1/3 of his own clock rate), and the destination clock by 22.5 years (3 times his own rate).

Before the journey the home clock reads (say) 0 years and the destination reads -10 years (they are synchronized).

After the journey the home clock reads 2.5 years and the destination clock reads 12.5 years (the time it takes to travel 10 ly at 0.8c, and the part I forgot to take into consideration!).

QED

I shall now write out 100 times: CLOCKS DO NOT "JUMP" IN THE DOPPLER ANALYSIS!

 

[seb7]

ok, so the light clock explanation helps to get a point across; but I'm correct in saying it can not be the actual reason for time dilation?
(surely if this were correct then wouldn't electrons have an elliptic orbit, and some kind of zero rest frame would have to exist where electrons had a perfectly circular orbit.)

They would see the length contracted to half, so the speed is 0.87 c.

meaning, as time halfs, distance to travel doubles?

If this were true, then what would happen if our ship circled the Earth 0.87c for a year (earth time). Both Earth and the ship occupants wouldn't both see a travel speed of 0.87c?
(ignoring gravitational effects on time)

 

[Nugatory]

ok, so the light clock explanation helps to get a point across; but I'm correct in saying it can not be the actual reason for time dilation?

If you want to understand the "actual reason" for time dilation, there are two possibilities:
1) Learn about the relativity of simultaneity, as has been suggested above. When we say that clock clock A and clock B are both running at the same rate, we're really saying that if we look at clock A twice, and both times we look at clock B at the same time, the difference between the two readings from clock A will be the same as the difference between the two readings from clock B. Because of the relativity of simultaneity, different observers will mean different things by that all-important phrase "at the same time", and depending on ho wthey define "at the same time" may find one clock running slower than the other.

It's worth repeating that this phenomenon is not the differential aging that appears in the twin paradox.

2) Recognize that it's just the way the geometry of spacetime works. There are two possible geometries: in one of them the speed of light is the same for all observers and relativistic effects such as time dilation appear; in the other the speed of light is constant relative to some hypothetical fixed point somewhere in the universe, everyone else measures the speed of light to be the difference between $c$ and their motion relative to that fixed point. Experiments show that we live in the first kind of universe.

(surely if this were correct then wouldn't electrons have an elliptic orbit, and some kind of zero rest frame would have to exist where electrons had a perfectly circular orbit.)

It's off-topic here, but electrons don't have any sort of orbit at all. The idea that electrons are particles moving in circles or ellipses (or any trajectory at all) around the nucleus was abandoned more than three-quarters of a century ago.

 

[Herman Trivilino]

ok, so the light clock explanation helps to get a point across; but I'm correct in saying it can not be the actual reason for time dilation?

The light clock demonstrates time dilation. Without a light clock you'd still have time dilation, so no, the light clock is not the reason for time dilation. The reason is that inertial reference frames are equivalent and the speed of light is the same in all of them.

(surely if this were correct then wouldn't electrons have an elliptic orbit, and some kind of zero rest frame would have to exist where electrons had a perfectly circular orbit.)

A hoop has a circular shape in its rest frame. The hoop is at rest relative to the observer. Another observer, moving relative to the first, could carry with him and exact duplicate of the hoop. Each would agree that their hoop is circular. Each could determine that the other's hoop is not circular.

meaning, as time halfs, distance to travel doubles?

No, you are talking about this as if all you need is one frame of reference to describe it. There are always (at least) two, otherwise time dilation and length contraction do not occur.

If this were true, then what would happen if our ship circled the Earth 0.87c for a year (earth time). Both Earth and the ship occupants wouldn't both see a travel speed of 0.87c? (ignoring gravitational effects on time)

At any instant, yes. Each would observe the other having a speed of 0.87 c.

 

[Herman Trivilino]

The ship's pilot is a trained individual who understands relativity. He planned the route before traveling, and knew full well (by calculating the spacetime interval as in #43) that at 0.8c he would age 7.5 years over the course of the trip, even without checking his clock at the destination.

2. $\tau^2 = t^2 - x^2 = x^2 / v^2 - x^2 = 100 * (1/0.64 - 1) = 56.25 = 7.5^2$

So, we have $\tau=7.5$ and $t=12.5$. But the time did not dilate from 7.5 to 12.5? How would you describe the relationship between those two amounts of time?

The traffic controller stationed on Earth is a trained individual who understands relativity, and knows full well (having done the same calculation) that at a speed of 0.8 c the clocks aboard the ship will register a time of 7.5 years upon arrival. He therefore reckons that the ship will travel a distance $$x'=vt=(0.8)(7.5)=6.$$ Previous experiments with light signals determined that the proper distance is $L=10$.

So, we have $L=10$ and $x'=6$. But the length did not contract from 10 to 6? How would you describe the relationship between those two amounts of distance?

 

[m4r35n357]

So, we have $\tau=7.5$ and $t=12.5$. But the time did not dilate from 7.5 to 12.5? How would you describe the relationship between those two amounts of time?

The traffic controller stationed on Earth is a trained individual who understands relativity, and knows full well (having done the same calculation) that at a speed of 0.8 c the clocks aboard the ship will register a time of 7.5 years upon arrival. He therefore reckons that the ship will travel a distance $$x'=vt=(0.8)(7.5)=6.$$ Previous experiments with light signals determined that the proper distance is $L=10$.

So, we have $L=10$ and $x'=6$. But the length did not contract from 10 to 6? How would you describe the relationship between those two amounts of distance?

Pardon the deleted post if you saw it, I was not happy with my initial answer.

On further reflection, I am just going to say that we agree with the relative times by doing things different ways. You are using a calculation of time dilation and I am using the Doppler shift (because I like to deal with things I can see). It looks like we are not going to agree on the approach, although we both get the same answers. In fact, since my calculation of space-time interval effectively contains gamma, we are not even using different mathematics.

As to length contraction, I maintain (unlike the ratio or difference between the two times) that there is nothing that is actually measurable at 6ly, so I have no interest or opinion on that particular relationship.

So, I have effectively come to the same conclusion as in my deleted post.

 

[Herman Trivilino]

On further reflection, I am just going to say that we agree with the relative times by doing things different ways.

Yes. And I agree that your way is just as valid as mine.

In fact, since my calculation of space-time interval effectively contains gamma, we are not even using different mathematics.

The two methods are equivalent. I have no issues with either of them.

As to length contraction, I maintain (unlike the ratio or difference between the two times) that there is nothing that is actually measurable at 6ly, so I have no interest or opinion on that particular relationship.

If the ship had left a trail of bread crumbs and you were in another ship constrained to travel with the same velocity, is there no way for you to determine its length using only measurements taken from your ship?

If there's an airport runway on the surface of planet Earth, and you fly parallel to it at a speed of 0.8 c is there no way for you to determine its length using only measurements taken aboard your ship?

 

[m4r35n357]

If the ship had left a trail of bread crumbs and you were in another ship constrained to travel with the same velocity, is there no way for you to determine its length using only measurements taken from your ship?

If there's an airport runway on the surface of planet Earth, and you fly parallel to it at a speed of 0.8 c is there no way for you to determine its length using only measurements taken aboard your ship?

That is the point at which my brain screams "enough!" and I lose the will to continue ;)
I'm sure if I chose to pursue it (and I have done in the past) I would agree with you. Thing is, I have a scaled spacetime diagram in front of me, and I see (sic) nothing on it which corresponds to a distance of 6ly. I just don't get what use it is to me that's all, and utility is what motivates me to work things out.
To labour the point, I know that space does not really contract, so at best it is just a calculation resulting from division of two numbers, and not even an optical illusion, since in reality it would actually appear to stretch in front of me and contract behind me.

 

[Herman Trivilino]

Thing is, I have a scaled spacetime diagram in front of me, and I see (sic) nothing on it which corresponds to a distance of 6ly. I just don't get what use it is to me that's all, and utility is what motivates me to work things out.

That's because on your spacetime diagram you're focusing on two events separated by an interval that is timelike. You can find a triangle on there with a hypotenuse of 7.5 y. One leg of that triangle is 10 ly, the other is 12.5 y. Time dilation is characterized by the ratio $\gamma=\frac{12.5}{7.5}$.

To labour the point, I know that space does not really contract, so at best it is just a calculation resulting from division of two numbers,

Suppose you and your friend are separated on a train by a distance of 7.5 light microseconds. The train zips through a tunnel at a speed of 0.8 c. At the same time you each reach out a window and place a mark on the tunnel wall. Later, when you're at rest relative to the tunnel, you measure the distance between the marks to be 12.5 light microseconds. Drawing this on a spacetime diagram you have a triangle with a hypotenuse of 7.5 light microseconds. One leg of that triangle is 10 microseconds, the other is 12.5 light microseconds. Length contraction is characterized by the ratio $\gamma=\frac{12.5}{7.5}$.

If space doesn't really contract, how is it that the distance between the marks is 12.5 when you measure it with a tape measure, yet when you use that same tape to measure the distance between you and your friend you get 7.5?

This time we're focused on two events separated by an interval that is spacelike.

 

[m4r35n357]

This time we're focused on two events separated by an interval that is spacelike.

[been guitaring this week!]
Yep, different scenario altogether. Your "time dilation" in the previous scenario was just the difference between two invariants (which I am fine with in principle), but the "space contraction" was coordinate dependent (it went away when relative motion ceased).

If your latest scenario is a "dual" of the previous one then I can conceive of calling the difference between two proper distances measured at various times "space contraction". I just don't find it a particularly instructive or satisfying concept (my choice).

As I have stated over and over, IMO what we can see/measure/observe trumps all.

 

[seb7]

If space doesn't really contract, how is it tha

isnt this just caused by the time dilation?

We can consider any object with velocity to also be at rest, so..
Two ships traveling side by side, already moving at 0.8c. One of the ships then accelerates again attempting to reach +0.8c from the ship they left behind. Does the ship behind observe them achieve another 0.8c?

 

[phinds]

isnt this just caused by the time dilation?

We can consider any object with velocity to also be at rest, so..
Two ships traveling side by side, already moving at 0.8c. One of the ships then accelerates again attempting to reach +0.8c from the ship they left behind. Does the ship behind observe them achieve another 0.8c?

Yes, but relative to someone in the frame of reference in which you state that they both are originally traveling at .8c, the first one continues at .8c and the second one gets up to something like .96c (I'm making up the number but it's in the ball-park). SO, the second one is not achieving ANOTHER .8c, he's just moving away from the first one at .8c In exactly the same way as the first one is moving away from the "fixed" observer at .8c.

If the first ship were to attempt to move away from the fixed observer at "another" .8c, he could not do so, he could only approach c asymptotically.

It is your choice of the word "another" that is confusing you about the scenarios.

 

[seb7]

so 0.96c(est) from the original fixed position, but is +0.8c beyond the ship. So whos to say we can't travel beyond light speed, it just can't ever be observed. Could it be light itself is infinite but can only be observed at light speed. Might help explain some of the weird effects of it apparently knowing the future.

----

This is how I think of spacetime, which is why when I saw the video, it made me rethink. Does this not work with the calculations? let me know if I'm just plain wrong.

I see four dimensions. I think of space as 3 dimensions, time as a 4 dimension. Time is like 90 degrees from space, but always 0 degrees in front of us. Every object (including us) are always moving through these 4 dimensions (spacetime) at the speed of light, it's that speed that is fixed, never changes.
ie. when we think we are still, we are moving through time at lightspeed. When a moving object changes direction, we are also changing the angle we were moving through time, this angle now being different to other objects traveling at different speeds. The time angle still appears to us to be front, but its now a different angle to other objects moving through time. This angle directly relates to the time and distance dilation observed.
This seems to work for all the calculations I've done.

 

[phinds]

so 0.96c(est) from the original fixed position, but is +0.8c beyond the ship. So whos to say we can't travel beyond light speed, it just can't ever be observed.

That's ridiculous.You keep ignoring reference frames. Beyond the speed of light RELATIVE TO WHAT?

Could it be light itself is infinite but can only be observed at light speed.

No it could not.

Might help explain some of the weird effects of it apparently knowing the future.

If you think you can know the future, you're off into crackpot land.

This is how I think of spacetime

spacetime does not care how we think of it, it just does what it does and the math tells the tale.

which is why when I saw the video, it made me rethink. Does this not work with the calculations? let me know if I'm just plain wrong.

I'm not sure what you are talking about but based on your other statements I'd hazard a guess that you are just plain wrong.

I see four dimensions. I think of space as 3 dimensions, time as a 4 dimension. Time is like 90 degrees from space, but always 0 degrees in front of us. Every object (including us) are always moving through these 4 dimensions (spacetime) at the speed of light, it's that speed that is fixed, never changes.
ie. when we think we are still, we are moving through time at lightspeed. When a moving object changes direction, we are also changing the angle we were moving through time, this angle now being different to other objects traveling at different speeds. The time angle still appears to us to be front, but its now a different angle to other objects moving through time. This angle directly relates to the time and distance dilation observed.
This seems to work for all the calculations I've done.

I don't follow you so can't comment but it sounds wrong. What does it even MEAN to have time at an angle? If you're talking about space-time diagrams, then I would suggest that you learn the standard terminology, then we can converse with you about them.

 

[phinds]

@seb7 you seem to be having a problem with the fact that there is no absolute reference frame. Velocity is only meaningful in terms of comparison with another object. That is the why it's called the theory of "relativity" and that part of it was known well before Einstein.

 

[Herman Trivilino]

isnt this just caused by the time dilation?

Not really. Time dilation and length contraction are both consequences of the postulates. Starting from them you can derive time dilation and then use that to derive length contraction. That's the usual route, but it's not the only way to do it.

We can consider any object with velocity to also be at rest, so.. Two ships traveling side by side, already moving at 0.8c.

When you say "already moving" what does that mean? As you said, you could consider them to be at rest.

One of the ships then accelerates again attempting to reach +0.8c from the ship they left behind. Does the ship behind observe them achieve another 0.8c?

You're saying that the second ship is "attempting" to move at speed 0.8c relative to the first. If he succeeds then yes, each ship will see the other as moving at speed 0.8c.

Look at the figure in Post #25: This is a view from the frame of reference in which both ships were previously moving at speed 0.8c relative to A. What we are calling the "first ship" is B, and it's at rest in this frame of reference. The "second ship" is C, and it has succeeded in moving at speed 0.8c relative to the first ship, B. Note that B will see the distance between A and C increase at a rate of 1.6c.

Now let's look at things from the frame of reference of A. In this frame both ships were originally moving at speed 0.8c. Afterwards, the second ship C has succeeded in pulling away from the first ship B at speed 0.8c.

 

[MachPrincipe]

I see four dimensions. I think of space as 3 dimensions, time as a 4 dimension. Time is like 90 degrees from space, but always 0 degrees in front of us. Every object (including us) are always moving through these 4 dimensions (spacetime) at the speed of light, it's that speed that is fixed, never changes.

Reference https://www.physicsforums.com/threads/time-dilation-reasoning.844227/page-3

You can check this PDF: http://www.thequantummachine.com/Quantummachine_4thspeed.pdf

For a peer-reviewed source, which is essentialy the same concept but applied in an ellaborate and formal way:

Representation of relativistic quantities by trigonometric functions, http://scitation.aip.org/content/contributor/AU0018829 , Am. J. Phys. 54, 536 (1986); http://dx.doi.org/10.1119/1.14557.

Quote: (from http://scitation.aip.org/content/aapt/journal/ajp/54/6/10.1119/1.14557)

"A ‘‘space‐time angle’’ φ is defined by setting v=c(sin φ). This leads to a form of Lorentz transformations which uses simple real trigonometric functions and yields a graphic correlation of important relativistic quantities for particles and for corresponding de Broglie waves. A number of relativistic relationships is obtained by the use of common trigonometric identities and formulas."

As adviced to you, first learn carefully about Minkowski diagrams. When you master them, you can follow with other non-standard diagrams (Bhreme, for example), and then revisit the links above.