How is the Minkowski 4-space equation connected to hyperbolic functions?

[stevmg]

Yes, you have seen https://www.physicsforums.com/blog.php?b=1911 but you forgot. I showed you how tanh(\phi)=\beta is derived. This is equivalent with cosh(\phi)=\frac{1}{\sqrt{1-tanh^2(\phi)}}=\gamma.

You are correct. I didn't use that derivation as I was trying to show what someone starting from total scratch would come up with - even if they did not know of that well known relationship you stated (and derived.) However, your derivation is logical and actually could be placed in "the chain of thinking" from "my beginning" to the final conclusion.

Steve

 

[stevmg]

If you differentiate x^{2} - c^{2} \, t^{2} = a^{2} with respect to t, considering x a function of t, then you ought to obtain:

<br /> 2 x \, \dot{x} - 2 \, c^{2} \, t = 0<br />

or

<br /> \beta \equiv \frac{\dot{x}}{c} = \frac{c \, t}{x}<br />

The definition of \gamma is:

<br /> \gamma = (1 - \beta^{2})^{-\frac{1}{2}} = \left[ 1 - \left(\frac{c \, t}{x}\right)^{2} \right]^{-\frac{1}{2}} = \left(\frac{x^{2} - c^{2} \, t^{2}}{x^{2}}\right)^{-\frac{1}{2}} = \frac{x}{a}<br />

I love that! That's freakin' neat! Good on you!

Of course it took me 25 years to figure out that \dot{x} = dx/dt but I'm a little slow.

Steve

 

[yossell]

.
I know Galilean transformations are "shear." That is - like a box squished to the left or right with non-perpendicular axes. Is that a form of rotation in itself? Or does the abscissa and ordinate axes remain perpendicular under Galilean transformations?

Is it the Lorentzian transformations that twist the ordinate axis off perpendicular to the x-axis?

I think I wouldn't worry about the fact that the Lorentz transformations are described as a rotation. I think that's only based on a formal (i.e. mathematical) similarity between matrices for Lorentz transformations and the matrices for rotations in standard Euclidean space, where the metric is positive definite (i.e. no negative contribution from any of the axes). If you're thinking in images, it's better to imagine something like a shear, as in Kev's diagrams - but with some funny stretching.

 

[Mentz114]

I think I wouldn't worry about the fact that the Lorentz transformations are described as a rotation. I think that's only based on a formal (i.e. mathematical) similarity between matrices for Lorentz transformations and the matrices for rotations in standard Euclidean space, where the metric is positive definite (i.e. no negative contribution from any of the axes).

True. Rotations mix spatial dimensions so x'=f(x,y) , y'=h(x,y) under a rotation. For boosts, space and time are mixed, t'=L(t,x), x'=L(t,x), hence apparent time-dilation and spatial contraction.

 

[stevmg]

I think I wouldn't worry about the fact that the Lorentz transformations are described as a rotation. I think that's only based on a formal (i.e. mathematical) similarity between matrices for Lorentz transformations and the matrices for rotations in standard Euclidean space, where the metric is positive definite (i.e. no negative contribution from any of the axes). If you're thinking in images, it's better to imagine something like a shear, as in Kev's diagrams - but with some funny stretching.

Linear algebra is a long way away from me...

Is it possible to give me a "quick and dirty" description of the kinds of transformations.

I know what "translation" is - that's moving the x or y origin to a new place but with the ordinate and abcissas still at right angles and no change in direction of either major axis.

Rotation was what we used to do in Analytic Geometry to the equations of conics to get rid of the xy term in the general conic equation Ax² + Bxy + Cy² + Dx + Ey + F = 0. In this case, we would rotate both axes and the new equation would have no xy term in it. We would then translate both x and y - wise to get a standard conic equation. The new x-axis and y-axis wre still perpendicular but angled off from the original x-y axes.

Now I have seen diagrams where the axes are no longer perpendicular (i.e., "squished") - is that "shear?" Is that what happens in Lorentz transforms? In Galilean transforms, don't the axes stay the same but the world lines (the t-vectors) even if straight are not at \pi/2 to the x or distance axes provided there is a non trivial v. I would presume that Galilean transformations would not alter the x, y, z versus t perpendicularity as these dimensions are independent of each other and no change in x, y or z does anything to t while in relativity, changes in x, y or z (or any other linear dimension if they exist) DOES alter the t component in itself.

So, we have translation - I think I covered that, rotation in the Analytic Geometry sense, but unexplained are rotation in the "squishing" sense or reflection, whatever that is.

Oh, btw there are alterations of axes where one changes the bases of each unit so that one can stretch or compress a dimension. Log paper or exponential paer are examples of that, I guess.

Can you give me a really q&d go-over of that? Am I on the right sheet of music?

 

[stevmg]

True. Rotations mix spatial dimensions so x'=f(x,y) , y'=h(x,y) under a rotation. For boosts, space and time are mixed, t'=L(t,x), x'=L(t,x), hence apparent time-dilation and spatial contraction.

Thanks for the reply. Please look at post #55 and you will see that I am more basic than that and I need to know the basics before we use terms like "boost" &tc. I really need the basics here but it will come back as I did take linear algebra probably before you were born.

Let's start with one thing... what is a "shear" when it comes to changing the axes? When is it used?

stevmg

 

[stevmg]

Getting back to steve's original question:

If you look at the diagram below:

You can see that the x' axis has been rotated anticlockwise while the t' axis has been rotated clockwise relative to the x and t axes respectively. The amount of rotation of the axes depends on the relative velocity between the F and F' frames. The relative velocity in the above diagram is 0.6c so the proper time t' =1 translates to t=0.8 and the proper distance x' =1 translates to x=0.8. Note that point P = (x',t') = (0,1) remains on the horizontal hyperbola in the left diagram, for all relative velocities and the point P = (x',t') = 1,0) remains on the vertical hyperbola in the right diagram, for all relative velocities.

Diagrams from http://hubpages.com/hub/Minkowski-Diagram

kev - I got to the website and saw all those diagrams that the article had. Now, how did you put that into your note? There are numerous diagrams on that website and you only used a couple.

stevmg

 

[stevmg]

Yes, you have seen https://www.physicsforums.com/blog.php?b=1911 but you forgot. I showed you how tanh(\phi)=\beta is derived. This is equivalent with cosh(\phi)=\frac{1}{\sqrt{1-tanh^2(\phi)}}=\gamma.

starthaus -

I didn't realize you were referring me straight to that article you wanted me to see.

I haven't yet gotten into your article yet...

That's why I was looking for the visual or geometric approach so I could better internalize the concepts. I am satisfied now with what I have done for my benefit unless someone can point out an error in the math.

stevmg

 

[yossell]

That's why I was looking for the visual or geometric approach so I could better internalize the concepts.

Kev's diagrams pretty much say it all. Perhaps they say too much. Let's just look at the first diagram. You may know a lot of this already, but maybe you'll find it helpful.

The first diagram plots life (space-time life) for inertial (unaccelerated) observer O who starts at the origin and (in his own frame). The t-axis is the blue line going up plots O's life as time ticks by - since there's no change in the x coordinate along this line (and we're suppressing y and z directions), this line shows O not going anywhere, just letting the time tick by. The blue notches along this diagram represent ticks of the clock, each notch a unit of time.

The x-axis is, in this diagram, the events that are simultaneous with t = 0, for Observer O. It's an unusual way of thinking about the x axis, but it's very helpful in space time. Each notch in this axis represents a unit distance. The horizontal blue line drawn at t = 1 represents all the events that are simultaneous with whatever happens at the point (1 0), according to O.

Now look at the steeper red line. This is the space-time path of some OTHER inertial observer, o', travels. O' is inertial too - he travels at a constant velocity hence he cuts out a straight line in space-time. The faster he travels, the more angled his world-line. But the speed of light is a limiting factor, so the angle that possible observers can travel is bounded: namely by lines that represent the speed of light. Often, units are chosen so that light can be represented by lines that lie at 45 degrees to the axis.

Ok - so that steeper red line represents the path of O' - it's HIS time axis, commonly written as his t' axis. But what about HIS x - axis? Well, his x-axis are those events that HE regards as simultaneous. These events appear on the same map - both O and O' are privy to the same events - they just disagree about time and simultaneity. Well, when you follow the Lorentz transformations, it turns out that the x-axis of O' is tilted up - so that it becomes the less tilted of the two red lines. So that red line marked x' axis is the events that observer O thinks happen at the same time as events at t' = 0. You can see that, apart from the origin, O and O' disagree about which events are simultaneous with which. Again, lines drawn parallel to this red line represent lines of simultaneity in the frame of O'.

If we're using units where light travels paths at 45 degrees, then for any frame F, the t axis and the x-axis make the same angle with the 45 degree line - fold the paper along a 45 degree line, and the t-axis and x-axis of a frame get mapped onto each other. Now, the faster something travels, the closer it is to a 45 degree angle - the closer O judges it as getting to the speed of light - and the more its x-axis is tilted over. If you were to imagine continuing the process, the x and t axes collapse onto each other at the speed of light.

This tells us the angles - but we would also like to know how the clocks and lengths of O' work - we would like to draw HIS notches - that is, we want to calibrate his clocks and lengths. What does HE regard as a tick? What does he regard as a unit distance? Well, because of time dilation, and lorentz contraction, the notches that he draws are different from ours. I think kev's second diagram represents the notches as seen by the second observer.

All that drives this is the Lorentz transformations. They're just linear transformations - it's just a matter of matrix multiplication, so, though they're abstract, you're probably more familiar with them than you realize.

As usual, this took longer to say - hope there aren't too many typos...

 

[starthaus]

Linear algebra is a long way away from me...

Is it possible to give me a "quick and dirty" description of the kinds of transformations.

I know what "translation" is - that's moving the x or y origin to a new place but with the ordinate and abcissas still at right angles and no change in direction of either major axis.

Rotation was what we used to do in Analytic Geometry to the equations of conics to get rid of the xy term in the general conic equation Ax² + Bxy + Cy² + Dx + Ey + F = 0. In this case, we would rotate both axes and the new equation would have no xy term in it. We would then translate both x and y - wise to get a standard conic equation. The new x-axis and y-axis wre still perpendicular but angled off from the original x-y axes.

Now I have seen diagrams where the axes are no longer perpendicular (i.e., "squished") - is that "shear?" Is that what happens in Lorentz transforms? In Galilean transforms, don't the axes stay the same but the world lines (the t-vectors) even if straight are not at \pi/2 to the x or distance axes provided there is a non trivial v. I would presume that Galilean transformations would not alter the x, y, z versus t perpendicularity as these dimensions are independent of each other and no change in x, y or z does anything to t while in relativity, changes in x, y or z (or any other linear dimension if they exist) DOES alter the t component in itself.

So, we have translation - I think I covered that, rotation in the Anaslytic Geometry sense, but unexplained are rotation in the "squishing" sense or reflection, whatever that is.

Oh, btw there are alterations of axes where one changes the bases of each unit so that one can stretch or compress a dimension. Log paper or exponential paer are examples of that, I guess.

Can you give me a really q&d go-over of that? Am I on the right sheet of music?

Here is a crash course.

1. Rotation in 2D plane is:

x&#039;=x*cos(\phi)+y*sin(\phi)
y&#039;=-x*sin(\phi)+y*cos(\phi)

You can convince yourself that this is correct by trying to plot a few images.

2. Lorentz transforms :

x&#039;=x*cosh(\phi)-(ct)*sinh(\phi)
(ct)&#039;=-x*sinh(\phi)+(ct)*cosh(\phi)


Because the Lorentz transforms look like the 2D transforms for rotation they are called, by abuse of language, "rotations in the hyperbolic plane". This is where the "rotation" comes from.

 

[stevmg]

Here is a crash course.

1. Rotation in 2D plane is:

x&#039;=x*cos(\phi)+y*sin(\phi)
y&#039;=-x*sin(\phi)+y*cos(\phi)

You can convince yourself that this is correct by trying to plot a few images.

2. Lorentz transforms :

x&#039;=x*cosh(\phi)-(ct)*sinh(\phi)
(ct)&#039;=-x*sinh(\phi)+(ct)*cosh(\phi)


Because the Lorentz transforms look like the 2D transforms for rotation they are called, by abuse of language, "rotations in the hyperbolic plane". This is where the "rotation" comes from.

It doesn't get any "crashier" than that.

There is some sort of axis rotation, though as shown in this diagram whose thumbnail is shown below. I still don't know how to post a damn picture into the text as others do and I do follow the instructions. It seems that the t' axis is rotated clockwise and the x' axis is rotated counterclockwise towards each other presumably as the v (or \beta) is increased as they squeeze onto the light cone.

But I see your point as the "rotation" was brought about by analogy to the analytical geometry or linear algebra rotating axes.

I assume you have taken chemistry in your studies. If you ever want to see "abusive notation" just take a course in biochemistry where they don't care about equation balance or anything of the like. They are just flow diagrams and not good ones at that (Krebs Cycle, etc.) but that is a different story for another distant distant time.

So rotation of both t and x axes ("squishing") is for Lorentz transforms. Rotation of ONE axis (the t-axis) is for Galilean transformations (no time dilation which causes the "upward" rotation of the x-axis in the "squishing" described above. I presume that is also called "shear" (rotation of t-axis alone.) The clockwise motion of the t'-axis in the moving FR would be caused by the same "shear" as seen in Galilean transformations but not as severe as the gamma factor contracts length and would shorten the angle of clockwise rotation. Click on the thumbnail and you will see more clearly what I am talking about.

"Shear" = Galilean
"Squishing" or stretching (the opposite of "squishing" is Lorentzian.

Of course, if we look at the O' FR by itself, both the t'axis and x'axis are perpendicular and it would be the O FR which would be out of whack (I guess "unsquished" the other way.)

?Right sheet of music?

 

[stevmg]

SOMEONE, how do I place a picture, not a thumbnail, onto a post. I've tried everything including using a website (www.photobucket.com) and nothing works. I even made the picture smaller so as to make sure it would fit in the box.

 

[DrGreg]

SOMEONE, how do I place a picture, not a thumbnail, onto a post. I've tried everything including using a website (www.photobucket.com) and nothing works. I even made the picture smaller so as to make sure it would fit in the box.

[noparse][PLAIN]http://InsertURLHere[/noparse]

 

[stevmg]

[noparse][PLAIN]http://InsertURLHere[/noparse][/QUOTE]

No luck.

DrGreg - will you give me a harmless URL that you have used and which worked so I can try it? Otherwise I am just floundering and getting nowhere. If I get one that works, I may learn something from it.

stevmg

 

[Dale]

Why don't you try one of kev's images from earlier in this thread:

[tryIMG]http://s1.hubimg.com/u/244448_f520.jpg[/tryIMG]

Just delete "try"

 

[stevmg]

Worked! I guess I cannot post from photobucket.

Know of any freebie sites or URL's someone like me can use for images (just for physics forums)

stevmg

 

[DrGreg]

No luck.

DrGreg - will you give me a harmless URL that you have used and which worked so I can try it? Otherwise I am just floundering and getting nowhere. If I get one that works, I may learn something from it.

stevmg

I have never used photobucket before, but I just went there and selected an image I found there at random.

The page I chose was http://media.photobucket.com/group/image/photography/EJBUOUE7H3/photography.jpg .

The code you need to copy & paste appears in a panel at the left of the page, "share this image" under "IMG code": [noparse]http://gi87.photobucket.com/groups/k132/EJBUOUE7H3/photography.jpg[/noparse]

http://gi87.photobucket.com/groups/k132/EJBUOUE7H3/photography.jpg

Or click on the "share this" button, "get link code", "IMG for bulletin board"

 

[stevmg]

Got it!

Thanks

Got it again with a larger image.

DrGreg, you are a true genius as as are you, too, DaleSpam.

stevmg

 

[stevmg]

Kev's diagrams pretty much say it all. Perhaps they say too much. Let's just look at the first diagram. You may know a lot of this already, but maybe you'll find it helpful. [continued below]

[N.B.] I think THIS is the diagram you want...

Yes... I was finally able to post that!

[yossell - continued] This diagram plots life (space-time life) for inertial (unaccelerated) observer O who starts at the origin and (in his own frame). The t-axis is the blue line going up plots O's life as time ticks by - since there's no change in the x coordinate along this line (and we're suppressing y and z directions), this line shows O not going anywhere, just letting the time tick by. The blue notches along this diagram represent ticks of the clock, each notch a unit of time.

The x-axis is, in this diagram, the events that are simultaneous with t = 0, for Observer O. It's an unusual way of thinking about the x axis, but it's very helpful in space time. Each notch in this axis represents a unit distance. The horizontal blue line drawn at t = 1 represents all the events that are simultaneous with whatever happens at the point (1 0), according to O.

Now look at the steeper red line. This is the space-time path of some OTHER inertial observer, o', travels. O' is inertial too - he travels at a constant velocity hence he cuts out a straight line in space-time. The faster he travels, the more angled his world-line. But the speed of light is a limiting factor, so the angle that possible observers can travel is bounded: namely by lines that represent the speed of light. Often, units are chosen so that light can be represented by lines that lie at 45 degrees to the axis.

Ok - so that steeper red line represents the path of O' - it's HIS time axis, commonly written as his t' axis. But what about HIS x - axis? Well, his x-axis are those events that HE regards as simultaneous. These events appear on the same map - both O and O' are privy to the same events - they just disagree about time and simultaneity. Well, when you follow the Lorentz transformations, it turns out that the x-axis of O' is tilted up - so that it becomes the less tilted of the two red lines. So that red line marked x' axis is the events that observer O thinks happen at the same time as events at t' = 0. You can see that, apart from the origin, O and O' disagree about which events are simultaneous with which. Again, lines drawn parallel to this red line represent lines of simultaneity in the frame of O'.

If we're using units where light travels paths at 45 degrees, then for any frame F, the t axis and the x-axis make the same angle with the 45 degree line - fold the paper along a 45 degree line, and the t-axis and x-axis of a frame get mapped onto each other. Now, the faster something travels, the closer it is to a 45 degree angle - the closer O judges it as getting to the speed of light - and the more its x-axis is tilted over. If you were to imagine continuing the process, the x and t axes collapse onto each other at the speed of light.

This tells us the angles - but we would also like to know how the clocks and lengths of O' work - we would like to draw HIS notches - that is, we want to calibrate his clocks and lengths. What does HE regard as a tick? What does he regard as a unit distance? Well, because of time dilation, and lorentz contraction, the notches that he draws are different from ours. I think kev's second diagram represents the notches as seen by the second observer.

All that drives this is the Lorentz transformations. They're just linear transformations - it's just a matter of matrix multiplication, so, though they're abstract, you're probably more familiar with them than you realize.

As usual, this took longer to say - hope there aren't too many typos...

Now, I can get back to serious business because I can post images of what I can draw with pencil and ruler to illustrate further questions...

Now I forgot what the devil I was asking about it the first place! But, thanks, again, for all your help.

stevmg