On Lorentz transforms.

[Mitcher]

Hello, I'am new here and in search of a simple explanation to a rather dull question : in the classical Lorentz first equation to translate from O to O' referentials we have x' = gamma.(x - vt). But v should be x/t, hence x = vt and we have x - vt = 0 which is meaningless. No amount of litterature I read on the subject could enlight me, I'am trying to write simple numerical exemples but this point is a non-goer.

[joigus]

Hi. Welcome to the forums. The v you see there is not x/t. x/t is the ratio between the x-coordinate of a point and the t-coordinate moving in an arbitrary way (mind you, not necessarily at a constant speed), while v is the rate at which O' moves with respect to O.

I hope that clarifies the question. Is that what's troubling you?

Edited August 17, 2022 by joigus

[Mitcher]

Hello, thank you for your answer but no, it doesn't clarify really.

Is O measuring the changing xt coordinates of a mobile moving at velocity V different from v, or is it the fixed coordinates of an event in O ?

Edited August 17, 2022 by Mitcher
typing error

[MigL]

The transforms relate different inertial frames of reference.
These frames have a velocity relative to each other, and that is the v you see ( as well as in the gamma factor ), relating the x position to the x' position.

[swansont]

3 hours ago, Mitcher said:

Hello, I'am new here and in search of a simple explanation to a rather dull question : in the classical Lorentz first equation to translate from O to O' referentials we have x' = gamma.(x - vt). But v should be x/t, hence x = vt and we have x - vt = 0 which is meaningless. No amount of litterature I read on the subject could enlight me, I'am trying to write simple numerical exemples but this point is a non-goer.

The x and t are the coordinates in the O frame, and v is the relative speed of the two frames. x and t can have any value; they are unrelated to v.

edit: xpost with MigL

[md65536]

19 hours ago, Mitcher said:

we have x' = gamma.(x - vt). But v should be x/t, hence x = vt and we have x - vt = 0 which is meaningless.

With x and t being anything, the case where x=vt describes a particle that is at the origin of O' at time t, and the meaning of gamma(x - vt) = 0 is that a distance of 0 will always be 0 no matter what the length contraction factor, or in other words if two things are at the same place and time, you can't change that just by transforming their coordinates. Conversely if you have 2 events 1 m apart, you can transform to another coordinate system where they're 0.5 m apart. Eg. let x=vt+1 be a particle that is 1 unit (according to O) away from the origin of O', then x' will depend on gamma.

[Mitcher]

Hi, sorry for the delay but I was blocked from answering earlier. Thanks for all your answers there but being as precise and specific as possible, is x the instant spatial coordinate of a single body moving at the same velocity v as O’, or is it the distance between two fixed points attached to O ?

[swansont]

19 minutes ago, Mitcher said:

Hi, sorry for the delay but I was blocked from answering earlier. Thanks for all your answers there but being as precise and specific as possible, is x the instant spatial coordinate of a single body moving at the same velocity v as O’, or is it the distance between two fixed points attached to O ?

If the body is at rest in O', then it is moving at v with respect to O

But since the transform is essentially looking at a snapshot, the speed of the object doesn't matter if we are just transforming a single point described by (x,t) in O and finding its coordinates in O'

Or you can get the transform the function x(t) and get x'(t') (this would have the object's velocity information in it)

 

[Mitcher]

33 minutes ago, swansont said:

If the body is at rest in O', then it is moving at v with respect to O

But since the transform is essentially looking at a snapshot, the speed of the object doesn't matter if we are just transforming a single point described by (x,t) in O and finding its coordinates in O'

Or you can get the transform the function x(t) and get x'(t') (this would have the object's velocity information in it)

 

To sum up, O is measuring an object moving along with O' at velocity v then, fair enough. But just earlier you wrote x and t can have any value; they are unrelated to v.

So I'am still confused.

[md65536]

1 hour ago, Mitcher said:

To sum up, O is measuring an object moving along with O' at velocity v then, fair enough. But just earlier you wrote x and t can have any value; they are unrelated to v.

The transformation doesn't tell you what x and t are, you can choose that and it works for anything.

If you make t a constant, then x(t) can describe a single event. You don't need objects at all.

x = vt+r for a constant r gives you the world line of an object that is at rest relative to O' (the x coordinate changes over time, but x' doesn't).

x = r gives you the world line of an object that is at rest relative to O (the x' coordinate changes over time). x = 2vt is an object not at rest relative to either. x can be independent of v. So: no, there's no implicit object that is moving at v, it is the reference frame O' that is moving at v.

[swansont]

2 hours ago, Mitcher said:

To sum up, O is measuring an object moving along with O' at velocity v then, fair enough. But just earlier you wrote x and t can have any value; they are unrelated to v.

So I'am still confused.

Let's say a person on a planet in space is in frame O, and a rocketship is carrying a passenger in frame O' The rocketship (O') is traveling at speed v with respect to O

They want to look at an event that's taking place (an explosion on an asteroid) at point x and time t in the O frame. They would use the Lorentz transform to find the location (x', t') in the O' frame.

Notice that the details of the motion of the asteroid (it's moving at speed w with respect to O) doesn't matter. We know it's at point x at time t

 

[Markus Hanke]

You can also look at this purely geometrically. If O and O’ are in uniform relative motion, then these two coordinate systems are related by a hyperbolic rotation in spacetime (ignore boosts for simplicity) - in other words, a Lorentz transformation is essentially just a simple rotation of the associated coordinates. The speed v is then directly related to the rotation angle by a simple equation; so you can express relative speed as an angle. This is called rapidity.

[studiot]

6 hours ago, Mitcher said:

Hi, sorry for the delay but I was blocked from answering earlier. Thanks for all your answers there but being as precise and specific as possible, is x the instant spatial coordinate of a single body moving at the same velocity v as O’, or is it the distance between two fixed points attached to O ?

 

I think you still haven't got the idea, perhaps this is because the discussion so far has been too general.

Perhaps looking at a specific example might help. So I agree with this and understand where you are coming from. I have been trying to generate some suitable numerical examples as this is obviously important to you.

On 8/17/2022 at 11:45 AM, Mitcher said:

 No amount of litterature I read on the subject could enlight me, I'am trying to write simple numerical exemples but this point is a non-goer.

However the first part of your first post contains a misconception which I have highlighted.

On 8/17/2022 at 11:45 AM, Mitcher said:

Hello, I'am new here and in search of a simple explanation to a rather dull question : in the classical Lorentz first equation to translate from O to O' referentials we have x' = gamma.(x - vt). But v should be x/t, hence x = vt and we have x - vt = 0 which is meaningless. No amount of litterature I read on the subject could enlight me, I'am trying to write simple numerical exemples but this point is a non-goer.

v is not   [math]\frac{x}{t}[/math]     but     [math]\frac{{dx}}{{dt}}[/math]

This is a convenient point to ask you if you have any calculus and if you have met the derivative before ?

 

[Mitcher]

1 hour ago, Markus Hanke said:

You can also look at this purely geometrically. If O and O’ are in uniform relative motion, then these two coordinate systems are related by a hyperbolic rotation in spacetime (ignore boosts for simplicity) - in other words, a Lorentz transformation is essentially just a simple rotation of the associated coordinates. The speed v is then directly related to the rotation angle by a simple equation; so you can express relative speed as an angle. This is called rapidity.

This is understood, thanks. Some times ago I tried to represent O by an angle 0<alpha<90 and gave it a try to see what it would translate to if I used the angle beta = (90 - alpha)/90 as O'. Even if it is conceptually wrong, for some reasons the results are very close to x' resulting from the Lorentz transform proper, a couple of degrees apart at most and spot on on the limits. Amazing.

20 minutes ago, studiot said:

 

I think you still haven't got the idea, perhaps this is because the discussion so far has been too general.

Perhaps looking at a specific example might help. So I agree with this and understand where you are coming from. I have been trying to generate some suitable numerical examples as this is obviously important to you.

However the first part of your first post contains a misconception which I have highlighted.

v is not   xt      but     dxdt

This is a convenient point to ask you if you have any calculus and if you have met the derivative before ?

 

Yes, I can do some math, that's why everything I asked before was about exactly what x and t represent and if they measure a fixed event or an interval because the numerical data I input simply don't close. When using v = 5 and c = 10, say, then I cannot get x'/t' = v for instance, and I can't find where its wrong.

[Markus Hanke]

39 minutes ago, Mitcher said:

Even if it is conceptually wrong, for some reasons the results are very close to x' resulting from the Lorentz transform proper, a couple of degrees apart at most and spot on on the limits. Amazing.

That’s because the rotation is a hyperbolic one, and thus rapidity uses the tanh  function - so the difference would only grow large once you have relative speeds close to the speed of light itself.

[studiot]

Thank you for your reply.

There are many different ways of presenting relativity, this way is very simple and appears to be the same as the path you are following.
With the exception that Semat uses K instead of gamma for the Lorenz factor.

What (I think) you call the Lorenz first equation appears as equation a on the left hand page.
I have ticked this.

I have also indicated where he explains these equations as

The Lorenz-Einstein equations of transformation of space and time coordinates

I have underlined the word 'coordinates' which in this case is the x coordinate in one coordinate system and the x' coordinate in another.

But x and x' are not distances. Distances are treated on the right hand page and called lengths.

Distances are coordinate differences.

Finally note that the left hand page contains two sets of equations, the forward transformation giving coordinates of the second coordinate system in terms of the axis variables of the first.

The second or reverse set gives the coordinates of the coordinates of the first coordinate system in terms of the axis variables of the second, dashed sytem.

 

If this looks familiar, we can proceed and examine why we need all of x, y,z and t in one system and x', y', z' and t' in the second.

 

I repeat these are the coordinate transformations.

The next step is to understand the how to apply these useful quantities like distance, time, velocity and so on.
Then we will get to some examples we can put numbers in.

 

[Mitcher]

17 minutes ago, Markus Hanke said:

That’s because the rotation is a hyperbolic one, and thus rapidity uses the tanh  function - so the difference would only grow large once you have relative speeds close to the speed of light itself.

I, fact no, the difference gets smaller and smaller for v approaching c. The maxima is found at about the 30 to 40 degrees range, where I get 2 degrees or so at most.

[studiot]

I don't know if your interest is course based or you are an amateur, but I note your other threads are about cosmology etc.

Here is a book (also available as a pdf I understand) that may be of interest as it develops the maths, and science to go with this stuff, written by someone who didn't like current textbooks so went out, taught himself, and then wrote one (at reasonable cost I might add.)

https://www.amazon.co.uk/Most-Incomprehensible-Thing-Introduction-Mathematics/dp/0957389469

 

Here are two pages from Peter Collier's book

The first describes the Lorens factor and transformatiosn, the second contains a nice numeric worked example.

[Mitcher]

27 minutes ago, studiot said:

Thank you for your reply.

There are many different ways of presenting relativity, this way is very simple and appears to be the same as the path you are following.
With the exception that Semat uses K instead of gamma for the Lorenz factor.

What (I think) you call the Lorenz first equation appears as equation a on the left hand page.
I have ticked this.

I have also indicated where he explains these equations as

The Lorenz-Einstein equations of transformation of space and time coordinates

I have underlined the word 'coordinates' which in this case is the x coordinate in one coordinate system and the x' coordinate in another.

But x and x' are not distances. Distances are treated on the right hand page and called lengths.

Distances are coordinate differences.

Finally note that the left hand page contains two sets of equations, the forward transformation giving coordinates of the second coordinate system in terms of the axis variables of the first.

The second or reverse set gives the coordinates of the coordinates of the first coordinate system in terms of the axis variables of the second, dashed sytem.

 

If this looks familiar, we can proceed and examine why we need all of x, y,z and t in one system and x', y', z' and t' in the second.

 

I repeat these are the coordinate transformations.

The next step is to understand the how to apply these useful quantities like distance, time, velocity and so on.
Then we will get to some examples we can put numbers in.

 

It is made clear here that O (or S), is contemplating the two ends of a rod moving at velocity v. Hence x is a moving interval relative to O. We are progressing but that's more or less what I had understood, howecer unable to match some real numbers into the equations.

8 minutes ago, studiot said:

I don't know if your interest is course based or you are an amateur, but I note your other threads are about cosmology etc.

Here is a book (also available as a pdf I understand) that may be of interest as it develops the maths, and science to go with this stuff, written by someone who didn't like current textbooks so went out, taught himself, and then wrote one (at reasonable cost I might add.)

https://www.amazon.co.uk/Most-Incomprehensible-Thing-Introduction-Mathematics/dp/0957389469

 

Here are two pages from Peter Collier's book

The first describes the Lorens factor and transformatiosn, the second contains a nice numeric worked example.

Hi ! when younger I did astrophysics and my predilection subject is cosmology. Thanks for the link, that's exactly what I have been trying to achieve with plain numbers. I don't conform that it's not possible to draw a simple working space-time diagram where Lorentz relationships, velocities addition and conserved ds would be obvious. I understand of course that you would need many more dimensions than a piece of paper can offer.

[Markus Hanke]

10 hours ago, Mitcher said:

I, fact no, the difference gets smaller and smaller for v approaching c. The maxima is found at about the 30 to 40 degrees range, where I get 2 degrees or so at most.

I may have misunderstood what you did, then. Apologies.

[Mitcher]

7 hours ago, Markus Hanke said:

I may have misunderstood what you did, then. Apologies.

No harm done. Interestingly it works almost exactly the same whether you use the angle itself or the sinus.

[studiot]

On 8/18/2022 at 11:45 PM, Mitcher said:

It is made clear here that O (or S), is contemplating the two ends of a rod moving at velocity v. Hence x is a moving interval relative to O. We are progressing but that's more or less what I had understood, howecer unable to match some real numbers into the equations.

Hi ! when younger I did astrophysics and my predilection subject is cosmology. Thanks for the link, that's exactly what I have been trying to achieve with plain numbers. I don't conform that it's not possible to draw a simple working space-time diagram where Lorentz relationships, velocities addition and conserved ds would be obvious. I understand of course that you would need many more dimensions than a piece of paper can offer.

Are you still interested in progressing this ?

Edited August 20, 2022 by studiot

[Mitcher]

On 8/19/2022 at 12:40 AM, studiot said:

I don't know if your interest is course based or you are an amateur, but I note your other threads are about cosmology etc.

Here is a book (also available as a pdf I understand) that may be of interest as it develops the maths, and science to go with this stuff, written by someone who didn't like current textbooks so went out, taught himself, and then wrote one (at reasonable cost I might add.)

https://www.amazon.co.uk/Most-Incomprehensible-Thing-Introduction-Mathematics/dp/0957389469

 

Here are two pages from Peter Collier's book

The first describes the Lorens factor and transformatiosn, the second contains a nice numeric worked example.

 

 

7 hours ago, studiot said:

Are you still interested in progressing this ?

Yes I'am working on it, thanks.

On 8/19/2022 at 12:40 AM, studiot said:

I don't know if your interest is course based or you are an amateur, but I note your other threads are about cosmology etc.

Here is a book (also available as a pdf I understand) that may be of interest as it develops the maths, and science to go with this stuff, written by someone who didn't like current textbooks so went out, taught himself, and then wrote one (at reasonable cost I might add.)

https://www.amazon.co.uk/Most-Incomprehensible-Thing-Introduction-Mathematics/dp/0957389469

 

Here are two pages from Peter Collier's book

The first describes the Lorens factor and transformatiosn, the second contains a nice numeric worked example.

Hi. With the help of the many answers I got from people here and the numerical exemple from Peter Collier’s book I was able to close this simple calculation for the first time, founding v’ = - v and ds = ds’ as it should be. Before that I was mixing v, x and t the wrong way. Only glitch is that he is using c = 1 and hence c squared also equals 1 but I think it will work if one use any number larger than 1 and v.

Edited August 20, 2022 by Mitcher

[studiot]

1 hour ago, Mitcher said:

Hi. With the help of the many answers I got from people here and the numerical exemple from Peter Collier’s book I was able to close this simple calculation for the first time, founding v’ = - v and ds = ds’ as it should be. Before that I was mixing v, x and t the wrong way. Only glitch is that he is using c = 1 and hence c squared also equals 1 but I think it will work if one use any number larger than 1 and v.

Glad you are puzzling this out for yourself now.

Much better that way.

Apologies about the use of 'Natural Units'   -  I should have warned you.

 

Here are some useful pdfs

https://www.seas.upenn.edu/~amyers/NaturalUnits.pdf

http://www.phys.ufl.edu/~korytov/phz5354/note_01_NaturalUnits_SMsummary.pdf

Edited August 20, 2022 by studiot

[Mitcher]

22 hours ago, studiot said:

Glad you are puzzling this out for yourself now.

Much better that way.

Apologies about the use of 'Natural Units'   -  I should have warned you.

 

Here are some useful pdfs

https://www.seas.upenn.edu/~amyers/NaturalUnits.pdf

http://www.phys.ufl.edu/~korytov/phz5354/note_01_NaturalUnits_SMsummary.pdf

Thanks for the .pdf's, very handy. I have no conceptual problem with c = 1 but within the frame of a numerical sheet it complicates the calculus making it necessary to fudge the equations as to avoid having dx' larger than dt', etc... I'am still working on it and I'am not sure somebody tried to show Lorentz transforms, ds invariance and velocities composition in a single coherent numerical piece of work.