Lorentz Transformation Puns

https://imgur.com/a/qcDnUnB

Here is the prompt (1.4 a) and my attempt. Am I doing this correctly so far? I am not sure if it is worth the effort to distribute algebra here.

UPDATE: Here is my second attempt at both 1.4 part a (Lorentz) and 1.4 part b (Galilean)

https://imgur.com/a/aN0R0vi

Was sitting in linear algebra today learning about change of basis which I understood as basically a shifting of the coordinate plane based on your perspective.

Is this really what goes on with Lorentz transformations with the new coordinate system being transformed based on your velocity?

UK resident here - just wandering at what level you normally get taught this. For example is there any of this in Further Maths A-Level, or are they 3rd year degree type difficulty?

Mny thks

"A cart rolls on a long table with velocity v. A smaller cart rolls on the first cart in the same direction with velocity v relative to the first cart. A third cart rolls on the second one, again in the same direction and with the same velocity v, and so on up to n carts. What is the velocity of the nth cart?"
I'm close to finishing this problem but need some help tying it all together.
Its easy to show that the rapidities sum together such that 𝜙_n = n𝜙 and since tanh(𝜙) = v (I've seen slightly different definitions but this is what we've used in class), 𝜙 = 1/2 ln((1 + v)/(1-v)).
My first task is to simplify this such that there are no logarithms or exponentials. I tried this but just ended up saying (1+v)/(1-v) = (1+v)/(1-v) which clearly isn't useful.
My second issue is finding the velocity. I know it should be u_n = c((1- (1-v)/(1+v))^n) /((1-+ (1-v)/(1+v))^n) , but I'm not sure how to put this together. Can anyone push me in the right direction? Thanks

I was messing around with the Lorentz Transformations in special relativity. I tried to do something but it didn't work and I can't figure out why. Here's my work and though process.

These are the Lorentz Transformations where only the time and the x direction are transformed.

t' = γ(t - (vx/c^2))

x' = γ(x - vt)

y' and z' are the same as y and z.

Next, I set up a scenario where there are 2 world lines traveling in the same direction at different speeds less than the speed of light. All world lines start at (0,0) in this scenario, so we can say the general equation for a world line is x = vt/c (light travels 45° relative to the x-axis). Since we have two world lines, we have two equations

x = v₀t/c

x = v₁t/c

I transform everything to the first equation where v = v₀. This gives these transformations:

t' = γ(t - (v₀x/c^2))

x' = γ(x - v₀t)

β in gamma also uses v₀. To transform the second equation, we replace x with x' and t with t', giving:

x' = v₁t'/c

I then replace t' and x' with the equations above:

γ(x - v₀t) = v₁γ(t - v₀x/c^2)/c

Doing a little bit of algebra and solving for x:

x = (v₀t+cv1t)/(c+v₀v1/c^2)

which should be the equation for the transformed line in xt space. Mathematically this seems to make sense, but graphing it doesn't work and I have no clue why. I came from a math background so there maybe some physics here that I'm missing. Any thoughts or comments are greatly appreciated.

Hi everyone, I've been enjoying thinking about the Lorentz transformation for objects traveling near the speed of light but I haven't been able to find anything that helps me visualize what would happen if objects move away from each other at those speeds.

For example if we were on planet/body A moving away from planet/body B at O.9c how would that affect our understanding of the time that had passed on planet B? Or if we were at point C observing A and B moving away from us perpendicular to our observation but away from one another at 0.9c?

Specifically I'm curious about the observed time dilation effects.

Many thanks for your consideration.

I am struggling getting from equation 1 to equation 2 HERE.

I explain the variables at the previous link just in case the notation isn't conventional. Basically I'm trying to derive this equation by repeatedly applying the properties of the Dirac matrices on equation 1 to get to equation 2.

I understand this is a relation between the Dirac matrices and the infinitesimal generator of S, the change of basis matrix guaranteed by Pauli's fundamental theorem under a Lorentz transformation... Finding T provides a way to obtain S explicitly. The problem is that I have no idea how my professor got there.

I've done some work but I really don't think it's going anywhere (calculations becoming humongous). Also I'm new to tensors so I'm not confident manipulating them quite yet.

Any help/guidance is welcome, thanks in advance!

If you calculate the eigenvalues of the Lorentz transformation matrix, you find that they are equal to the relativistic Doppler shift factors, is there a reason why this is true (I.e, not mathematically, but physically/intuitively)? Furthermore, could you justify that this ‘proves’ the shift factors somehow? (The math of the eigenvectors produce the equations ct= +/- x)

I'm studying special relativity at an introductory level and my textbook only defines one dimensional motion lorentz transformations. So a particle moving horizontally. It stays that x'=gamma*(x-vt), t being time, x being the position relative to reference S, x' position relative to reference S', and v being the velocity of reference S'.

If a particle moved along a diagonal line, would this be true for y and y'? And would this also mean this is true for the lorentz velocity transformations?

Can someone please tell me the eigenvalues and eigenvectors of this tensor? I’m just trying to verify my answer!

picture

Hello,

i am sorry if this question is maybe malformed or confusing. I am a complete layman when it comes to physics and am mainly interested in getting the general concepts correct in my mind.

I have recently learned that the one-way speed of light is impossible to measure and that it is only due to a simplifying convention (Einstein synchronization) that is is assumed, that the one-way speed of light is the same as the two-way speed.
Of course a theoretical impossibility to measure a value x means, that it makes no difference for any physical process what the real value of x is. Because if it did make a difference then it would not be theoretically impossible to measure the value of x.
According to what i read, this means that the one-way speed of light could be infinite in one direction and 1/2 c in the return direction.

Then i watched this video by PBS Space Time https://www.youtube.com/watch?v=msVuCEs8Ydo .
In it, the presenter claims that the Lorentz Transformations result in an absolute necessity for a cosmic speed limit. Without such a limit the universe as we know it could not exist.

Does this not mean, that the one-way speed of light can not be infinite? Or do the Lorentz Transformations only require there to be a limit on the two-way speed of light?
Are physicists always talking about the two-way speed of light when they are talking about c or is there any currently accepted theory in physics, that requires the one-way speed of light to not be infinite in order to work?

Thanks in advance

So I am a second-year physics student and I'm working as a research assistant with a professor this summer. He currently has me reading fourth-year class notes about coordinate transformations and contravariant/ covariant tensors. He gave me an old assignment from the class that he wants me to work on but I've been having a lot of trouble conceptualizing coordinate transforms and tensors.

One question from the assignment has a scalar field of x^(u) from 0-3, on which I am supposed to perform a Lorentz boost in the x^(3) direction, but I don't know how to do it. I made some attempts but they were incorrect.

Not sure if to call it orthogonality, but apparently, for some Lorentz transformation a^mu_nu , this holds (1st pic).

Now I don't know how this works, I tried my hand at it but I think I'm failing at tensors, could someone help me please? Here's my current work (pic 2).

I'm a total newbie at tensors. I also have to prove that

partial_mu= a^mu_nu (partial_nu)'

I don't think I'll be able to prove that for Lorentz transforms considering my knowledge of how to use tensors.

From An Introduction to Mechanics by Kleppner and Kolenkow:

"Here is one model to justify the assumption that y = y' and z = z': consider two trains on parallel tracks. Each train has an observer holding a paint brush at the same height in their system, say at 1 m above the floor of the train. Each train is close to a wall. The trains approach at relative speed v, and each observer holds the brush to the wall, leaving a stripe. Observer 1 paints a blue stripe and observer 2 paints a yellow stripe.

Suppose that observer 1 sees that the height of observer 2 has changed, so that the blue stripe is below the yellow stripe. Observer 2 would have to see the same phenomenon except that it is now the yellow stripe that is below the blue stripe. Because their conclusions are contradictory they cannot both be right. Since there is no way to distinguish between the systems, the only conclusion is that both stripes are at the same height. We conclude that distance perpendicular to the direction of motion is unchanged by the motion of the observer."

Since the tracks are parallel, I understand why z = z'. What I don't understand is how the observers see the other's heights changing? It can't be a physical change, or else y would not be equal to y'. But I'm having trouble visualising the change in perspectives that would cause the description here. Can anyone help me out? Thanks.

What is the precise extent to which 3D objects change their geometry when undergoing relativistic effects, when moving close to the speed of light (as predicted by the Lorentz transformation)?

I assume that despite the changes to shape, some fundamental geometric features of 3D objects are preserved no matter the frame of reference. Imagine a solid 3D cube at rest. Now imagine that this cube is traveling close to the speed of light. I assume that a cube will preserve some of its fundamental geometry. Namely, a cube undergoing Lorentz contraction will not become a torus or vice-versa. (I might be wrong about this)

What is the formal limit to these changes? Will the affine structure of a 3D object undergoing Lorentz contractions change or would its topology also change?

Thank you.

A spaceship has a life support system that can keep its crew alive for 800 hours the crew. The crew wants to travel a distance of 1 x 10^15 m (as measured in the rest frame of the start and end points). a.) It takes off balls of light approximately 927 hours to travel the same distance. does this mean that the ship will have to travel faster then light? Explain why or why not. b.) How fast will the ship have to go so that it can travel 10^15 m without depleting its life support system?Express your answer as a fraction of the speed of light. c.) how long did the trip take according to an observer at rest at the destination?

Wasnt able to make much progress on my own, using the standard v = d/t formula for part a suggests that the ship does need to travel faster than light but i doubt if thats the case as part b requires to express the answer as a fraction of the speed of light.

Im think i can transform time using t' = gamma * (t-v/c^2*x) for part c but the main catch is finding the velocity of the ship

https://gyazo.com/54964cdb76f6833edb44bd86ab484090

Hi, I'm wondering is it possible to construct a transformation analogous to Lorentz transformation but in curved/non-flat space time? For example, will mass/energy be perceived differently between someone who is near and far away from a massive object? Thank you so much

When we observe a ball moving at high speed we would see it squashed flat into an oblate spheroid. Is the contraction of the spatial dimension due to it being "rotated" into the unseen time dimension, or is that just a mathematical description?

If no object has any "thickness" or extent in the time dimension, then when time "rotates" in to space, things should appear flat, while the spatial thickness rotates away where it is not observable.

If this rotation has any basis in reality, then I'm not sure what it would mean for a spatial thickness to be rotated into time. It would give objects a "duration" or a span of time corresponding to the rotated dimension. Or does time dilation account for this?

Or is rotation just an analogy and this is where that analogy breaks down?

I am trying to follow Yale's Prof Shankar's Youtube video about Lorentz Transformations. Pretty simple stuff I'm led to believe - he says so himself - but it's still got me beat.

Around about 6:05 he says that xt = ct and I can see that easily enough.

But then he goes on to say that for the 'other' party x prime also = ct.

And it seems to me that should be (xprime + ut) = ct.

Can anyone clear this up for me?

p.s. it's in channel 'Yale Courses' and titled '13. Lorentz Transformations'

As the hype and questions have died down, this one will stay up for a full week. We'll see if there's demand for this the next week.

As always, sorted by new.

If we postulate: natural transformations (of coordinates) are linear transformations that preserve the Minkowski metric. Then natural transformations coincide precisely with Lorentz Transformations (mod reflections). This is a very common derivation of Lorentz Transformations.

However if we postulate: natural transformations are linear transformations that preserve the speed of light. This is not sufficient, but Lorentz Transformations are very, very often also derived from the invariance of the speed of light.

Why is it not sufficient? Say light travels from (0,0) to (x,t). Then we'll have x/t=c. Now chose any Lorentz Transformation L. Similarly if (x',t')=L(x,t), then x'/t'=c. However! Now if you chose the transformation L scalled by 6, then (x'',t'')=(6L)(x,t)=(6x',6t'). And (6x')/(6t')=x'/t'=c. So 6L also preserves the speed of light! In fact it preserves all speeds,

Therefore we need an extra postulate right? But often no extra postulate is mentioned, that's why I'm confused. Which hidden postulate am I missing?

(edit: I think I know what the postulate is somewhat. Ignoring some issues to do with the origin and translation. First there's an important correspondence velocity <--> observer <--> "NaturalTransformation". (ie NatTran is a function of velocity) And second, this correspondence is "very symmetric". That implies this "NatTran" is a Lorentz Transformation. I'm being terse since I'm more interested in what others have to say. Also tbc this "NatTran." has nothing to do with category theory)

Forgive me if this is not the correct place for questions about interpretation, or if this has been asked before.
I wondered how the Lorentz transformations would work in an expanding (such as ours), so I made a little calculation in Mathematica. I imagined a 2D plot where the x-axis would represent the "distance" between an observer and another body, while the y-axis would represent the apparent time of each body (given by the Lorentz transformation). I assumed, since the expansion rate is constant, that therefore the velocity with which any body is moving away from the observer can be calculated by multiplying the distance between them (from the point of view of the observer) and a constant value.
Anyway, having the observer stand on t = 0, and taking an instant photograph of what he sees, renders this plot. The immediate downward parabola around the observer makes sense, the farthest you are from the observer, the higher the velocity, and therefore, the higher the difference in perceived time. However, what is happening at the "extremes", when the difference in time starts to decrease? Are "very far away" objects perceived as being "in time" with the observer?
The plot ends when the velocity reaches the speed of light (which in this calculations is arbitrary). Would this mean that the edges of the observable universe are "in time" with the observer?
I also made this gif for the plot changing through time. The plot shows the observer "reaching" the time of bodies a certain radius away from it, then reach the time of the edges at t = 0, and finally (yet kind of weirdly) become an upward parabola and leaving the rest behind, while the edges remain fixed at t = 0.
Does anyone have any insight into this? Is there a name for it, or am I wrong somewhere? I am not a physicist, and only made this out of curiosity, so any answers will be happily welcomed, so long as it teaches me something. Also, excuse my english. Not native and long time without use.

So I was looking at some videos online which show how a matrix transformation would look like from a vector perspective, and it reminded me of another video which showed how a coordinate system would transform under the Lorentz transformations.

So my question is can this be done? If so then wouldn't it be more compact to write it that way?

How do you derive the Lorentz transform using linear algebra? I did some reading on this late night after I found this article (https://drive.google.com/file/d/1bNy8pX9ZcAjTgKWFvUpxGYRjEXF_Ufpn/view?usp=drivesdk) explaining it, but I could only follow up to a point. I understand everything till the end of page 2, but then they say something about all the points being the same distance from the time axis and start doing matrix multiplication that doesn't make sense, because the dimensions don't match and I just got lost at that point. I'm not sure if whoever wrote this article didn't know what they were doing of I'm just completely misunderstanding. Anyway, I don't expect anyone to read the whole article and explain it to me; I just wanted to explain what I'd read so far and give some context for my confusion. I live linear algebra and I really want to understand SR in terms of it. Any help is greatly appreciated! ;)

https://gyazo.com/f575c78743e161f46bccc9c2ab870e2e

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