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### Course: Physics library>Unit 16

Lesson 3: Lorentz transformation

# Algebraically manipulating Lorentz transformation

The forms in which we've introduced the Lorentz transformations are really nice in that they reveal the symmetry of the two axes of spacetime. But there are other representations: some are just more common in practical use, while others reveal other interesting similarities with classical mechanics!

## Want to join the conversation?

• Why is it so important to see the symmetry?
• It's not important to see the symmetry, as long as you understand EXACTLY what spacetime is and how it behaves. In other words, seeing the symmetry just makes it easier to not only see why the spacetime diagram looks the way it does, but also to understand spacetime as a concept in the first place. I also think that it's easier to remember and makes more sense than the manipulated forms of the equation, which may seem arbitrary.
• If v approaches c would the Lorentz FActor equal infinite?
• As v approaches c we get something that approaches 1/0 (from the left side of zero), so in theory, yes, we get an infinity.

In the case of time, if we're not too rigorous, we can think of it as being T = t - xc/c /e
: . T = (t-x)/e, where e is some very small numer close to zero... so, T would essentially equal infinity.
• Must a symmetry be the invariance of a particular laws form under some transformation(a change of something the law is described interms of, or can it be the invariance of something else? Like the speed of the light(the electromagnetic wave the laws form specifies.
Symmetries( when taken as invariances of laws forms),according to Noethers theorem imply and are implied by conservation laws. Each law that is invariant is linked to a conserved quantity and vise-versa.The symmetry (invariance ) of an X law undere transformationalchange Y correspons by Noethers theorem to the conservation of a quantity(WHAT QUANTITY??). I this quantity related to Y and or X and if so HOW is the conserved quantity related to the law that in invariant or the transformation under which it is so unchanged?
Given the symmetry of the transformation equations are x'=Y(x-Bct) and
ct'=Y(ct-Bx) show that space and time are really not separate but unify into one homogenized concept called "spacetime"(ST)what is the law in spacetime that is invariant? and under what transformations does it hold for? I think the law that is invariant is the speed of light(electromagnetic radiation which results from Maxwells laws of the electromagnetic field)[we could also say that what is invariant in the spacetime is its interval structure] and the transformations it is invariant under is the transformations of coordinates due to the "relative uniform motion" of the frames(motion between so called inertial frames).
• Lets start off with non-Relativistic mechanics for simplicity.

Noethers theorem is based on the concept of something called action. Action is a function which is the time integral of the total kinetic energy minus the total potential energy of the system. (The laws of Newtonian mechanics can be derived using the principle of least action)

In the Wikipedia article on Noethers theorem there information about translational, rotational and time invarance as well as a generic derivation based on a single independent variable, the multi-variable version is much more complex.

Within Relativistic mechanics you don't have separate translational and time invarance they become a single invarance of space-time translation. What this translates to is something like conservation of liner momentum but it becomes the conservation of something called 4-momentium.
(1 vote)
• was so amazing I can't even describe it. my question: Would it be accurate to say the Lorentz transformation is just a more informed and accurate method of Galilean transformations because we're following quantum rules?
(1 vote)
• It would literally be the most accurate statement.

Why? It was explained to me like this.
As physics gets older and older, our equations and predictions get more and more accurate. The discovery of special relativity, therefore, does not fundamentally change previous equations, it just adjusts them.

The best part? The some of equations we have now for the universe are not only incomplete, but show EXTREMELY SMALL inaccuracies. In other words, that equation itself might have another factor, or a slightly modified version. Good luck finding it!
• x' = gamma(x-beta*ct)
so if lorentz factor gamma is incredible large then x' will be incredibly bigger than x.
does this affect our reality? or what is the benefit of x' being incredibly larger than x.?
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• In what way is the function Y =[1/sqrt(1-v^2/c^2)] in the x scaling of the Galilean transformation seen as analogous to the projection operator functions cos Q evaluated at Q=tan-1 (v/c) and the Yv function analogous to the circular function sin, for projecting the x and t coordinates onto the x' tilted(at Q=tan-1(v/c) axis ? How do we see Y as a projection function analogous to the way we see cos as a projection function?
(1 vote)
• At 5.03 where you say you are going to divide both sides by c to find t' why don't we divide Y by c also. Why don't we get (Y/c)[t-(v/c^2)x] instead of Y[t-(v^2/x^2)]?
(1 vote)
• Any mistakes? Every time I look at it I notice something wrong.
E^2=p^2*c^2+m^2*c^4
γ=cos(arctan(i*v/c))
p=m*v*γ=m*c*sin(arctan(i*v/c))
E^2=(m*c*i*sinh(arctanh(v/c)))^2*c^2+m^2*c^4=m^2*c^4*(i*sinh(arctanh(v/c)))^2+m^2*c^4=m^2*c^4(1-sinh(arctanh(v/c))^2)
sqrt(E^2)=sqrt(m^2*c^4(1-sinh(arctanh(v/c))^2))
E=m*c^2*cosh(arctanh(v/c))
E=m*c^2*γ
for completion sake, if v>c then
E^2=(m*c*i*sin(arctan(i*c/v)))^2*c^2+m^2*c^4
E=m*c^2*sin(arctan(i*v/c))
E=m*c^2*γv*i/c
E=m*c
γ*v*i

E becomes negative when v>c when it is squared (E^2). There is a transformation that takes place when v/c > 1.

x/sqrt(x^2-y^2)+i*y/sqrt(x^2-y^2)
x/(i*sqrt(x^2-y^2))+i*y/(i*sqrt(x^2-y^2))
-i*x/sqrt(y^2-x^2)+y/sqrt(y^2-x^2)
(1 vote)
• If the primed frame is the one that is moving (usually by convention this is how it is I have found from textbooks and such) shouldn't x'=x1/gamma as "moving rods shrink" and we know that gamma is equal to or greater than 1? or in this video is S' frame stationary and S is the movign frame?
(1 vote)
• Given that the symmetry of spacetime gives us the Lorentz transformations between x and x'; ct and ct' as x'=Y(x - B ct)
and ct'=Y(ct - B x)
and that this shows that space and time are not separate but are unified into one homogenized concept called spacetime(ST). But what is the law in the spacetime that is invariant, and under what transformations does this symmetry(invariance) hold(what change is this symmetry applicable for)?
I think the law that is invariant is the speed of light law of electromagnetic radiation (Maxwells electromagnetic field law) and the transformation it is invariant under is the transformations of coordinates between uniformly moving systems(of one inertial frame relative to another inertial frame).
What quantity is conserved between the systems(is it the interval ds^2(but is this physical or just mathematical)?
Why does ST possesses such Symmetry?