- #36
PeterDonis
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olgerm said:I want the angles between, coordinatelines not between representation of coordinatelines on euclidean sheet
What's the difference?
olgerm said:I want the angles between, coordinatelines not between representation of coordinatelines on euclidean sheet
You mean, you want the angle between the ##t## axis in spacetime and the ##t'## axis in spacetime, not the angle between the representation of the ##t## axis on the diagram and the representation of the ##t'## axis on the diagram? Then you need to use hyperbolic geometry, not trigonometry.olgerm said:I know that axes and coordinatelines are same thing, but I meant, that I want the angles between, coordinatelines not between representation of coordinatelines on euclidean sheet.
Sorry I got confused. I did not realease that angels in spacetime and euclidean representation are diferent, when I started thread.Ibix said:You mean, you want the angle between the ##t## axis in spacetime and the ##t'## axis in spacetime, not the angle between the representation of the ##t## axis on the diagram and the representation of the ##t'## axis on the diagram? Then you need to use hyperbolic geometry, not trigonometry.
There is a nice relationship between the Euclidean angle ##\alpha## and the rapidity ##\rho##olgerm said:Sorry I got confused. I did not realease that angels in spacetime and euclidean representation are diferent, when I started thread.
I do not know how to do that.Ibix said:That should let you solve for the various angles.
olgerm said:And really in spacetime
Ibix said:That should let you solve for the various angles.
Should I use ##\vec{v_1}\cdot\vec{v_2}=|\vec{v_1}|*|\vec{v_2}|*cosh(\alpha)## instead of ##\vec{v_1}\cdot\vec{v_2}=|\vec{v_1}|*|\vec{v_2}|*cos(\alpha)##?PeterDonis said:The quantities you get from the inverse hyperbolic functions
olgerm said:Should I use ##\vec{v_1}\cdot\vec{v_2}=|\vec{v_1}|*|\vec{v_2}|*cosh(\alpha)## instead of ##\vec{v_1}\cdot\vec{v_2}=|\vec{v_1}|*|\vec{v_2}|*cos(\alpha)##?
olgerm said:In euclidean space
I meant 4-vectors by ##\vec{v_1}## and ##\vec{v_2}##.PeterDonis said:Note that in relativity, the dot product of 3-vectors is not an invariant; it changes when you change frames. Only the dot product of 4-vectors is invariant.
olgerm said:I meant 4-vctirs by ##\vec{v_1}## and ##\vec{v_2}##.
olgerm said:Now I am quite sure, that it is ##\vec{v_1}\cdot\vec{v_2}=|\vec{v_1}|*|\vec{v_2}|*cos(\alpha)##,
These are quantities that describe location in SR-spacetime. I am not sure what kind of answer you espect.PeterDonis said:If they are 4-vectors, you are wrong. You haven't answered the questions I asked about what these vectors mean, physically. If you don't know the answer to that you aren't going to understand anything else.
Can you answer the question, what do the 4-vectors ##\vec{v_1}## and ##\vec{v_2}## mean physically? Or don't you know? If you don't know, then you need to figure that out first before even trying to guess about what their dot product and its relationship with the rapidity ##\alpha## might be.
olgerm said:These are quantities that describe location in SR-spacetime.
olgerm said:I am not sure what kind of answer you espect.
PeterDonis said:But ##\alpha## here, as has been said, is the rapidity; it is not an "angle in spacetime between the worldlines" in any useful sense, since it is not limited to the range ##0## to ##2 \pi##.
Fair point. But I think it is fair to say that rapidity is analogous to angle in the same way that interval is analogous to distance. Like the angle in Euclidean geometry, rapidity is a measure of how far off parallel two lines are and is additive. But it can take any value, not just ##0-2\pi##, fundamentally because it is not periodic. As rapidity, I think it's also only usually defined for timelike vectors since they are the only things that correspond to something moving. It's certainly possible to write the inner product of a timelike and a spacelike vector, but if you try to derive a rapidity from it you'll get a complex number. Not necessarily a bad thing, but maybe goes some way to explaining why it doesn't get used so much.PeterDonis said:I don't think rapidity can be correctly called an "angle". It certainly is not an "angle between axes in spacetime" in the sense the OP seems to want to interpret that; rapidities are not limited to the range ##0## to ##2 \pi##. See my previous post in response to the OP.
No! No! No! You should use the correct math. You are completely mislead by whatever book. A Minkowski plane is not a euclidean plane! For a simple introduction, seeolgerm said:Should I use ##\vec{v_1}\cdot\vec{v_2}=|\vec{v_1}|*|\vec{v_2}|*cosh(\alpha)## instead of ##\vec{v_1}\cdot\vec{v_2}=|\vec{v_1}|*|\vec{v_2}|*cos(\alpha)##?
No. Introducing that factor of ##i## allows many equations to take on the same form as they do for Euclidean space, but the similarity is superficial and just in the mathematical formalism. The space is still non-Euclidean in ways that cannot be avoided - for example, a straight line is not, in general, the shortest distance between two points.olgerm said:it is poosible to choose base metric is euclidean just ##\vec{e_t'}=\sqrt{-1}*\vec{e_t}## and ##\vec{e_x'}=\vec{e_x}##.
As @Nugatory says, not really. It's a mistake to try in my opinion. One of the key points for clear thinking is to make assumptions and differences between concepts as clear as possible. The ##ict## approach has always struck me as attempting to bury something quite subtle, and it comes back to bite when you move on to GR.olgerm said:it is poosible to choose base metric is euclidean just ##\vec{e_t'}=\sqrt{-1}*\vec{e_t}## and ##\vec{e_x'}=\vec{e_x}##. Then ##\vec{v_1}\cdot\vec{v_2}=|\vec{v_1}|*|\vec{v_2}|*cos(\alpha)##
then:
##\frac{e_t'}{i}\cdot e_x'=|e_t'/i|*|e_x'|*cos(\alpha_{t'-x'})##
##e_t\cdot e_x=|e_t'/i|*|e_x|*cos(\alpha_{t-x})##
##0=|e_t'/i|*|e_x|*cos(\alpha_{t-x})##
##\alpha_{t-x}=arccos(0)##
##\alpha_{t-x}=\frac{2\pi}{4}##
olgerm said:Can you confirm these are correct?
olgerm said:I had to edit ##\alpha_{t-t'}## and ##\alpha_{x-x'}##.
The equation ##\alpha=atan(v/c)## is commonly used in science to calculate the angle of deflection of a particle or object traveling at a high speed. It is also used in special relativity to determine the observed angle of an object's motion from a different frame of reference.
The equation ##\alpha=atan(v/c)## is derived from the Lorentz transformation equations, which describe how measurements of space and time differ between two frames of reference moving relative to each other at a constant velocity. By solving for the angle of deflection in these equations, we arrive at the equation ##\alpha=atan(v/c)##.
No, the equation ##\alpha=atan(v/c)## is only applicable to objects moving at relativistic speeds, meaning speeds close to the speed of light. At lower speeds, the angle of deflection can be calculated using the simpler equation ##\alpha=v/c##.
The equation ##\alpha=atan(v/c)## is a fundamental component of Einstein's theory of special relativity. It helps to explain the effects of time dilation and length contraction at high speeds and is crucial in understanding the behavior of objects in different frames of reference.
The equation ##\alpha=atan(v/c)## has many practical applications in fields such as particle physics, astrophysics, and engineering. It is used to calculate the trajectories of particles in particle accelerators, the bending of light in gravitational lensing, and the design of spacecraft trajectories. It also has implications for technologies such as GPS systems and satellite communication.