Trajectory of Sun's motion

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Hungry4info
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Re: Trajectory of Sun's motion

Post #21by Hungry4info » 04.04.2008, 14:29

I believe the ' is a reference to the derivative of a function.

d/dx f[x] = f'(x).
Where d/dx (x^n) = nx^(n-1).
Thus, if f(x)=x^3, f'(x)=3x^2.
And if f(x)=2x^2, f'(x)=4x.

Underscores denote subscripts, I believe.

If any of that's wrong, please, somebody tell me.
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Re: Trajectory of Sun's motion

Post #22by t00fri » 04.04.2008, 14:53

Epimetheus, Hungry4info,

Epimetheus wrote:
t00fri wrote:
Consider the components of the vector X in 2 such frames, this means

X.X=X^2 = (c*t)^2 - x^2 -y^2 -z^2 = (c*t')^2 - x'^2 -y'^2 - z'^2

Forgive my budding mathematical knowledge ( intermediate algebra and intro to discrete math), Dr. Shrempp, but in the above equation are we using ' as logical not operator? The right-hand side is the inverse? :roll:

...
Epimetheus

Sorry if this prime notation was unclear. It is very simple. The primed quantities are merely labelling the components of the 4-vector in another relativistic reference frame.

In the original, unprimed frame, the position 4-vector had the form

X = (c*t, x,y,z)

after a Lorentz transformation to another reference frame (the "primed" one), all components of our vector change into

X' = (c*t', x',y',z')

Note the most fundamental feature that is at the basis of all this: the speed of light c stays the same in all possible frames (it gets NO prime)!!

Despite all components of the 4-vector change, the length of that 4-vector must remain the same, hence we get the equation that I wrote down above:

X.X=X^2 = (c*t)^2 - x^2 -y^2 -z^2 = (c*t')^2 - x'^2 -y'^2 - z'^2 = X'^2

Got it?

F.
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Re: Trajectory of Sun's motion

Post #23by Epimetheus » 04.04.2008, 15:12

t00fri wrote:Note the most fundamental feature that is at the basis of all this: the speed of light c stays the same in all possible frames (it gets NO prime)!!

Despite all components of the 4-vector change, the length of that 4-vector must remain the same, hence we get the equation that I wrote down above:

X.X=X^2 = (c*t)^2 - x^2 -y^2 -z^2 = (c*t')^2 - x'^2 -y'^2 - z'^2 = X'^2

Got it?
F.

With all due respect, sir, I can't even remember where I left my keys! :? I bet they're following one of those worldlines and I'm on the wrong plane of reference!

BTW, doc, does X.X represent a multi-dimensional matrix? The velocity of light remains constant in ALL reference frames, regardless of an observer at rest at any cartesian coordinate within any hyper-polytropal dimension, correct?
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Re: Trajectory of Sun's motion

Post #24by Hungry4info » 04.04.2008, 15:18

Yeah, this is the kind of stuff that I find fascinating, but know nothing about. Is there a comprehensive website that I can read about this?
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Re: Trajectory of Sun's motion

Post #25by Epimetheus » 04.04.2008, 15:52

Just follow the good doctor's links in a prior post here:

viewtopic.php?f=4&t=11959

:)
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Re: Trajectory of Sun's motion

Post #26by t00fri » 04.04.2008, 16:03

Epimetheus wrote:
t00fri wrote:Note the most fundamental feature that is at the basis of all this: the speed of light c stays the same in all possible frames (it gets NO prime)!!

Despite all components of the 4-vector change, the length of that 4-vector must remain the same, hence we get the equation that I wrote down above:

X.X=X^2 = (c*t)^2 - x^2 -y^2 -z^2 = (c*t')^2 - x'^2 -y'^2 - z'^2 = X'^2

Got it?
F.

With all due respect, sir, I can't even remember where I left my keys! :? I bet they're following one of those worldlines and I'm on the wrong plane of reference!

BTW, doc, does X.X represent a multi-dimensional matrix? The velocity of light remains constant in ALL reference frames, regardless of an observer at rest at any cartesian coordinate within any hyper-polytropal dimension, correct?

Well, this forum is not well suited for math notation, so we should really activate the LaTex feature ASAP!

X.X is to indicate the scalar product of the two vectors. To be precise,
if X is a row vector, like so, X=(c*t, x,y,z) then the other X should be a transposed, i.e. /column/ vector. But columns are hard to draw here ;-) . There is another notation in terms of the vector components. But it is also hard to display here. It uses greek indices and the usual convention that repeating indices are to be summed over.

Something like

X_mu * X^mu; mu = 0,1,2,3

So X.X is a product that contracts a vector and it's transpose into a scalar (hence scalar product) , i.e. it is just one dimensionful number, the length^2 of our vector.

I wasn't explicit here since you mentioned somewhere that you were taking math classes at University, these days?

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Re: Trajectory of Sun's motion

Post #27by Epimetheus » 04.04.2008, 16:59

t00fri wrote:Well, this forum is not well suited for math notation, so we should really activate the LaTex feature ASAP!

X.X is to indicate the scalar product of the two vectors. To be precise,
if X is a row vector, like so, X=(c*t, x,y,z) then the other X should be a transposed, i.e. /column/ vector. But columns are hard to draw here ;-) . There is another notation in terms of the vector components. But it is also hard to display here. It uses greek indices and the usual convention that repeating indices are to be summed over.

Something like

X_mu * X^mu; mu = 0,1,2,3

So X.X is a product that contracts a vector and it's transpose into a scalar (hence scalar product) , i.e. it is just one dimensionful number, the length^2 of our vector.

I wasn't explicit here since you mentioned somewhere that you were taking math classes at University, these days?

F.

Yes sir! I believe I know what you're speaking of - Big O, theta, sigma and the like? The type of symbols you use for proofs by induction, say the giant S looking symbol that would state n=0 as n tends to infinity = {0(n+1) + 1(n+2) ... }
Representing the summation of a series, or pi to represent the multiplicative of a series, and so on. I was just introduced to proof by induction, propositional logic, infinities and greater infinities, number theory, Euler and Hamilton Paths and the like in Intro to Discrete Math and struggled with the concepts a great deal. Not a natural math head, just a motivated student. We'll see where I am in a year or two with all this! 8)

Love reading your posts, doc!

p.s. yes, a LateX or mathmatica port would be nice for this subject matter. Hmmmm! :idea:
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Epimetheus
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Re: Trajectory of Sun's motion

Post #28by Epimetheus » 07.04.2008, 09:23

t00fri wrote:Epimetheus, Hungry4info,

Epimetheus wrote:
t00fri wrote:
Consider the components of the vector X in 2 such frames, this means

X.X=X^2 = (c*t)^2 - x^2 -y^2 -z^2 = (c*t')^2 - x'^2 -y'^2 - z'^2

Forgive my budding mathematical knowledge ( intermediate algebra and intro to discrete math), Dr. Shrempp, but in the above equation are we using ' as logical not operator? The right-hand side is the inverse? :roll:

...
Epimetheus

Sorry if this prime notation was unclear. It is very simple. The primed quantities are merely labelling the components of the 4-vector in another relativistic reference frame.

In the original, unprimed frame, the position 4-vector had the form

X = (c*t, x,y,z)

after a Lorentz transformation to another reference frame (the "primed" one), all components of our vector change into

X' = (c*t', x',y',z')

Note the most fundamental feature that is at the basis of all this: the speed of light c stays the same in all possible frames (it gets NO prime)!!

Despite all components of the 4-vector change, the length of that 4-vector must remain the same, hence we get the equation that I wrote down above:

X.X=X^2 = (c*t)^2 - x^2 -y^2 -z^2 = (c*t')^2 - x'^2 -y'^2 - z'^2 = X'^2

Got it?

F.

It's starting to sink in. So the primed part happens after the event! ( e.g. The event at (x, y, z, t) in frame S has coordinates (x?, y?, z?, t?) in frame S?. ) How would one apply the Lorentz Transformation to a 4-D vector? If you want to model this behavior in virtual space would you use the Matrix Form and convert to cartesian coords?

It's interesting how this understanding of hyperspace is starting to show up in game programming. For example, the use of quarternions and slerp for rotations and camera angles, which is much smoother and no "gimbal lock," utilizes the same 4-D vectors for camera views (reference frames ;) ) that you've laid out here for understanding spacetime and worldlines and the like. Fascinating stuff!
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