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23 May 2003

Vectors Four Mu

by George Hrabovsky, President of MAST

News from MAST

This week's news is not really from MAST in specific. I would like to report the joining of the President and Treasurer of MAST. Dianna LaVigne is now Dianna Hrabovsky. I also want to give special thanks to Sheldon for his nice announcement last week. Here is the blushing bride,

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Here is our wedding.

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Here is our cake.

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Also, I would like to report a successful storm chase expedition on 10 May at approximately 9:00 PM CDT, Dianna, Rocky Wenz, and myself intercepted a tornado within a mile of us north of Peoria, IL. We did not actually see the tornado, but we did catch it breaking powerlines (we were so close you could see the arcing of the powerline flashes). This was Rocky's first successful chase in five years, and it was Dianna's first real chase.

Four-Scalars

Thus far, for the sake of simplicity, we have been considering spacetime as having only one spatial dimension. Of course, we know that in the real world there are three apparent spatial dimensions. This gives us a total of four dimensions, so the spacetime interval is actually,

Δ s^2 = - Δ t^2 + Δ x^2 + Δ y^2 + Δ z^2 .

Of course, we do not always want to deal with cartesian coordinates, so we generalize this to

Δ s^2 = - Δ x _ 0^2 + Δ x _ 1^2 + Δ x _ 2^2 + Δ x _ 3^2 .

Adopting the convention that a Latin subscript of superscript is summed from 1 to 3, we can rewrite this

Δ s^2 = - Δ x _ 0^2 + Δ x _ n^2 .

This is a four-dimensional quantity that is invariant. Such a quantity is called a 4-scalar.
    

Four-Vectors

We can consider the four-dimensional Lorentz transformations

Overscript[x, ~] _ 0 = γ (x _ 0 - v x _ 1)

Overscript[x, ~] _ 1 = γ (x _ 1 - v x _ 0)

Overscript[x, ~] _ 2 = x _ 2

Overscript[x, ~] _ 3 = x _ 3 .

We can make a 4 × 4 matrix of the coefficients,

(γ -γ v 0 0 ) . -γ v γ 0 0 0 0 1 0 0 0 0 1

We can name this matrix α _ μν, where the Greek subscripts of superscripts are summed from 0 to 3.
    If we have a four-dimensional vector that undergoes a Lorentz transformation, we call it a 4-vector. A 4-vector, Overscript[V, ~] _ μ, has the form

Overscript[V, ~] _ μ = α _ μν V _ ν .

We will explore some of the ramifications of 4-vectors next time.

Answer to the Theory Challenge from Last Time

The task was to simplify the result,

FormBox[RowBox[{x, , =, , RowBox[{x _ 0 e^(i ω t), =, , RowBox[{RowBox[{(Re[A _ 0 &nbs ...            ]}], (1)}]}]}], TraditionalForm]

We can always express a complex number in polar form,

x + i y = r e^(i θ)

where

r = (x^2 + y^2)^1/2 .

and

i θ = i tan^(-1) y/x .

Converting the denominator of (1) into this form, we have

r = [(ω _ 0^2 - ω ^2)^2    + 4 γ^2 ω^2]^1/2,

and

i θ = i (2 γ ω)/(ω _ 0^2 - ω ^2) .

so,

r e^(i θ) =    [(ω _ 0^2 - ω ^2)^2    + 4 γ^2 ω^2]^1/2 e^(i tan^(-1) (2 γ i ω)/(ω _ 0^2 - ω ^2)) .

If we define an angle

β = θ^(-1)

     = tan^(-1) (ω _ 0^2 - ω ^2)/(2 γ ω) .

then we have,

r e^(i θ) =    [(ω _ 0^2 - ω ^2)^2    + 4 γ^2 ω^2]^1/2 e^(-i β) .

then the real solution becomes

x = (Re[A _ 0    e^(i    θ _ 0)] e^(i ω t)/m)/( [(ω _ 0^2 - ω ^2)^2    + 4 γ^2 ω^2]^1/2 e^(-i β) )

     = (Re[A _ 0    e^(i    θ _ 0)] e^(i ω t))/(m [(ω _ 0^2 - ω ^2)^2    + 4 γ^2 ω^2]^1/2 e^(-i β) )

     = Re[A _ 0    e^(i    θ _ 0)]/(m [(ω _ 0^2 - ω ^2)^2    + 4 γ^2 ω^2]^1/2 e^(-i β) ) e^(i ω t)

     = Re[A _ 0 ]/m [(ω _ 0^2 - ω ^2)^2    + 4 γ^2 ω^2]^1/2    e^(i ω t) e^(i    θ _ 0) e^(i β)

     = Re[A _ 0 ]/m [(ω _ 0^2 - ω ^2)^2    + 4 γ^2 ω^2]^1/2    e^(i (ω t + θ _ 0 + β)) .

Theory Challenge

What happens if we add this solution to the solution for an underdamped oscillator?

Books That I Like

Bernard F. Schutz (1990), A First Course in General Relativity, Cambridge University Press. This has a nice introduction to spacetime diagrams and derives how to view other coordinate systems.

Ray D'Inverno (1992), Introducing Einstein's Relativity, Oxford University Press. This is my favorite introduction to relativity.

Online Resources

4-Vectors:

For a brief description go to:

http://www-physics.mps.ohio-state.edu/~cleo/hep/explain/4vector_1.html

For a slightly more advanced approach,

http://www.pact.cpes.sussex.ac.uk/users/markh/RQF1/node12.html

Forced Oscillators

Here is a fairly elementary site that presents the topic in general terms.

http://www.pinkmonkey.com/studyguides/subjects/physics/chap10/p1010601.asp

Here is a more challenging site that gets similar results to those above.

http://colos1.fri.uni-lj.si/~colos/COLOS/TUTORIALS/JAVA/JAVAXYZET/RESONANCE/HTML/resonance_6.html

Converted by Mathematica  (May 23, 2003)