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Frame of Reference Transformation
Using the Lorentz transformations and the index notation,
{t′=γ(t−vz/c2)x′=x′y′=y′z′=γ(z−vt)
[x′0x′1x′2x′3]=[γ(x0−vx3/c)x1x2γ(x3−vx0)]=[γ(x0−βx3)x1x2γ(x3−vx0)]
Where β≡vc
This can be expressed in matrix form as
[x′0x′1x′2x′3]=[γ00−γβ01000010−γβ00γ]⋅[x0x1x2x3]
Letting the indices run from 0 to 3, we can write
x′μ=∑3ν=0(Λμν)xν
Where Λ is the Lorentz transformation matrix for motion in the z direction.
Using the Einstein convention, this can be written as
x′μ=Λμνxν
If we take the 4-vector quantities to be on an infinitesimally small scale, then there exists a linear relationship between the transformation. Following the rules of partial differentiation,
dt^' \equiv \frac{\partial t'}{\partial t} dt+\frac{\partial t'}{\partial x} dx + \frac{\partial t'}{\partial y} dy+ \frac{\partial t'}{\partial z} dz \Rightarrow dx^{'0} \equiv \frac{\partial x^{'0}}{\partial x^0} dx^0+\frac{\partial x^{'0}}{\partial x^1} dx^1 + \frac{\partial x^{'0}}{\partial x^2} dx^2+ \frac{\partial x^{'0}}{\partial x^3} dx^3
dx^' \equiv \frac{\partial t'}{\partial t} dt+\frac{\partial x'}{\partial x} dx + \frac{\partial x'}{\partial y} dy+ \frac{\partial x'}{\partial z} dz\Rightarrow dx^{'1} \equiv \frac{\partial x^{'1}}{\partial x^0} dx^0+\frac{\partial x^{'1}}{\partial x^1} dx^1 + \frac{\partial x^{'1}}{\partial x^2} dx^2+ \frac{\partial x^{'1}}{\partial x^3} dx^3
dy^' \equiv \frac{\partial y'}{\partial t} dt+\frac{\partial y'}{\partial x} dx + \frac{\partial y'}{\partial y} dy+ \frac{\partial y'}{\partial z} dz\Rightarrow dx^{'2} \equiv \frac{\partial x^{'2}}{\partial x^0} dx^0+\frac{\partial x^{'2}}{\partial x^1} dx^1 + \frac{\partial x^{'2}}{\partial x^2} dx^2+ \frac{\partial x^{'2}}{\partial x^3} dx^3
dz^' \equiv \frac{\partial z'}{\partial t} dt+\frac{\partial z'}{\partial x} dx + \frac{\partial z'}{\partial y} dy+ \frac{\partial z'}{\partial z} dz\Rightarrow dx^{'3} \equiv \frac{\partial x^{'3}}{\partial x^0} dx^0+\frac{\partial x^{'3}}{\partial x^1} dx^1 + \frac{\partial x^{'3}}{\partial x^2} dx^2+ \frac{\partial x^{'3}}{\partial x^3} dx^3
Expressing this in matrix form
[dx′0dx′1dx′2dx′3]=[∂x′0∂x0∂x′0∂x1∂x′0∂x2∂x′0∂x3∂x′1∂x0∂x′1∂x1∂x′1∂x2∂x′1∂x3∂x′2∂x0∂x′2∂x1∂x′2∂x2∂x′2∂x3∂x′3∂x0∂x′3∂x1∂x′3∂x2∂x′3∂x3]⋅[dx0dx1dx2dx3]
Again, using a summation over the indicies
dx′μ=∑3ν=0∂x′μ∂xνdxν
Using the Einstein convention
dx′μ=∂x′μ∂xνdxν
The Lorentz transformations are also invariant in that they are just a rotation, i.e. Det Λ=1. The inner product is preserved,
ΛμνημνΛνμ=ημν
[γ00−γβ01000010−γβ00γ]⋅[10000−10000−10000−1]⋅[γ00−γβ01000010−γβ00γ]T=[10000−10000−10000−1]
[γ2−β2γ20000−10000−10000−γ2+β2γ2]=[10000−10000−10000−1]
[γ2(1−β2)0000−10000−10000−γ2(1−β2)]=[10000−10000−10000−1]
Where γ≡1√1−β2
[γ2γ20000−10000−10000−γ2γ2]=[10000−10000−10000−1]
[10000−10000−10000−1]=[10000−10000−10000−1]
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