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Units, Constants and Useful Formulas
Lorentz Invariance
------------------
A quantity that remains unchanged by a Lorentz
transformation is said to be Lorentz invariant.
This implies that the laws of physics are
independent of the orientation or the boost
velocity of the reference frame through space.
Lorentz invariance is achieved if the laws of
physics are cast in terms of 4-vector inner
products. Let's see this for a rotation of
cartesian axes.
If we rotate a set of cartesian axes so that
x -> x' and y -> y' then from Pythagoras we
can show that for any given point:
x'^{2} + y'^{2} = x^{2} + y^{2}.
The quantity x^{2} + y^{2} is invariant under the
rotation.
For spacetime under a Lorentz transformation
(t -> t' and x -> x'), the invariant quantity
is instead given by:
x'^{2} + y'^{2} + z'^{2} - ct'^{2} = x^{2} + y^{2} + z^{2} - ct^{2}
Proof:
We will consider only one spacial dimension
and perform a Lorentz transformation of the
2-vector (t,x) - ordinarily it would be a
4-vector:
X = γ(x - vt)
T = γ(t - vx/c^{2})
γ = 1/√(1 - v^{2}/c^{2})
s^{2} = -c^{2}T^{2} + X^{2}
= -γ^{2}c^{2}(t - vx/c^{2})^{2} + γ^{2}(x - vt)^{2}
= -γ^{2}{c^{2}t^{2} - 2xvt + v^{2}x^{2}/c^{2}} + γ^{2}{x^{2} - 2xvt + v^{2}t^{2}}
= -γ^{2}{c^{2}t^{2} - 2xvt + v^{2}x^{2}/c^{2} - x^{2} + 2xvt - v^{2}t^{2}}
= -γ^{2}{c^{2}t^{2} + v^{2}x^{2}/c^{2} - x^{2} - v^{2}t^{2}}
= -γ^{2}{(c^{2} - v^{2})t^{2} + (v^{2}/c^{2} - 1)x^{2}}
= -γ^{2}{(1 - v^{2}/c^{2})c^{2}t^{2} - (1 - v^{2}/c^{2})x^{2}}
= -c^{2}t^{2} + x^{2} - the invariant interval
We can also construct the invariant interval ds^{2}
by taking the dot product of the covariant and
contravariant components of a 4-vector. A 4-vector
by itself is not invariant under a Lorentz
transformation.
dx^{μ} = (cdt,dx^{1},dx^{2},dx^{3}) - contravariant
dx_{μ} = (-cdt,dx_{1},dx_{2},dx_{3}) - covariant
ds^{2} = Σdx_{μ}dx^{μ}
= dx_{μ}dx^{μ}
= -c^{2}(dt)^{2} + (dx)^{2}/c^{2} + (dy)^{2}/c^{2} + (dz)^{2}/c^{2}
The result is invariant under rotations in space
and a Lorentz transformation. This can also be
written as:
ds^{2} = dx_{μ}dx^{μ} = η_{μν}dx^{μ}dx^{ν} where η_{μν} is the MINKOWSKI
METRIC.
- -
_{ } |-1 0 0 0 |
η_{μν} = | 0 1 0 0 |
_{ } | 0 0 1 0 |
_{ } | 0 0 0 1 |
- -
Note, the time-coordinate has different sign than
the space coordinates. This simple change of sign
leads to completely new hyperbolic geometry that
is different from conventional Euclidean geometry.