special relativity

trees/def/def-001H.tree

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+\title{Timelike Separated Events}
+\taxon{Definition}
+\p{
+    An event #{S} is said to be timelike separated if
+    ##{
+        (\Delta x^0)^2 > (\Delta x^1)^2 + (\Delta x^2)^2 + (\Delta x^3)^2
+    }
+    or briefly #{\Delta s^2 > 0}. The spatial separation is less than the distance light travels.
+}

trees/def/def-001I.tree

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+\title{Lightlike Separated Events}
+\taxon{Definition}
+\p{
+    Events connected by the world-line of a \strong{photon} are said to be \strong{lightlike separated}.
+    For which #{\Delta s^2 = 0}.
+}

trees/def/def-001J.tree

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+\title{Spacelike Separated Events}
+\taxon{Definition}
+\p{
+    Two events for which #{\Delta s^2 < 0} are said to be \strong{spacelike separated}.
+    Events that are simultaneous in a Lorentz frame but in different position are spacelike separated.
+}

trees/def/def-001K.tree

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+\title{Einstein's Summation Convention}
+\taxon{Definition}
+\p{
+    When index variable appears twicec in a single term and
+    is not otherwise defined, it is assumed to be summed over.
+    If the indices can range over #{\{1,2,3\}},then
+    ##{
+        s = \sum_{i=1}^3 a_i b_i = a_1 b_1 + a_2 b_2 + a_3 b_3
+    }
+    can be simplified to
+    ##{
+        s = a_i b_i
+    }
+}

trees/def/def-001L.tree

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+\title{Minkowski Metric}
+\taxon{Definition}
+\p{
+    The \strong{Minkowski Metric}, aka \strong{Minkowski Tensor}, is a tensor #{\eta_{\mu\nu}} whose elements are defined by the matrix
+    ##{
+        \eta_{\mu\nu} = \begin{pmatrix}
+            -1 & 0 & 0 & 0 \\
+            0 & 1 & 0 & 0 \\
+            0 & 0 & 1 & 0 \\
+            0 & 0 & 0 & 1 
+        \end{pmatrix}
+    }
+    where #{\mu} and #{\nu} are Lorentz indices run over #{0,1,2,3}.
+}

trees/def/def-001M.tree

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+\title{Symmetric and Antisymmetric Part}
+\taxon{Definition}
+\p{
+    Any square matrix #{A} can be written as the sum of a symmetric and antisymmetric matrix:
+    ##{
+        A = A_S + A_A
+    }
+    where #{A_S} is the \strong{symmetric part} and #{A_A} is the \strong{antisymmetric part}.
+    ##{
+        A_S = \frac{1}{2}(A + A^T)
+    }
+    where #{A_S} is a [symmetric matrix](def-001N)
+    ##{
+        A_A = \frac{1}{2}(A - A^T)
+    }
+    where #{A_A} is an [antisymmetric matrix](def-001O)
+}

trees/def/def-001N.tree

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+\title{Symmetric Matrix}
+\taxon{Definition}
+\p{
+    A square matrix #{A} is symmetric if it is equal to its transpose:
+    ##{
+        A = A^T
+    }
+    This also implies #{A^{-1} A^T = I}
+}

trees/def/def-001O.tree

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+\title{Antisymmetric Matrix}
+\taxon{Definition}
+\p{
+    A square matrix #{A} is antisymmetric if it is equal to the negative of its transpose:
+    ##{
+        A = -A^T
+    }
+    This also implies #{A^{-1} A^T = -I}
+}

trees/def/def-001P.tree

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+\title{Kronecker Delta}
+\taxon{Definition}
+\p{
+    The \strong{Kronecker delta} is defined as
+    ##{
+        \delta_{i}^j = \begin{cases}
+            1 & \text{if } i = j \\
+            0 & \text{if } i \neq j
+        \end{cases}
+    }
+}

trees/def/def-001Q.tree

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+\title{Lorentz Transformations}
+\taxon{Definition}
+\p{
+    Consider a frame #{S} and #{S'} which is moving along the #{+x} direction of the #{S} frame
+    with a velocity #{v}.
+    Assume that the origins of the two frames coincide at #{t=t'=0} and coordinate axes are parallel.
+
+    We say that #{S'} is boosted along the #{x} direction with velocity parameter #{\beta\equiv\frac{v}{c}}.
+    The \strong{Lorentz transformations} are defined by a set of equations that relate the coordinates of an event in the two frames.
+    ##{
+        \begin{align*}
+            x' &= \gamma(x-\beta ct) \\
+            y' &= y \\
+            z' &= z \\
+            ct' &= \gamma(ct-\beta x)
+        \end{align*}
+    }
+    where #{\gamma\equiv\dfrac{1}{\sqrt{1-\beta^2}} = \dfrac{1}{\sqrt{1-\frac{v^2}{c^2}}}} is the \strong{Lorentz factor}.
+    The coordinates orthogonal to the #{x} direction remains unchanged.
+}
+\p{
+    Lorentz transformations are the linear transformations of coordinates that remains the #{\Delta s^2} unchanged. 
+    We can write the Lorentz transformations in matrix form:
+    ##{
+        \begin{pmatrix}
+            ct' \\
+            x' \\
+            y' \\
+            z' 
+        \end{pmatrix}
+        =
+        \begin{pmatrix}
+            \gamma & -\beta\gamma & 0 & 0 \\
+            -\beta\gamma & \gamma & 0 & 0 \\
+            0 & 0 & 1 & 0 \\
+            0 & 0 & 0 & 1 
+        \end{pmatrix}
+        \begin{pmatrix}
+            ct \\
+            x \\
+            y \\
+            z 
+        \end{pmatrix}
+    }
+    Or in a more compact form:
+    ##{
+        x'^\mu = L^\mu_\nu x^\nu
+    }
+    where #{L^\mu_\nu} is the \strong{Lorentz transformation matrix} presented above.
+}

trees/def/def-001R.tree

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+\title{Light-cone Coordinates}
+\taxon{Definition}
+\p{
+    The \strong{light-cone coordinates} can be defined as
+    two independent [linear combinations](def-000L) of the time 
+    and a chosen spatial coordinate (conventionally #{x^1}):
+    ##{
+        \begin{align*}
+            x^+ \equiv \frac{1}{\sqrt{2}} (X^0 + X^1) \\
+            x^- \equiv \frac{1}{\sqrt{2}} (X^0 - X^1)
+        \end{align*}
+    }
+    while other spatial coordinates remain unchanged. Thus the complete set of 
+    light-cone coordinates is #{(x^+,x^-,x^2,x^3)}.
+}