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{"id":379,"date":"2020-08-08T15:19:43","date_gmt":"2020-08-08T15:19:43","guid":{"rendered":"https:\/\/mcpab.com\/?page_id=379"},"modified":"2021-02-13T00:02:06","modified_gmt":"2021-02-13T00:02:06","slug":"on-the-transformation-of-time-and-space","status":"publish","type":"page","link":"https:\/\/physicsandmusic.com\/on-the-transformation-of-time-and-space\/","title":{"rendered":"The Transformation of Time and Space"},"content":{"rendered":"\n
In his original article, Einstein elegantly derives the Lorentz transformation of time and space between systems that move at constant velocity relative to each other. Einstein simplified <\/strong>the mathematical derivation by assuming that the relative velocity of the two systems is in the direction of the $x$ axis, and implicitly assumed <\/strong>(without explicitly discussing it) that the measurement of length, perpendicular to the velocity direction, is the same in both systems. In this section, we will follow the original reasoning of Einstein, and we will<\/p>\n\n\n\n
extend the derivation of the Lorentz transformation to the case of a general boost<\/strong>, where a vector of arbitrary<\/strong> direction represents the relative velocity of the two systems<\/li>
explicitly discuss the conditions on the general form of the Lorentz transformation that preserve the length perpendicular to the velocity.<\/li><\/ul>\n\n\n\n
We will first derive the transformation of time from the synchronization<\/strong> of clocks like Einstein did in his original paper. <\/p>\n\n\n\n
Then, using the invariance<\/strong> of the speed of light along with the formalism of bra-ket vectors, we will derive the Lorentz transformation<\/strong> of both time and space in a relatively concise way.<\/p>\n\n\n\n
We will assume that the reader is familiar with advanced calculus, linear algebra and the fundamental principles of Special Relativity.<\/p>\n\n\n\n
Before getting into the details we define a few concepts and notations that will come in handy as we proceed.<\/p>\n\n\n\n
Lorentz Transformation and Events<\/h2>\n\n\n\n
We consider two systems of reference \\(F\\) and \\(F\\) moving at constant velocity $\\vb v$ with respect to each other, where $\\vb v$ is a vector of arbitrary direction. The system $F$ has spatial coordinates $x_1$, $x_2$, $x_3$ and time $t$, and $F’$ has spatial coordinates $x_1’$, $x_2’$, $x_3’$ and time $t’$.<\/p>\n\n\n\n
An observer in $F$ will identify an event happening at a given place and time by a four-vector $\\vb* X$:<\/p>\n\n\n\n
and an observer in $F’$ will record the same event as $\\vb* X’$:<\/p>\n\n\n\n
${\\vb* X’}= (x’_1,x’_2,x’_3,c t’)$<\/p>\n\n\n\n
where $c$ is the speed of light. We will use sometimes the shorthand version of a four-vector as <\/p>\n\n\n\n
${\\vb* X}= ({\\vb* x},c t)$<\/p>\n\n\n\n
where $\\vb* x$ is the vector of the spatial coordinates.<\/p>\n\n\n\n
The Lorentz transformation describes how the coordinates and the time of the same event observed in $F$ and $F’$ are related to each other, and it is assumed to be linear<\/p>\n\n\n\n
where $\\vb \\Lambda({\\vb v})$ is a matrix depending only on the velocity vector $\\vb v$.<\/p>\n\n\n\n
bra-ket notation<\/h2>\n\n\n\n
We introduce now the bra<\/strong><\/em>–ket<\/strong><\/em> notation for vectors and linear transformations, which will greatly simplify the derivation of the space transformation. The scalar product of two N-dimensional vectors $\\vb{a}$ and $\\vb{b}$ is written in bra-ket notation as<\/p>\n\n\n\n
$ \\bra{\\vb a}\\ket{\\vb b}$<\/p>\n\n\n\n
In matrix form the bra<\/em><\/strong> vector $\\bra{\\vb a}$ is represented as a row vector, and the ket <\/em><\/strong>vector $\\ket{\\vb b}$ is a column vector. A linear transformation in bra-ket form reads<\/p>\n\n\n\n
$ \\ket{\\vb a} \\bra{\\vb b}$<\/p>\n\n\n\n
which applied to the ket vector $\\ket{\\vb c}$ acts as multiplying the vector $\\vb a$ by the scalar product of $\\vb b$ and $\\vb c$<\/p>\n\n\n\n
In Special Relativity, different observers in the same system of reference looking at the same event will agree that the event has happened at the same time only if their clocks are synchronized. Clocks synchronization is a powerful and central concept in the theory, and it is defined<\/strong> through a thought experiment. We consider two observers in the same system of reference located at different positions A<\/strong> and B<\/strong>. The observer in A<\/strong> sends a photon at time $t_A$ to the observer in B<\/strong>. The photon arrives in B<\/strong> at $t_B$, and the observer in B<\/strong> reflects it to A<\/strong>, where it finally comes at time $t’_A$. The clocks in A<\/strong> and B<\/strong> are synchronized if<\/p>\n\n\n\n
The synchronization of clocks is the key to get to the transformation of time. Let us now look again at the thought experiment above, but now both from $F$ and $F’$,<\/p>\n\n\n\n
View from $F’$<\/h3>\n\n\n\n
a photon is emitted at a time $t’_{0}$ from the origin of $F’$ along an arbitrary direction<\/li>
it is reflected at time $t’_{1}$ by a mirror posed at an arbitrary length from the source<\/li>
it arrives back at the origin of $F’$ at time $t’_{2}$<\/li><\/ul>\n\n\n\n
Since we assume that the clocks in $F’$ are synchronized, the following relation holds true:<\/p>\n\n\n\n\\begin{equation}\n \\frac{1}{2}(t’_{0}+t’_{2}) = t’_{1} \\label{sync}\n\\end{equation}\n\n\n\n
View from $F$<\/h3>\n\n\n\n
We now look at the three events, emission, reflection and arrival at the source from the perspective of $F$<\/p>\n\n\n\n
Emission: <\/strong>The photon is emitted from the source at time $t_{0}$ and at the position<\/li><\/ul>\n\n\n\n
${\\vb v}\\, t_{0}$<\/p>\n\n\n\n
where $\\vb v$ is the velocity of $F’$ measured from $F$.<\/p>\n\n\n\n
Reflection: <\/strong>The photon travels to the mirror with speed $\\vb c_{up}$ and reaches the mirror at a time $t_{1}$<\/li><\/ul>\n\n\n\n
where $\\vb c_{down}$ is the photon velocity on its way back to the source. The position of the photon when it meets the source is,<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n
where ${\\vb p}$ is a vector transforming the spatial coordinates and $\\beta$ is the coefficient transforming time, we arrive at the linear equation in ${\\vb p}$ and $\\beta$<\/p>\n\n\n\n\\begin{equation}\n\t {\\vb p}\\vdot [ {\\vb v} (\\,\\|{\\vb c_{up}-\\vb v} \\|^{-1}+ \\,\\|{\\vb c_{down}-\\vb v} \\|^{-1}) – 2 {\\vb c_{up}}\\|{\\vb c_{up}-\\vb v} \\|^{-1}] +\n\t \\beta(\\|{\\vb c_{up}-\\vb v} \\|^{-1} – \\,\\|{\\vb c_{down}-\\vb v} \\|^{-1}) =0\n\\label{eq2}\\end{equation}\n\n\n\n
\n
\n
Equation (\\ref{eq2}) can be further simplified if we look now more in detail to the direction of the vectors ${\\vb v}$, ${\\vb c_{down}}$ and ${\\vb c_{up}}$. <\/p>\n\n\n\n
Since the photon in $F’$ follows the same direction going to the mirror and back to the source, ${\\vb c_{down} -\\vb v}$ and ${\\vb c_{up}-\\vb v}$ are parallel, but have different magnitude; hence ${\\vb c_{up}}$, ${\\vb c_{down}}$ and ${\\vb v}$ are all in the same plane, see Fig (1). In the system of reference of three mutually orthogonal unit vectors ${\\vb e}_1$, ${\\vb e}_2$, and ${\\vb e}_3$, with ${\\vb e}_1$ and ${\\vb e}_2$ in the same plane of ${\\vb c{up}}$ and ${\\vb v}$, and ${\\vb e}_1$ in the same direction of $\\bf v$. the vector $\\bf p$ has components, $ {\\vb e}_1 = {\\vb v}\/{\\parallel v\\parallel}$, and equation (\\ref{eq2}) simplifies to<\/p>\n\n\n\n