Frame of Reference Transformation - Revision history

Vanwdani: /* Frame of Reference Transformation */

2019-01-01T19:04:36Z

Frame of Reference Transformation

Vanwdani at 18:47, 15 May 2018

2018-05-15T18:47:39Z

Vanwdani at 04:07, 10 July 2017

2017-07-10T04:07:20Z

Vanwdani at 04:00, 10 July 2017

2017-07-10T04:00:20Z

Vanwdani at 04:00, 10 July 2017

2017-07-10T04:00:05Z

Vanwdani at 03:58, 10 July 2017

2017-07-10T03:58:19Z

Vanwdani at 03:49, 10 July 2017

2017-07-10T03:49:01Z

Vanwdani at 03:40, 10 July 2017

2017-07-10T03:40:19Z

Vanwdani at 03:23, 10 July 2017

2017-07-10T03:23:55Z

Vanwdani at 03:19, 10 July 2017

2017-07-10T03:19:41Z

@@ Line 85: / Line 85: @@
-<center><math>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</math></center>
+<center><math>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}</math></center>
-<center><math>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</math></center>
+<center><math>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}</math></center>
-<center><math>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</math></center>
+<center><math>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}</math></center>
-<center><math>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</math></center>
+<center><math>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}</math></center>

@@ Line 1: / Line 1: @@
-<center><math>\textbf{\underline{Navigation}}</math>
+<center><math>\underline{\textbf{Navigation}}</math>
 [[4-momenta|<math>\vartriangleleft </math>]]
@@ Line 242: / Line 242: @@
-<center><math>\textbf{\underline{Navigation}}</math>
+<center><math>\underline{\textbf{Navigation}}</math>
 [[4-momenta|<math>\vartriangleleft </math>]]

@@ Line 1: / Line 1: @@
 Using the Lorentz transformations and the index notation,
@@ Line 231: / Line 240: @@
 & 0 & 0 & -1
 \end{bmatrix}</math></center>

← Older revision		Revision as of 04:00, 10 July 2017
Line 123:		Line 123:
	Again, using a summation over the indicies		Again, using a summation over the indicies

−	<center><math>d\mathbf x^{'\mu}=\sum_{\mu=0}^3 \frac{\partial x^{'\mu}}{\partial x^{\nu}}d\mathbf x^{\nu}</math></center>	+	<center><math>d\mathbf x^{'\mu}=\sum_{\nu=0}^3 \frac{\partial x^{'\mu}}{\partial x^{\nu}}d\mathbf x^{\nu}</math></center>

@@ Line 123: / Line 123: @@
 Again, using a summation over the indicies
-<center><math>d\mathbf x^{'\mu}=\sum_{\mu=0}^3 \frac{\partial x^{'\mu}}{\partial x^{\mu}}d\mathbf x^{\mu}</math></center>
+<center><math>d\mathbf x^{'\mu}=\sum_{\mu=0}^3 \frac{\partial x^{'\mu}}{\partial x^{\nu}}d\mathbf x^{\nu}</math></center>
 Using the Einstein convention
-<center><math>d\mathbf x^{'\mu}= \frac{\partial x^{'\mu}}{\partial x^{\mu}}d\mathbf x^{\mu}</math></center>
+<center><math>d\mathbf x^{'\mu}= \frac{\partial x^{'\mu}}{\partial x^{\nu}}d\mathbf x^{\nu}</math></center>

@@ Line 119: / Line 119: @@
 dx^3
 \end{bmatrix}</math></center>
 The Lorentz transformations are also invariant in that they are just a rotation, i.e. Det <math>\Lambda=1</math>.  The inner product is preserved,

@@ Line 93: / Line 93: @@
 <center><math>\begin{bmatrix}
 dx'^0 \\
 dx'^1 \\
 dx'^2\\
 dx'^3
 \end{bmatrix}=
 \begin{bmatrix}
   \frac{\partial x^{'0}}{\partial x^0} & \frac{\partial x^{'0}}{\partial x^1} & \frac{\partial x^{'0}}{\partial x^2} & \frac{\partial x^{'0}}{\partial x^3}  \\
   \frac{\partial x^{'1}}{\partial x^0} & \frac{\partial x^{'1}}{\partial x^1} & \frac{\partial x^{'1}}{\partial x^2} & \frac{\partial x^{'1}}{\partial x^3}  \\
   \frac{\partial x^{'2}}{\partial x^0} & \frac{\partial x^{'2}}{\partial x^1} & \frac{\partial x^{'2}}{\partial x^2} & \frac{\partial x^{'2}}{\partial x^3}  \\
   \frac{\partial x^{'3}}{\partial x^0} & \frac{\partial x^{'3}}{\partial x^1} & \frac{\partial x^{'3}}{\partial x^2} & \frac{\partial x^{'3}}{\partial x^3}
 \end{bmatrix}
@@ Line 106: / Line 112: @@
 \begin{bmatrix}
 dx^0  \\
 dx^1 \\
 dx^2 \\
 dx^3
 \end{bmatrix}</math></center>

← Older revision		Revision as of 03:23, 10 July 2017
Line 88:		Line 88:


		+
		+	Expressing this in matrix form
		+
		+	<center><math>\begin{bmatrix}
		+	dx'^0 \\
		+	dx'^1 \\
		+	dx'^2\\
		+	dx'^3
		+	\end{bmatrix}=
		+	\begin{bmatrix}
		+	\frac{\partial x^{'0}}{\partial x^0} & \frac{\partial x^{'0}}{\partial x^1} & \frac{\partial x^{'0}}{\partial x^2} & \frac{\partial x^{'0}}{\partial x^3} \\
		+	\frac{\partial x^{'1}}{\partial x^0} & \frac{\partial x^{'1}}{\partial x^1} & \frac{\partial x^{'1}}{\partial x^2} & \frac{\partial x^{'1}}{\partial x^3} \\
		+	\frac{\partial x^{'2}}{\partial x^0} & \frac{\partial x^{'2}}{\partial x^1} & \frac{\partial x^{'2}}{\partial x^2} & \frac{\partial x^{'2}}{\partial x^3} \\
		+	\frac{\partial x^{'3}}{\partial x^0} & \frac{\partial x^{'3}}{\partial x^1} & \frac{\partial x^{'3}}{\partial x^2} & \frac{\partial x^{'3}}{\partial x^3}
		+	\end{bmatrix}
		+	\cdot
		+	\begin{bmatrix}
		+	dx^0 \\
		+	dx^1 \\
		+	dx^2 \\
		+	dx^3
		+	\end{bmatrix}</math></center>

	The Lorentz transformations are also invariant in that they are just a rotation, i.e. Det <math>\Lambda=1</math>. The inner product is preserved,		The Lorentz transformations are also invariant in that they are just a rotation, i.e. Det <math>\Lambda=1</math>. The inner product is preserved,

@@ Line 76: / Line 76: @@
-<center><math>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} dt+\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</math></center>
+<center><math>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</math></center>
-<center><math>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</math></center>
+<center><math>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</math></center>
-<center><math>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</math></center>
+<center><math>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</math></center>
-<center><math>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</math></center>
+<center><math>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</math></center>