Lorentz Transformations - Revision history

Karson at 04:27, 29 November 2022

2022-11-29T04:27:40Z

← Older revision		Revision as of 00:27, 29 November 2022
Line 1:		Line 1:
	The Lorentz Transformation is a transformation that allows one to shift between different coordinate systems. Namely, it allows one to transform the cartesian coordinate system of a stationary reference frame to another cartesian coordinate system of a reference frame that is moving with constant velocity <math>v</math> with respect to the stationary reference frame. It is named after Hendrik Antoon Lorentz, who derived the transformation in 1904 while working on developing the theory of electrodynamics for moving bodies. The transformation is as follows		The Lorentz Transformation is a transformation that allows one to shift between different coordinate systems. Namely, it allows one to transform the cartesian coordinate system of a stationary reference frame to another cartesian coordinate system of a reference frame that is moving with constant velocity <math>v</math> with respect to the stationary reference frame. It is named after Hendrik Antoon Lorentz, who derived the transformation in 1904 while working on developing the theory of electrodynamics for moving bodies (2). The transformation is as follows

	<math>\begin{align}x'&=\gamma(x-vt)\\ y'&=y\\ z'&=z\\ t'&=\gamma(t-\frac{xv}{c^2})\end{align}</math>		<math>\begin{align}x'&=\gamma(x-vt)\\ y'&=y\\ z'&=z\\ t'&=\gamma(t-\frac{xv}{c^2})\end{align}</math>
Line 5:		Line 5:
	where <math>\gamma=\frac{1}{\sqrt{1-v^2/c^2}}</math>.		where <math>\gamma=\frac{1}{\sqrt{1-v^2/c^2}}</math>.

	The Lorentz Transformation can be considered a more general formulation of of the ideas of time dilation and length contraction as it gives not just the changes in length and in time when going from one reference frame to another, but rather it gives the full coordinates of objects in space and time when going from one reference frame to another. Because it is a more general formulation, Einstein's equations for time dilation and length contraction can be derived from the Lorentz Transformation (shown in exercise 2.2). Going further, we can also use the Lorentz Transformation to relate observed speeds in different reference frames, which is done by calculating differentials from the Lorentz Transformation (shown in exercise 2.3).		The Lorentz Transformation can be considered a more general formulation of of the ideas of time dilation and length contraction as it gives not just the changes in length and in time when going from one reference frame to another, but rather it gives the full coordinates of objects in space and time when going from one reference frame to another. Because it is a more general formulation, Einstein's equations for time dilation and length contraction can be derived from the Lorentz Transformation (shown in exercise 2.2). Going further, we can also use the Lorentz Transformation to relate observed speeds in different reference frames, which is done by calculating differentials from the Lorentz Transformation (shown in exercise 2.3)(2).
	==Derivation of the Lorentz Transformation==		==Derivation of the Lorentz Transformation==
	The following derivation is from (4).		The following derivation is from (4).

Karson: /* Derivation of the Lorentz Transformation */

2022-11-29T04:24:32Z

Derivation of the Lorentz Transformation

← Older revision		Revision as of 00:24, 29 November 2022
Line 41:		Line 41:
	The general solution solved for the moving reference frame to the above equation is		The general solution solved for the moving reference frame to the above equation is

	<math>x'=xcosh\alpha+ctsinh\alpha\;\;\;\;\;\;\;\;\;\;\; ct'=xsinh\alpha+ctcosh\alpha</math>		<math>x'=xcosh\;\alpha+ctsinh\;\alpha\;\;\;\;\;\;\;\;\;\;\; ct'=xsinh\;\alpha+ctcosh\;\alpha</math>

	where <math>\alpha\in\mathbb{R}</math> (this is shown to be the general solution to the above equation in section 1.2). Our next step in deriving the Lorentz Transformation is to find expressions for <math>sinh\alpha</math> and <math>cosh\alpha</math>. This can be done by considering another thought experiment: consider tracking the origin of the stationary reference frame from the perspective of the moving reference frame. This simplifies the above general solution to the following since <math>x=0</math>.		where <math>\alpha\in\mathbb{R}</math> (this is shown to be the general solution to the above equation in section 1.2). Our next step in deriving the Lorentz Transformation is to find expressions for <math>sinh\alpha</math> and <math>cosh\alpha</math>. This can be done by considering another thought experiment: consider tracking the origin of the stationary reference frame from the perspective of the moving reference frame. This simplifies the above general solution to the following since <math>x=0</math>.

	<math>x'=ctsinh\alpha\;\;\;\;\;ct'=ctcosh\alpha</math>		<math>x'=ctsinh\;\alpha\;\;\;\;\;ct'=ctcosh\;\alpha</math>

	Now we can find expressions for <math>sinh\alpha</math> and <math>cosh\alpha</math>, which we can then plug back into our general equation for <math>x'</math> and <math>ct'</math> to get the Lorentz Transformation. Note: to find expressions for <math>sinh\alpha</math> and <math>cosh\alpha</math>, we use the identities stated in section 1.1.		Now we can find expressions for <math>sinh\;\alpha</math> and <math>cosh\;\alpha</math>, which we can then plug back into our general equation for <math>x'</math> and <math>ct'</math> to get the Lorentz Transformation. Note: to find expressions for <math>sinh\;\alpha</math> and <math>cosh\;\alpha</math>, we use the identities stated in section 1.1.

	<math>		<math>
	\frac{x'}{ct'}=\frac{-v}{c}=tanh\alpha\;\;\;\; \Rightarrow \;\;\;\;sinh\alpha=\frac{-v/c}{\sqrt{1-v^2/c^2}} \;\;\;\; cosh\alpha=\frac{1}{\sqrt{1-v^2/c^2}}		\frac{x'}{ct'}=\frac{-v}{c}=tanh\;\alpha\;\;\;\; \Rightarrow \;\;\;\;sinh\;\alpha=\frac{-v/c}{\sqrt{1-v^2/c^2}} \;\;\;\; cosh\;\alpha=\frac{1}{\sqrt{1-v^2/c^2}}
	</math>		</math>

Line 56:		Line 56:

	<math>		<math>
	x'=xcosh\alpha+ctsinh\alpha\;\; \Rightarrow \;\; x\frac{1}{\sqrt{1-v^2/c^2}}+ct\frac{-v/c}{\sqrt{1-v^2/c^2}}=\frac{x-vt}{\sqrt{1-v^2/c^2}}=\gamma(x-vt)		x'=xcosh\;\alpha+ctsinh\;\alpha\;\; \Rightarrow \;\; x\frac{1}{\sqrt{1-v^2/c^2}}+ct\frac{-v/c}{\sqrt{1-v^2/c^2}}=\frac{x-vt}{\sqrt{1-v^2/c^2}}=\gamma(x-vt)
	</math>		</math>

	<math>		<math>
	ct'=xsinh\alpha+ctcosh\alpha=x\frac{-v/c}{\sqrt{1-v^2/c^2}}+ct\frac{1}{\sqrt{1-v^2/c^2}}=\frac{ct-xv/c}{\sqrt{1-v^2/c^2}}=c\gamma(t-xv/c^2)		ct'=xsinh\;\alpha+ctcosh\;\alpha=x\frac{-v/c}{\sqrt{1-v^2/c^2}}+ct\frac{1}{\sqrt{1-v^2/c^2}}=\frac{ct-xv/c}{\sqrt{1-v^2/c^2}}=c\gamma(t-xv/c^2)
	</math>		</math>

Karson: /* A Computational Model */

2022-11-29T04:08:13Z

A Computational Model

← Older revision		Revision as of 00:08, 29 November 2022
Line 154:		Line 154:

	===A Computational Model===		===A Computational Model===
	[https://commons.wikimedia.org/wiki/File:Relativistic_wheels.gif Visualization of a wheel rotating at relativistic speeds by Thierry Dugnolle]		[https://commons.wikimedia.org/wiki/File:Relativistic_wheels.gif Visualization of a wheel rotating at relativistic speeds by Thierry Dugnolle] (also shown to the right)

	[[File:Relativistic wheels.gif\|thumb\|Relativistic wheels]]		[[File:Relativistic wheels.gif\|thumb\|Relativistic wheels]]

Karson: /* A Computational Model */

2022-11-29T04:07:11Z

A Computational Model

← Older revision		Revision as of 00:07, 29 November 2022
Line 155:		Line 155:
	===A Computational Model===		===A Computational Model===
	[https://commons.wikimedia.org/wiki/File:Relativistic_wheels.gif Visualization of a wheel rotating at relativistic speeds by Thierry Dugnolle]		[https://commons.wikimedia.org/wiki/File:Relativistic_wheels.gif Visualization of a wheel rotating at relativistic speeds by Thierry Dugnolle]

			[[File:Relativistic wheels.gif\|thumb\|Relativistic wheels]]

	==Examples==		==Examples==

Karson: /* Difficult */

2022-11-29T04:05:50Z

Difficult

← Older revision		Revision as of 00:05, 29 November 2022
Line 210:		Line 210:
	Show that <math>v_x'=\frac{v_x-u}{1-v_xu/c^2}</math>, <math>v_y'=\frac{v_y}{\gamma(1-v_xu/c^2)}</math>, and <math>v_z'=\frac{v_z}{\gamma(1-v_xu/c^2)}</math> are the velocity transformation formulae between reference frames where <math>u</math> is the speed of the moving reference frame from the perspective of the stationary reference frame (this was formerly <math>v</math> in our derivation of the Lorentz Transformation).		Show that <math>v_x'=\frac{v_x-u}{1-v_xu/c^2}</math>, <math>v_y'=\frac{v_y}{\gamma(1-v_xu/c^2)}</math>, and <math>v_z'=\frac{v_z}{\gamma(1-v_xu/c^2)}</math> are the velocity transformation formulae between reference frames where <math>u</math> is the speed of the moving reference frame from the perspective of the stationary reference frame (this was formerly <math>v</math> in our derivation of the Lorentz Transformation).

	Solution:		Solution (3):

	We begin (as stated in the introduction) by calculating the differentials of the Lorentz Transformation, which is shown below.		We begin (as stated in the introduction) by calculating the differentials of the Lorentz Transformation, which is shown below.

Karson: /* Examples */

2022-11-29T03:59:28Z

Examples

@@ Line 167: / Line 167: @@
 <math>
 x'=\gamma(x-vt)=\frac{1.0\;m-.75c(1.2\;s)}{\sqrt{1-.75^2c^2/c^2}}=\frac{1.0\;m-.75c(1.2\;s)}{\sqrt{1-.75^2}}=-4.1\times 10^8 \;m
 </math>

Karson: /* Examples */

2022-11-29T03:55:17Z

Examples

← Older revision		Revision as of 23:55, 28 November 2022
Line 162:		Line 162:

	Solution:		Solution:

			We are given the full coordinates in space and time for the event in the unprimed, stationary, reference frame. So we can get the coordinates of the event in space and time in the primed, moving, reference frame simply by applying the Lorentz Transformation to each coordinate.

			<math>
			x'=\gamma(x-vt)=\frac{1.0\;m-.75c(1.2\;s)}{\sqrt{1-.75^2c^2/c^2}}=\frac{1.0\;m-.75c(1.2\;s)}{\sqrt{1-.75^2}}=-4.1\times 10^8 \;m
			</math>

	===Middling===		===Middling===

Karson: /* Derivation of the Lorentz Transformation */

2022-11-29T00:22:00Z

Derivation of the Lorentz Transformation

← Older revision		Revision as of 20:22, 28 November 2022
Line 79:		Line 79:
	<math>\begin{align}x&=\gamma(x'+vt')\\ y&=y'\\ z&=z'\\ t&=\gamma(t'+\frac{x'v}{c^2})\end{align}</math>		<math>\begin{align}x&=\gamma(x'+vt')\\ y&=y'\\ z&=z'\\ t&=\gamma(t'+\frac{x'v}{c^2})\end{align}</math>

	This is also sometimes referred to as the Inverse Lorentz Transformation, and it is equivalent to the regular Lorentz Transformation except that the relative motion between the reference frames is switched (i.e. here we consider the primed reference frame moving away from the unprimed one in the positive <math>x ~~direction~~</math>, where as for the previous derivation we considered the unprimed reference frame moving away from the primed one in the negative <math>x'</math> direction).		This is also sometimes referred to as the Inverse Lorentz Transformation, and it is equivalent to the regular Lorentz Transformation except that the relative motion between the reference frames is switched (i.e. here we consider the primed reference frame moving away from the unprimed one in the positive <math>x</math> direction, where as for the previous derivation we considered the unprimed reference frame moving away from the primed one in the negative <math>x'</math> direction).

	===Hyperbolic Trigonometric Identities===		===Hyperbolic Trigonometric Identities===

Karson: /* Derivation of the Lorentz Transformation */

2022-11-29T00:21:20Z

Derivation of the Lorentz Transformation

← Older revision		Revision as of 20:21, 28 November 2022
Line 79:		Line 79:
	<math>\begin{align}x&=\gamma(x'+vt')\\ y&=y'\\ z&=z'\\ t&=\gamma(t'+\frac{x'v}{c^2})\end{align}</math>		<math>\begin{align}x&=\gamma(x'+vt')\\ y&=y'\\ z&=z'\\ t&=\gamma(t'+\frac{x'v}{c^2})\end{align}</math>

	This is also sometimes referred to as the Inverse Lorentz Transformation, and it is equivalent to the regular Lorentz Transformation except that the relative motion between the reference frames is switched.		This is also sometimes referred to as the Inverse Lorentz Transformation, and it is equivalent to the regular Lorentz Transformation except that the relative motion between the reference frames is switched (i.e. here we consider the primed reference frame moving away from the unprimed one in the positive <math>x direction</math>, where as for the previous derivation we considered the unprimed reference frame moving away from the primed one in the negative <math>x'</math> direction).

	===Hyperbolic Trigonometric Identities===		===Hyperbolic Trigonometric Identities===

Karson: /* Verification of the General Solution */

2022-11-29T00:18:12Z

Verification of the General Solution

← Older revision		Revision as of 20:18, 28 November 2022
Line 100:		Line 100:
	\begin{align}		\begin{align}
	(ctcosh\;\alpha+xsinh\;\alpha)^2-(xcosh\;\alpha+ctsinh\;\alpha)^2&=c^2t^2cosh^2\;\alpha+x^2sinh^2\;\alpha+2ctxcosh\;\alpha\;sinh\;\alpha-(x^2cosh^2\;\alpha+c^2t^2sinh^2\;\alpha+2ctxcosh\;\alpha\;sinh\;\alpha)\\		(ctcosh\;\alpha+xsinh\;\alpha)^2-(xcosh\;\alpha+ctsinh\;\alpha)^2&=c^2t^2cosh^2\;\alpha+x^2sinh^2\;\alpha+2ctxcosh\;\alpha\;sinh\;\alpha-(x^2cosh^2\;\alpha+c^2t^2sinh^2\;\alpha+2ctxcosh\;\alpha\;sinh\;\alpha)\\
	&=(c^2t^2cosh^2\;\alpha-c^2t^2sinh^2\;\alpha)-(x^2cosh^2\;\alpha-x^2sinh^2\;\alpha)+(2ctxcosh\;\alpha\;~~sing~~\;\alpha-2ctxcosh\;\alpha\;~~sing~~\;\alpha)\\		&=(c^2t^2cosh^2\;\alpha-c^2t^2sinh^2\;\alpha)-(x^2cosh^2\;\alpha-x^2sinh^2\;\alpha)+(2ctxcosh\;\alpha\;sinh\;\alpha-2ctxcosh\;\alpha\;sinh\;\alpha)\\
	&=c^2t^2(cosh^2\;\alpha-sinh^2\;\alpha)-x^2(cosh^2\;\alpha-sinh^2\;\alpha)\\		&=c^2t^2(cosh^2\;\alpha-sinh^2\;\alpha)-x^2(cosh^2\;\alpha-sinh^2\;\alpha)\\
	&=c^2t^2-x^2		&=c^2t^2-x^2