Initial CM Frame 4-momentum components - Revision history

Vanwdani at 18:53, 15 May 2018

2018-05-15T18:53:32Z

Vanwdani at 00:56, 16 June 2017

2017-06-16T00:56:27Z

Vanwdani: /* Initial CM Frame 4-momentum components */

2017-06-16T00:55:55Z

Initial CM Frame 4-momentum components

Vanwdani: /* Initial CM Frame 4-momentum components */

2017-06-16T00:54:30Z

Initial CM Frame 4-momentum components

Vanwdani: /* Initial CM Frame 4-momentum components */

2017-06-16T00:51:38Z

Initial CM Frame 4-momentum components

Vanwdani at 22:31, 15 June 2017

2017-06-15T22:31:50Z

Vanwdani at 22:29, 15 June 2017

2017-06-15T22:29:02Z

Vanwdani: /* Initial CM Frame 4-momentum components */

2017-06-14T02:14:18Z

Initial CM Frame 4-momentum components

Vanwdani: Created page with "=Initial CM Frame 4-momentum components= 400 px '''Figure 2: Definition of variables in the Center of Mass Frame''' …"

2017-05-31T17:54:55Z

Created page with "=Initial CM Frame 4-momentum components= <center>400 px</center> <center>'''Figure 2: Definition of variables in the Center of Mass Frame'''</center> …"

New page

=Initial CM Frame 4-momentum components=

<center>[[File:400px-CMcopy.png |400 px]]</center>
<center>'''Figure 2: Definition of variables in the Center of Mass Frame'''</center>

Starting with the definition for the total relativistic energy:

<center><math>E^2\equiv p^2c^2+m^2c^4</math></center>

<center><math>\Longrightarrow {E^2}-p^2c^2=(mc^2)^2</math></center>
Since we can assume that the frame of reference is an inertial frame, it moves at a constant velocity, the mass should remain constant.

<center><math>\frac {d\vec p}{dt}=0\Rightarrow \frac{d(m\vec v)}{dt}=\frac{c\ dm}{dt}\Rightarrow \frac{dm}{dt}=0</math></center>

<center><math> \therefore m=const</math></center>

We can use 4-momenta vectors, i.e. <math>{\mathbf P}\equiv \left(\begin{matrix} E\\ p_x \\ p_y \\ p_z \end{matrix} \right)=\left(\begin{matrix} E\\ \vec p \end{matrix} \right)</math> ,with c=1, to describe the variables in the CM Frame.

Using the fact that the scalar product of a 4-momenta with itself,

<center><math>{\mathbf P_1}\cdot {\mathbf P^1}=P_{\mu}g_{\mu \nu}P^{\nu}=\left(\begin{matrix} E\\ p_x \\ p_y \\ p_z \end{matrix} \right)\cdot \left( \begin{matrix}1 & 0 & 0 & 0\\0 & -1 & 0 & 0\\0 & 0 & -1 & 0\\0 &0 & 0 &-1\end{matrix} \right)\cdot \left(\begin{matrix} E & p_x & p_y & p_z \end{matrix} \right)</math></center>

<center><math>{\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2=s</math></center>

is invariant.

Using this notation, the sum of two 4-momenta forms a 4-vector as well

<center><math>{\mathbf P_1}+ {\mathbf P_2}= \left( \begin{matrix}E_1+E_2\\\vec p_1 +\vec p_2 \end{matrix} \right)= {\mathbf P}</math></center>

The length of this four-vector is an invariant as well

<center><math>{\mathbf P^2}=({\mathbf P_1}+{\mathbf P_2})^2=(E_1+E_2)^2-(\vec p_1 +\vec p_2 )^2=(m_1+m_2)^2=s</math></center>

@@ Line 1: / Line 1: @@
-<center><math>\textbf{\underline{Navigation}}</math>
+<center><math>\underline{\textbf{Navigation}}</math>
 [[Initial_Lab_Frame_4-momentum_components|<math>\vartriangleleft </math>]]
 [[VanWasshenova_Thesis#Initial_4-momentum_Components|<math>\triangle </math>]]
@@ Line 61: / Line 60: @@
-<center><math>\textbf{\underline{Navigation}}</math>
+<center><math>\underline{\textbf{Navigation}}</math>
 [[Initial_Lab_Frame_4-momentum_components|<math>\vartriangleleft </math>]]

← Older revision		Revision as of 00:56, 16 June 2017
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	<center><math>{\mathbf P^2}=({\mathbf P_1}+{\mathbf P_2})^2=(E_1+E_2)^2-(\vec p_1 +\vec p_2 )^2=(m_1+m_2)^2=s</math></center>		<center><math>{\mathbf P^2}=({\mathbf P_1}+{\mathbf P_2})^2=(E_1+E_2)^2-(\vec p_1 +\vec p_2 )^2=(m_1+m_2)^2=s</math></center>
		+
		+
		+
		+	----
		+
		+
		+	<center><math>\textbf{\underline{Navigation}}</math>
		+
		+	[[Initial_Lab_Frame_4-momentum_components\|<math>\vartriangleleft </math>]]
		+	[[VanWasshenova_Thesis#Initial_4-momentum_Components\|<math>\triangle </math>]]
		+	[[Special_Case_of_Equal_Mass_Particles\|<math>\vartriangleright </math>]]
		+
		+	</center>

← Older revision		Revision as of 00:55, 16 June 2017
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−	<center><math>{\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2~~=\sqrt{s}~~</math></center>	+	<center><math>{\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2</math></center>

← Older revision		Revision as of 00:54, 16 June 2017
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−	<center><math>{\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2</math></center>	+	<center><math>{\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2=\sqrt{s}</math></center>

← Older revision		Revision as of 00:51, 16 June 2017
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−	<center><math>\frac {d\vec p}{dt}=0\Rightarrow \frac{d(m\vec v)}{dt}=\frac{c\ dm}{dt}\Rightarrow \frac{dm}{dt}=0</math></center>	+	<center><math>\frac {d\vec p}{dt}=0\Rightarrow \frac{d(m\vec v)}{dt}=\frac{v\ dm}{dt}\Rightarrow \frac{dm}{dt}=0</math></center>

← Older revision		Revision as of 22:29, 15 June 2017
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		+	<center><math>\textbf{\underline{Navigation}}</math>
		+
		+	[[Limits_based_on_Mandelstam_Variables\|<math>\vartriangleleft </math>]]
		+	[[VanWasshenova_Thesis#Initial_4-momentum_Components\|<math>\triangle </math>]]
		+	[[Special_Case_of_Equal_Mass_Particles\|<math>\vartriangleright </math>]]
		+
		+	</center>
		+
		+
	=Initial CM Frame 4-momentum components=		=Initial CM Frame 4-momentum components=

← Older revision		Revision as of 02:14, 14 June 2017
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−	<center><math>{\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2=s</math></center>	+	<center><math>{\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2</math></center>