New IAC Wiki - User contributions [en]

Results

2019-05-30T03:34:13Z

Vanwdani:

Unweighted Occupancy

[[File:UnweightedOccupancy_twoCylinders.png]]

Weighted Occupancy for each sector based on just initial S1 LUND file

[[File:S1occupancy80nA.png]]

[[File:S2occupancy80nA.png]]

[[File:S3occupancy80nA.png]]

[[File:S4occupancy80nA.png]]

[[File:S5occupancy80nA.png]]

[[File:S6occupancy80nA.png]]

Combining all sectors to create Region Occupancy representing LUND files for each sector
Is Region 1 a match? https://clasweb.jlab.org/wiki/index.php/File:DCocclH2CylO6_GEMC4.3.0CJ5.7.4_.png

[[File:RegionalOccupancy80nA.png]]

Vertex Points for just initial S1 LUND file

[[File:Vertex_twoCylinders.png]]

Results

2019-05-30T03:31:02Z

Vanwdani:

Unweighted Occupancy

[[File:UnweightedOccupancy_twoCylinders.png]]

Weighted Occupancy for each sector based on just initial S1 LUND file

[[File:S1occupancy80nA.png]]

[[File:S2occupancy80nA.png]]

[[File:S3occupancy80nA.png]]

[[File:S4occupancy80nA.png]]

[[File:S5occupancy80nA.png]]

[[File:S6occupancy80nA.png]]

Combining all sectors to create Region Occupancy representing LUND files for each sector

[[File:RegionalOccupancy80nA.png]]

Vertex Points for just initial S1 LUND file

[[File:Vertex_twoCylinders.png]]

File:RegionalOccupancy80nA.png

2019-05-30T03:30:36Z

Vanwdani:

Results

2019-05-30T03:14:09Z

Vanwdani:

Unweighted Occupancy

[[File:UnweightedOccupancy_twoCylinders.png]]

Weighted Occupancy for each sector based on just initial S1 LUND file

[[File:S1occupancy80nA.png]]

[[File:S2occupancy80nA.png]]

[[File:S3occupancy80nA.png]]

[[File:S4occupancy80nA.png]]

[[File:S5occupancy80nA.png]]

[[File:S6occupancy80nA.png]]

Combining all sectors to create Region Occupancy representing LUND files for each sector

Vertex Points for just initial S1 LUND file

[[File:Vertex_twoCylinders.png]]

Results

2019-05-30T03:10:20Z

Mlr Summ TF

2019-04-11T03:09:58Z

Vanwdani: /* Latest Stuff */

[[VanWasshenova_Thesis#Mlr_Summ_TF]]

=Moller Summary=

==Scattering Xsect==

As derived earlier, in the [[Differential_Cross-Section]] section

In the Ultra-relativistic limit as <math> E \approx p</math>

<center><math>\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{4E^{*2}\sin^4{\theta}}\left( \cos^4{\theta}+6\cos^2{\theta}+9\right)=\frac{\alpha ^2\left(3+\cos^2{\theta}\right)^2}{4E^{*2}\sin^4{\theta}} </math></center>

{|style="margin: 0 auto;"
| [[File:MolThetaCM_Theory.png |thumb | border | center |500 px |alt=Theory Frame Moller CM Frame |'''Figure 3a:''' A plot of the number of Moller scattering angle theta in the center of mass frame versus the theoretical differential cross section. The width of the bins is 0.001 degrees for the angles in the center of mass frame corresponding to angles of 5 to 40 degrees in the lab frame. A weight has been assigned for each value in theta which will give the theoretical differential cross section when applied.]]
| [[File:MolThetaLab_Theory.png |thumb | border | center |500 px |alt=Theory Lab Frame Moller Frame |'''Figure 3b:''' A plot of the number of Moller scattering angle theta in the lab frame versus the theoretical differential cross section. The width of the bins is 0.5 degrees for the angles in the lab frame. A weight has been assigned for each value in theta which will give the theoretical differential cross section when applied.]]
|}

Weight the E-vs-Theta plot with Xsect

[[File:MollerEThetaSigma_Full.png|frame|center|alt=Moller Electron Energy vs Angle Theta in Lab Frame|'''Figure 2:''' Using the theoretical differential cross section, the distribution of Moller electrons for a CM energy of approximately 53Mev can be distributed through the CM scattering angle Theta. Using Lorentz transformations, these distributions can be transformed to the lab frame. At around 60 degrees in the lab the Moller electron has an energy of close to 1 MeV. Such a low energy does not allow the Moller electron to leave the constrains of the target where they are created.]]

Moller electron radius -vs- Momentum (Full solenoid and relativistic)

== Baseline==
== Moller events using an lH2 target geometry No Raster==

===DC hits -vs- Solenoid ===

Starting the clas12 Moller simulation with a simple configuration, most components downstream with respect to the drift chambers are taken out of the gcard file. The remaining components, the Drift Chamber (DC), the Solenoid, and the Magnetic Components (CAD) are all that remain as shown below. The Torus field is held at zero Tesla, and the Solenoid field strength is increased from 0 to it's max of 5T.

[[File:ChangingRates_S1_PhiThetaHits_Full.png||200 px|frame|center|alt=Moller Electron Energy vs Angle Theta in Lab Frame|'''Figure 2:''' With the Torus strength at zero Telsa, the solenoid is varied to show the hits within the DC moving off the faces of R1. These hits consist of primary Moller electrons, as well as secondary electrons and photons created by the Moller electrons.]]

As the Solenoid field strength increases, the Moller electrons are forced to into a helical path of decreasing radius, effectively "rotating" off the DC face. The relative value of the field changes the direction field, which in turn causes slightly more hits to be deposited on one side versus the other. This effect is only noticeable in that the neighboring sectors to S1 were not simulated.

[[File:CompareOppositeFieldsFULL_S1_PhiThetaHits.png|200 px|frame|center|alt=Moller Electron Energy vs Angle Theta in Lab Frame|'''Figure 2:''' The effect of the sign of the solenoid magnetic field is apparent when neighboring sectors are not simulated.]]

However, as the field reaches maximum, there are still particles which are found at higher values of <math>\theta</math> that would have been expected to have remained after the effects of the Solenoid. To understand the causes of this phenomena, the hits on Region 1 can be broken into the most common types of particles that are detected; Primary Moller electrons and secondary electrons and photons caused by the Moller electron. Looking more closely at this particle makeup as the solenoid field is increased, we can see that the ratio of primary electrons to total hits decreases while the ratio of photons to total hits increases.

[[File:DC_MAG_rates_barPlot.png|600 px|frame|center|alt=Moller Electron Energy vs Angle Theta in Lab Frame|'''Figure 2:''' The rate of primary Moller electron traversing S1R1 decreases as the solenoid field strength is increased.]]

====Without Magnet Components====
To examine the possibility that scattering is the cause of the noise, the simulation is further simplified by removing the magnetic components stored in the CAD directory.

[[File:ComparingMagnetComponentsFULL_S1_PhiThetaHits.png|600 px|frame|center|alt=Moller Electron Energy vs Angle Theta in Lab Frame|'''Figure 2:''' With the Torus strength at zero Telsa, the solenoid is at full strength and the magnet components in the CAD directory are removed. Due to scattering effects, the rate for 50nA is reduced by a factor of almost 20.]]

To effectively determine the cause of the remaining particles in S1R1 the hit particles' vertex points can be recorded producing a tomography plot. A changing field is still useful in that the lack of number of hits at high solenoid field strength would make determining the physical cause difficult.

=====Photons Hits in R1=====

[[File:Screen_ShotMollerGammaPhiThetaBins.png]]

[[Merging background hits]]

=====Secondary Electron Hits in R1=====

=====Primary Moller Electron Hits in R1=====

[[File:ComponentStudy.png]]

==CLAS12 Conditions==

Summarize with picture photo rates -vs- change and location of photons

Summarize secondary moller electron rate location

===[[Determining and Verify Shield Limits]]===

===FTOn ShieldIn===

For the Moller electron,

[[File:FTOn_ShieldIn_S1R1Moller_withTomography.png]]

For the scattered electron, there are no secondary hits.

===FTOff ShieldIn===

For secondary hits from the Moller electron

[[File:FTOff_ShieldIn_S1R1Moller_withTomography.png]]

===FTOn ShieldOut===

[[File:FTOn_ShieldOut_S1R1Moller_withTomography.png]]

===FTOff ShieldOut===

[[File:FTOff_ShieldOut_S1R1Moller_withTomography.png]]
==New Cone==

===FTOn===
From beamline text file
<pre>
1589.27-238.8=1350.47
</pre>

<pre>
PbCylinder | root | Pb pipe on beamline | 0*mm 0.0*mm 1350.47*mm | 0 0 0 | 999966 | Cons | 34*mm 36*mm 111.2*mm 113.2*mm 441.3*mm 0.0*deg 360*deg | beamline_w | no | 1 | 1 | 1 | 1 | 1 | no | no | no
</pre>

'''At Standard Vertex Position:'''

High Cone Position:

<math>1350.47+441.3=1791.77\ mm</math>

<math>\theta_{high}=\arctan \left( \frac{113.2}{1791.77} \right)=3.61^{\circ}</math>

Low Cone Position:

<math>1350.47-441.3=909.17\ mm</math>

<math>\theta_{low}=\arctan \left( \frac{113.2}{909.17} \right)=2.39^{\circ}</math>

====[[standard_R1_36_38_R2_111_113]]====

'''At forward Vertex Position:''' ''(subtract 40mm from standard vertex distance)''

High Cone Position:

<math>1791.77-40=1751.77\ mm</math>

<math>\theta_{high}=\arctan \left( \frac{113.2}{1751.77} \right)=3.69^{\circ}</math>

Low Cone Position:

<math>909.17-40=869.17\ mm</math>

<math>\theta_{low}=\arctan \left( \frac{113.2}{869.17} \right)=2.5^{\circ}</math>

<math>\rightarrow \theta_{high}=5^{\circ} \rightarrow 1751.77 \arctan 5^{\circ} = 153.26\ mm </math>

<math>\rightarrow \theta_{low}=5^{\circ} \rightarrow 869.17 \arctan 5^{\circ} = 76.04\ mm </math>

====[[standard_R1_74_76_R2_151_153]]====

====Summary Tables====
{| class="wikitable"
|+ FTOn from 0,0,0 Vertex
! 50nA
! S1R1 2ndryMoller e- rate
! S1R1 2ndryMoller gamma rate
! S1R1 2ndryMoller particle rate
! Effective Shield Rate
|-
! R1_36_38_R2_36_38
| 111 Hz
| 10652 Hz
| 37 Hz
| ?
|-
! R1_36_38_R2_50_54
| 108 Hz
| 10668 Hz
| 31 Hz
| ?
|-
! R1_36_38_R2_70_72
| 109 Hz
| 11053 Hz
| 45 Hz
| ?
|-
! R1_36_38_R2_75_77
| Hz
| Hz
| Hz
| ?
|-
! R1_36_38_R2_80_82
| Hz
| Hz
| Hz
| ?
|-
! R1_36_38_R2_90_92
| Hz
| Hz
| Hz
| ?
|-
! R1_36_38_R2_95_97
| Hz
| Hz
| Hz
| ?
|-
! R1_36_38_R2_100_102
| Hz
| Hz
| Hz
| ?
|-
! R1_36_38_R2_111_113
| Hz
| Hz
| Hz
| ?
|-
! R1_36_38_R2_116_118
| Hz
| Hz
| Hz
| ?
|-
! R1_36_38_R2_121_123
| Hz
| Hz
| Hz
| ?
|-
! R1_74_76_R2_151_153
| Hz
| Hz
| Hz
| ?
|-
! R1_36_38_R2_500_503
| 150 Hz
| 12097 Hz
| 46 Hz
| ?
|}

{| class="wikitable"
|+ FTOn from (0,0,0) Vertex w/o FT
! 50nA
! S1R1 2ndryMoller e- rate
! S1R1 2ndryMoller gamma rate
! S1R1 2ndryMoller particle rate
! Effective Shield Rate
|-
! R1_0_524.0_R2_0_1034.47
| ?
| ?
| ?
| ?

|}

==Moller Electrons below 5 degrees==
[[File:ReallyFullMolThetaMomLabWeighted.png | 300 px]]

[[CLAS12_MollerDataTable_12-16-2018]]

== Moller events using an dual polarized target geometry with Raster==

===Photon Hits in R1 when Raster size has radius of 0.2 cm===

==Moller rate -vs- length of a single taerget==

===0.5 cm radius -vs- Z===

Target is a one 0.5 cm radius cylinder of length Z.

By how much does the moller rate change at full field ?

=Latest Stuff=

[[File:UnweightedOccupancy_need_to_fix.png]]

[[VanWasshenova_Thesis#Mlr_Summ_TF]]

= References=

[https://misportal.jlab.org/mis/physics/clas12/viewFile.cfm/2016-006.pdf?documentId=32 CLASNOTE 2016-06 Moller shield simulations: comparison of the GEMC-optimized layout and the engineering design]

File:UnweightedOccupancy need to fix.png

2019-04-11T03:09:41Z

Vanwdani:

Monte Carlo Binary Collision Approximation

2019-02-26T04:22:50Z

Vanwdani:

<center><math>\underline{\textbf{Navigation}}</math>

[[U-Channel|<math>\vartriangleleft </math>]]
[[VanWasshenova_Thesis#Moller_Scattering|<math>\triangle </math>]]
[[https://wiki.iac.isu.edu/index.php/Limit_of_Energy_in_Lab_Frame|<math>\vartriangleright </math>]]

</center>

=Limits based on Mandelstam Variables=

Since the Mandelstam variables are the scalar product of 4-momenta, which are invariants, they are invariants as well. The sum of these invariant variables must also be invariant as well. Find the sum of the 3 Mandelstam variables when the two particles have equal mass in the center of mass frame gives:

<center><math>s+t+u=(4(m^2+ p \ ^{*2}))+(-2 p \ ^{*2}(1-cos\ \theta))+(-2 p \ ^{*2}(1+cos\ \theta))</math></center>

<center><math>s+t+u \equiv 4m^2</math></center>

Since

<center><math>s \equiv 4(m^2+\vec p \ ^{*2})</math></center>

This implies

<center><math>s \ge 4m^2</math></center>

In turn, this implies

<center><math> t \le 0 \qquad u \le 0</math></center>

At the condition both t and u are equal to zero, we find

<center><math> t = 0 \qquad u = 0</math></center>

<center><math>-2 p \ ^{*2}(1-cos\ \theta) = 0 \qquad -2 p \ ^{*2}(1+cos\ \theta) = 0</math></center>

<center><math>(-2 p \ ^{*2}+2 p \ ^{*2}cos\ \theta) = 0 \qquad (-2 p \ ^{*2}-2 p \ ^{*2}cos\ \theta) = 0</math></center>

<center><math>2 p \ ^{*2}cos\ \theta = 2 p \ ^{*2} \qquad -2 p \ ^{*2}cos\ \theta = 2 p \ ^{*2}</math></center>

<center><math>\cos\ \theta = 1 \qquad \cos\ \theta = -1</math></center>

<center><math>\Rightarrow \theta_{t=0} = \arccos \ 1=0^{\circ} \qquad \theta_{u=0} = \arccos \ -1=180^{\circ}</math></center>

Holding u constant at zero we can find the minimum of t

<center><math>s+t_{max} \equiv 4m^2</math></center>

<center><math>\Rightarrow t_{max}=4m^2-s</math></center>

<center><math>t_{max}=4m^2-4m^2- 4p \ ^{*2}</math></center>

The maximum transfer of momentum would be

<center><math>t_{max}=-4p \ ^{*2}</math></center>

<center><math>-2 p \ ^{*2}(1-cos\ \theta_{t=max})=-4p \ ^{*2}</math></center>

<center><math>(1-cos\ \theta_{t=max})=2</math></center>

<center><math>-cos\ \theta_{t=max}=1</math></center>

<center><math> \theta_{t=max} \equiv \arccos -1</math></center>

The domain of the arccos function is from −1 to +1 inclusive and the range is from 0 to π radians inclusive (or from 0° to 180°). We find as expected for u=0 at <math>\theta=180^{\circ}</math>

<center><math>\theta_{t=max}=180^{\circ}</math></center>

However, from the definition of s being invariant between frames of reference

<center><math>s \equiv \overbrace{\left({\mathbf P_1^*}+ {\mathbf P_2^{*}}\right)^2=\left({\mathbf P_1^{'*}}+ {\mathbf P_2^{'*}}\right)^2}^{CM\ FRAME}=\overbrace{\left({\mathbf P_1}+ {\mathbf P_2}\right)^2 = \left({\mathbf P_1^{'}}+ {\mathbf P_2^{'}}\right)^2}^{LAB\ FRAME}</math></center>

In the center of mass frame of reference,

<center><math>E_1^*=E_2^*=E^* \quad and \quad \vec p \ _1^*=-\vec p \ _2^*= \vec p \ ^*</math></center>

<center><math>s \equiv 2m^2+2(E_1^{*2}+\vec p \ ^{*2} )</math></center>

Using the relativistic energy equation

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

<center><math>s \equiv 2m^2+2((m^2+\vec p \ ^{*2})+\vec p \ ^{*2})</math></center>

<center><math>s=4m^2+4 \vec p \ ^{*2})</math></center>

<center><math>\frac{s-4m^2}{4}= \vec p \ ^{*2}</math></center>

<center><math>t=-2 p \ ^{*2}(1-cos\ \theta)=\frac{-2(s-4m^2)}{4}(1-cos\ \theta)</math></center>

<center><math>\frac{-2t}{s-4m^2}=(1-cos\ \theta)</math></center>

<center><math>cos\ \theta=1-\frac{-2t}{s-4m^2}</math></center>

----

<center><math>\underline{\textbf{Navigation}}</math>

[[U-Channel|<math>\vartriangleleft </math>]]
[[VanWasshenova_Thesis#Moller_Scattering|<math>\triangle </math>]]
[[https://wiki.iac.isu.edu/index.php/Limit_of_Energy_in_Lab_Frame|<math>\vartriangleright </math>]]

</center>

Frame of Reference Transformation

2019-01-01T19:04:36Z

Vanwdani: /* Frame of Reference Transformation */

<center><math>\underline{\textbf{Navigation}}</math>

[[4-momenta|<math>\vartriangleleft </math>]]
[[VanWasshenova_Thesis#Moller_Scattering|<math>\triangle </math>]]
[[4-gradient|<math>\vartriangleright </math>]]

</center>

=Frame of Reference Transformation=
Using the Lorentz transformations and the index notation,

<center><math>
\begin{cases}
t'=\gamma (t-vz/c^2) \\

x'=x' \\

y'=y' \\

z'=\gamma (z-vt)
\end{cases}
</math></center>

<center><math>\begin{bmatrix}
x'^0 \\
x'^1 \\
x'^2\\
x'^3
\end{bmatrix}=
\begin{bmatrix}
\gamma (x^0-vx^3/c) \\
x^1 \\
x^2 \\
\gamma (x^3-vx^0)
\end{bmatrix}
=
\begin{bmatrix}
\gamma (x^0-\beta x^3) \\
x^1 \\
x^2 \\
\gamma (x^3-vx^0)
\end{bmatrix}</math></center>

<center>Where <math>\beta \equiv \frac{v}{c}</math></center>

This can be expressed in matrix form as

<center><math>\begin{bmatrix}
x'^0 \\
x'^1 \\
x'^2\\
x'^3
\end{bmatrix}=
\begin{bmatrix}
\gamma & 0 & 0 & -\gamma \beta \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
-\gamma \beta & 0 & 0 & \gamma
\end{bmatrix}
\cdot
\begin{bmatrix}
x^0 \\
x^1 \\
x^2 \\
x^3
\end{bmatrix}</math></center>

Letting the indices run from 0 to 3, we can write

<center><math>\mathbf x'^{\mu}=\sum_{\nu=0}^3 (\Lambda_{\nu}^{\mu}) \mathbf x^{\nu}</math></center>

<center>Where <math>\Lambda</math> is the Lorentz transformation matrix for motion in the z direction.</center>

Using the Einstein convention, this can be written as

<center><math>\mathbf x'^{\mu}= \Lambda_{\nu}^{\mu} \mathbf x^{\nu}</math></center>

If we take the 4-vector quantities to be on an infinitesimally small scale, then there exists a linear relationship between the transformation. Following the rules of partial differentiation,

<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>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>

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>

Again, using a summation over the indicies

<center><math>d\mathbf x^{'\mu}=\sum_{\nu=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^{\nu}}d\mathbf x^{\nu}</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,

<center><math>\Lambda_{\nu}^{\mu} \eta_{\nu}^{\mu} \Lambda_{\mu}^{\nu}=\eta_{\nu}^{\mu}</math></center>

<center><math>
\begin{bmatrix}
\gamma & 0 & 0 & -\gamma \beta \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
-\gamma \beta & 0 & 0 & \gamma
\end{bmatrix}\cdot
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 &-1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{bmatrix}\cdot
\begin{bmatrix}
\gamma & 0 & 0 & -\gamma \beta \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
-\gamma \beta & 0 & 0 & \gamma
\end{bmatrix}^T=
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 &-1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{bmatrix}</math></center>

<center><math>
\begin{bmatrix}
\gamma^2-\beta^2 \gamma^2 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -\gamma^2+\beta^2 \gamma^2
\end{bmatrix}=
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 &-1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{bmatrix}</math></center>

<center><math>
\begin{bmatrix}
\gamma^2(1-\beta^2) & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -\gamma^2(1-\beta^2)
\end{bmatrix}=
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 &-1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{bmatrix}</math></center>

<center>Where <math>\gamma \equiv \frac{1}{\sqrt{1-\beta^2}}</math></center>

<center><math>
\begin{bmatrix}
\frac{\gamma^2}{\gamma^2} & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -\frac{\gamma^2}{\gamma^2}
\end{bmatrix}=
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 &-1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{bmatrix}</math></center>

<center><math>
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{bmatrix}=
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 &-1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{bmatrix}</math></center>

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