# 2.9 Lightlike compactification, b

A First Course in String Theory

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Consider coordinates $\displaystyle{(ct', x')}$ related to $\displaystyle{(ct, x)}$ by a boost with velocity parameter $\displaystyle{\beta}$. Express the identifications in terms of the primed coordinates.

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\displaystyle{ \begin{aligned} \begin{bmatrix} c t' \\ x' \end{bmatrix} &= \begin{bmatrix} \gamma & -\beta \gamma \\ -\beta \gamma & \gamma \\ \end{bmatrix} \begin{bmatrix} c\,t \\ x \end{bmatrix} \\ \end{aligned}}

\displaystyle{ \begin{aligned} \begin{bmatrix} ct \\ x \\ \end{bmatrix} &\sim \begin{bmatrix} ct - 2 \pi R \\ x + 2 \pi R \\ \end{bmatrix} \end{aligned}}

\displaystyle{ \begin{aligned} \begin{bmatrix} c t' \\ x' \end{bmatrix} &\sim \begin{bmatrix} c t' \\ x' \end{bmatrix} + \begin{bmatrix} - \gamma - \beta \gamma \\ \gamma + \beta \gamma \\ \end{bmatrix} 2 \pi R \\ \end{aligned}}

— Me@2021-03-30 08:38:16 PM

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# Conscious time, 2

If no one has any kind of date, records, memories, or evidence about the past, retro-diction MAY be the same as prediction. But in such a case, it is by definition not our “past” any more.

— Me@2013-08-08 3:11 PM

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If no one has any kind of date, records, memories, or evidence about the past, then consciousness ceases to exist.

We, as conscious beings, cannot exist anymore.

— Me@2021-03-30 4:08 PM

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# Serialize, 2

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~ 安排

~ stably list

~ serialize

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An in-series path gives no choice.

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— Me@2011.08.13

— Me@2021-03-29

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# 光速飛行

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— Me@2021-03-24 07:36:49 AM

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# Ex 1.20 Sliding pendulum

Structure and Interpretation of Classical Mechanics

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Consider a pendulum of length $l$ attached to a support that is free to move horizontally, as shown in figure 1.4. Let the mass of the support be $m_1$ and the mass of the pendulum be $m_2$. Formulate a Lagrangian and derive Lagrange’s equations for this system.

~~~

[guess]


(define ((F->C F) local)
(->local (time local)
(F local)
(+ (((partial 0) F) local)
(* (((partial 1) F) local)
(velocity local)))))

;

(define ((q->r l y1) local)
(let ((q (coordinate local)))
(let ((x1 (ref q 0))
(theta (ref q 1)))
(let ((x2 (+ x1 (* l (sin theta))))
(y2 (- y1 (* l (cos theta)))))
(up x1 y1 x2 y2)))))

(show-expression
((q->r 'l 'y_1)
(up 't
(up 'x_1 'theta)

;

(define (KE m vx vy)
(* 1/2 m (+ (square vx) (square vy))))

(define ((T-rect m1 m2) local)
(let ((q (coordinate local))
(v (velocity local)))
(let ((x1dot (ref v 0))
(y1dot (ref v 1))
(x2dot (ref v 2))
(y2dot (ref v 3)))
(+ (KE m1 x1dot y1dot)
(KE m2 x2dot y2dot)))))

(show-expression
((T-rect 'm_1 'm_2)
(up 't
(up 'x_1 'y_1 'x_2 'y_2)
(up 'xdot_1 'ydot_1 'xdot_2 'ydot_2))))

;

(define ((U-rect g m1 m2) local)
(let* ((q (coordinate local))
(y1 (ref q 1))
(y2 (ref q 3)))
(* g (+ (* m1 y1)
(* m2 y2)))))

(show-expression
((U-rect 'g 'm_1 'm_2)
(up 't
(up 'x_1 'y_1 'x_2 'y_2)
(up 'xdot_1 'ydot_1 'xdot_2 'ydot_2))))

;

(define (L-rect g m1 m2)
(- (T-rect m1 m2) (U-rect g m1 m2)))

(show-expression
((L-rect 'g 'm_1 'm_2)
(up 't
(up 'x_1 'y_1 'x_2 'y_2)
(up 'xdot_1 'ydot_1 'xdot_2 'ydot_2))))

;

(define (L-l l y1 g m_1 m_2)
(compose
(L-rect g m_1 m_2) (F->C (q->r l y1))))

(show-expression
((L-l 'l 'y_1 'g 'm_1 'm_2)
(->local 't
(up 'x_1 'theta)

;

(show-expression
(((Lagrange-equations
(L-l 'l 'y_1 'g 'm_1 'm_2))
(up
(literal-function 'x_1)
(literal-function 'theta)))
't))

;



[guess]

— Me@2021-03-26 08:22:01 PM

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# Meta-time 7

Two dimensional time, 6

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y is meta x

~ y is the next dimension of x

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y is meta-time

~ y is the second dimension of time

— Me@2017-07-10 06:12:08 PM

— Me@2021-03-26 06:00:45 PM

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# Unfulfilled potential

What are you most scared of?

The Circle 2017 film

— Me@2017-02-02 07:22:47 AM

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2021.03.26 Friday ACHK

# 姻緣伏線 1.1

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1. 同一個人會隨時間轉變；

2. 同一個人會有超過一個自我，而各個自我，可以有不同性格；

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「不刻意去找」，不代表「刻意不去找」。

— Me@2021-03-24 07:36:49 AM

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# 1984, 2.2

— Photomyne colorization

<= 1984

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# Lightlike compatification

A First Course in String Theory

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2.9 Lightlike compatification

(a) Rewrite this identification using light-cone coordinates.

\begin{aligned} \begin{bmatrix} x \\ ct \end{bmatrix} &\sim \begin{bmatrix} x \\ ct \end{bmatrix} + 2 \pi \begin{bmatrix} R \\ -R \end{bmatrix} \end{aligned}

~~~

\begin{aligned} \begin{bmatrix} x^+ \\ x^- \end{bmatrix} &= \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \\ \end{bmatrix} \begin{bmatrix} x^0 \\ x^1 \end{bmatrix} \\ \end{aligned}

\begin{aligned} \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \\ \end{bmatrix} \begin{bmatrix} x^0 \\ x^1 \\ \end{bmatrix} &\sim \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \\ \end{bmatrix} \begin{bmatrix} x^0 \\ x^1 \\ \end{bmatrix} + \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \\ \end{bmatrix} \begin{bmatrix} - 2 \pi R \\ 2 \pi R \\ \end{bmatrix} \\ \end{aligned}

\begin{aligned} \begin{bmatrix} x^+ \\ x^- \\ \end{bmatrix} &\sim \begin{bmatrix} x^+ \\ x^- \\ \end{bmatrix} + \frac{1}{\sqrt{2}} \begin{bmatrix} 0 \\ - 4 \pi R \\ \end{bmatrix} \\ \end{aligned}

— Me@2021-03-22 06:06:10 PM

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# Particle indistinguishability is the major source of quantum effects, 1.05

If there is no particle indistinguishability, every particle has a well-defined identity, then every particle has a well-defined trajectory.

Then even if no detector is installed, there is a well-defined difference between go-left and go-right in the double-slit experiment.

Then there will be no indistinguishability of cases, aka quantum superposition states.

— Me@2021-02-04 7:04 AM

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# 冰橋

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— Me@2021-03-20 10:45:31 PM

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# Ex 1.19 A two-bar linkage, 2

Structure and Interpretation of Classical Mechanics

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… Formulate a Lagrangian that describes the system and find the Lagrange equations of motion …

~~~

[guess]


(define ((U-rect g m1 m2 m3) local)
(let* ((q (coordinate local))
(y1 (ref q 1))
(y2 (ref q 3))
(y3 (ref q 5)))
(* g (+ (* m1 y1)
(* m2 y2)
(* m3 y3)))))

;

(define (L-rect g m1 m2 m3)
(- (T-rect m1 m2 m3) (U-rect g m1 m2 m3)))

(show-expression
((L-rect 'g 'm_1 'm_2 'm_3)
(up 't
(up 'x_1 'y_1 'x_2 'y_2 'x_3 'y_3)
(up 'xdot_1 'ydot_1 'xdot_2 'ydot_2 'xdot_3 'ydot_3))))

;

(define (L-l l1 l2 g m_1 m_2 m_3)
(compose
(L-rect g m_1 m_2 m_3) (F->C (q->r l1 l2))))

(show-expression
((L-l 'l_1 'l_2 'g 'm_1 'm_2 'm_3)
(->local 't
(up 'x_2 'y_2 'theta 'phi)

;

(show-expression
(((Lagrange-equations
(L-l 'l_1 'l_2 'g 'm_1 'm_2 'm_3))
(up
(literal-function 'x_2)
(literal-function 'y_2)
(literal-function 'theta)
(literal-function 'phi)))
't))

;



[guess]

— Me@2021-03-20 05:01:21 PM

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# Event Realism 6.2

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Lee’s event realism ~ past realism

should be transcended by now-realism, in which the past is part of the present.

— Me@2013-07-20 03:38 PM

— Me@2021-03-19 11:02 PM

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# 魔法學堂

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（問：但其中的「魔法」，十分奇幻。那真的可以稱為「科學」嗎？）

— Me@2011.08.12

— Me@2021-03-19

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# 山盟海誓 3

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— Me@2021-03-19 03:34:58 PM

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