SRACC_01

(%i1)

kill ( all ) ;

done

Figure 1:

https://en.wikipedia.org/wiki/Acceleration_(special_relativity)#Curved_world_lines

Bob's t(T) coordinate time (T=Alice's time)

(%i1)

t ( T ) : = ( c / g ) · log ( sqrt ( 1 + ( g · T ( t ) / c ) ^ 2 ) + ( g · T ( t ) / c ) ) ;

t (T) := \frac{c}{g} \cdot \log (\sqrt{1 + {(\frac{g \cdot T (t)}{c})}^{2}} + \frac{g \cdot T (t)}{c})

Bob's x(T) coordinate position (T=Alice's time)

(%i2)

x ( T ) : = ( c ^ 2 / g ) · ( sqrt ( 1 + ( g · T ( t ) / c ) ^ 2 ) − 1 ) ;

x (T) := \frac{c^{2}}{g} \cdot (\sqrt{1 + {(\frac{g \cdot T (t)}{c})}^{2}} - 1)

(%i3)

dxT : diff ( x ( T ) , T ( t ) , 1 ) ;

\frac{g \cdot T (t)}{\sqrt{\frac{g^{2} \cdot {T (t)}^{2}}{c^{2}} + 1}}

(%i4)

dtT : diff ( t ( T ) , T ( t ) , 1 ) ;

\frac{c \cdot (\frac{g^{2} \cdot T (t)}{c^{2} \cdot \sqrt{\frac{g^{2} \cdot {T (t)}^{2}}{c^{2}} + 1}} + \frac{g}{c})}{g \cdot (\sqrt{\frac{g^{2} \cdot {T (t)}^{2}}{c^{2}} + 1} + \frac{g \cdot T (t)}{c})}

Bob's coordinate velocity

(%i5)

ratsimp ( dxT / dtT ) ;

g \cdot T (t)

Is a Geodesic!

(%i6)

v ( t ) : = g · T ( t ) ;

v (t) := g \cdot T (t)

g_00 metric term

(%i7)

g_00 ( x ) : = ( 1 + g · x ( t ) / c ^ 2 ) ^ 2 ;

g_00 (x) := {(1 + \frac{g \cdot x (t)}{c^{2}})}^{2}

For changing the curve "time" parameter. Two possibilities?

(%i8)

T1 ( t ) : = ( c / g ) · sinh ( g · t / c ) ;

T1 (t) := \frac{c}{g} \cdot \sinh (\frac{g \cdot t}{c})

(%i9)

T2 ( t ) : = ( c / g ) · tanh ( g · t / c ) ;

T2 (t) := \frac{c}{g} \cdot \tanh (\frac{g \cdot t}{c})

Red pill or Blue pill? Chose one

(%i10)

T ( t ) : = T2 ( t ) ;

T (t) := T2 (t)

(%i11)

Γ : integrate ( sqrt ( g_00 ( x ( t ) ) − v ( t ) ^ 2 / c ^ 2 ) , t ) ;

t

what???

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