relativity on rotated graph paper: calculations with causal … · 2018-01-12 · relativity on...
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Relativity on Rotated Graph Paper:
Calculations with Causal Diamonds
Rob Salgado ([email protected])
Dept of Physics, U Wisconsin La Crosse
“Relativity on rotated graph paper”
Am.J.Phys. 84, 344 (2016)
https://doi.org/10.1119/1.4943251
Relativity on Rotated Graph Paper an ordinary Minkowski spacetime diagram
emphasizing light-signals allowing “ticks of a light-clock” to be visualized
physically motivated:traced out by the light-signals in a ticking light-clock
method of calculation:
count boxes (“clock diamonds”) and do simple algebra
the visualization encodes many relativistic effectsand lends itself to numerous physical interpretations
first developed for use in algebra-based introductory courses
new methods more appropriate for more advanced students
Can you see the “4 ticks” on a spacetime diagram?
“Ticks” (a.k.a. “clock diamonds”)
are constructed using the
light-signals in a longitudinal light-clock
2 2
4
5 3
=
−
5
3
formulahyperbolic graph paper
with distinguished center
two-observer graph paper,
with distinguished β2=(3/5)rotated graph paper
??4
Inertial observer Alice at rest and
inertial observer Bob traveling with 3
5β =
Light-Clock Diamonds – as units of displacement
timelike
“ticks”
spacelike
“sticks”
Alice’s Clock Diamonds
worldlines of Alice’s
light-clock mirrors
Alice’s
worldline
Bob’s Clock Diamonds?
• DIAGONAL along Bob’s
worldline
• SAME AREA as Alice’s
Clock Diamond
lightlike
“links”
timelike
“ticks”
spacelike
“sticks”
lightlike
“links”
Alice’s Clock Diamonds
worldlines of Alice’s
light-clock mirrors
Alice’s
worldline
Bob’s Clock Diamonds!!!
• DIAGONAL along Bob’s
worldline
• SAME AREA as Alice’s
Clock Diamond
timelike
“ticks”
spacelike
“sticks”
lightlike
“links”
Light-Clock Diamonds – as units of displacement
• Intersection of (the future light cone of event O) and (the past light cone of event Q).
“events that can be influenced by O and can then influence Q”
• area of the diamond (in units of clock diamonds)= squared-interval
• aspect ratio of the diamond= square of the Doppler Factor(encodes velocity )
Doppler 1
1k
ββ
+=−
2
2
1
1
k
kβ −=
+
2 (width )(height )s u v=
2 (width )
(height )
uk
v=
( / )V cβ =
Causal Diamonds
O
Q
O
Q
β (4) -1 3= =
(4)+1 5
s2 2= (8)(2) = 16 = ( )42 2(8)
= = 4 = (2)(2)
k
Causal Diamonds with clock-diamond components
2
2
1
1
k
kβ −=
+
2(width )(height )s u v=
2 (width )
(height )
uk
v=
β (1) -1= = 0
(1) +1
s2 2= (4)(4) = 16 = ( )42 2(4)
= =1= (1)(4)
k
Alice says:
Bob says:
The
“4”!
The Clock Effect/Twin Paradox
2
2
( ) 100
( )
OZ
==10
2
2
( ) 16
( )
OQ == 4
2
2
( ) 16
( )
QZ == 4
10
4
4
O
Q
Z
O
Q
Z
Collision (in Energy-Momentum Space)
Moore (Six Ideas that Shaped Physics) R10S.315
17
13
8
5
12
8
6
10
2 fP
1 fP
2iP
1iP
1 1 1
15 38
17 5i fm β β= = = −
2 2 2
512
13i fm β β= = − = ?
• verify m2f = 12
• compute β2f
side note:
there are 4 frames
of reference on this
spacetime diagram
1 1 1
15 38
17 5i fm β β= = = −
2 2 2
512
13i fm β β= = − = ?
Collision (in Energy-Momentum Space)
Moore (Six Ideas that Shaped Physics) R10S.315
17
13
8
5
12
8
6
10
• verify m2f = 12
• compute β2f
(9) -1 8= =
(9) +1 10β
s2 2= (36)(4) = 144 = ( )12
k2 (36)
= = 9(4)
20
16
Relativity on Rotated Graph Paper
“Relativity on rotated graph paper”
Am.J.Phys. 84, 344 (2016)
https://doi.org/10.1119/1.4943251
get folks to use the method
chapter for a Modern Physics text?
uniformly-accelerated observers
other mathematical properties
physicsforums.com/insights/relativity-rotated-graph-paper/
geogebra.org/robphy