Relativity on Rotated Graph Paper:

Calculations with Causal Diamonds

Rob Salgado (rsalgado@uwlax.edu)

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