p = m v;

vs. rest energy

momentum: μ Sinh[θ] μ β

energy: μ Cosh[θ]

energy: μ Cosh[θ]

Series[Cosh[β],{β,0,4}]

In[]:=

1+++

2

β

2

4

β

24

5

O[β]

Out[]=

μ = m c

If KE is μ v^2/c^2

If energy is m c^2 ( 1 - β^2 + ...)

Series[mc^2/Sqrt[1-v^2/c^2],{v,0,6}]

In[]:=

2

c

m

2

v

2

3m

4

v

8

2

c

5m

6

v

16

4

c

7

O[v]

Out[]=

##

E^2 - p^2

For a speed 0 edge, E(β) = μ c Cosh[β] (

Factor of c

Between two elementary time hypersurfaces, difference in the spatial position for the edge is c Δt

We are counting causal edges per unit area defined by the causal graph

Measuring a flux through timelike surfaces we have to account for the direction of the flow;

for the spacelike surface we don’t

for the spacelike surface we don’t

Out[]=

vertical distance in every square is Δt; horizontal distance is c Δt

1 elementary light cone filled with stuff has spatial extent c Δt Cosh[β] / Δt

Δt Sinh[β] / Δt

Δt Sinh[β] / Δt

Imagine the vertical flux is μ ( per elementary light cone ) [ number of causal edges inside a single elementary light cone ]

Series[Cosh[β],{β,0,2}]

In[]:=

1++O()

2

β

2

3

β

Out[]=

Series[Sinh[β],{β,0,4}]

In[]:=

β++

3

β

6

5

O[β]

Out[]=

E = c μ ( 1 + 1/2 v^2/c^2 + ...)

p = μ v/c

μ = m c

(mCosh[b])^2-c^2(cm)^2Sinh[b]^2//FullSimplify

In[]:=

2

m

2

Cosh[b]

4

c

2

Sinh[b]

Out[]=

Sinh[ArcTanh[b]]//FullSimplify

In[]:=

b

1-

2

b

Out[]=