Mathematical Principles of Theoretical Physics

2.2. LORENTZ INVARIANCE 53

Second, the 4-D acceleration is defined by

aμ=

duμ

ds

= (a^0 ,a^1 ,a^2 ,a^3 ),

a^0 =

1

c

√

1 −v^2 /c^2

d

dt

(

1

√

1 −v^2 /c^2

)

(2.2.27) ,

ak=

1

c^2

√

1 −v^2 /c^2

d

dt

(

vk

√

1 −v^2 /c^2

)

for 1≤k≤ 3.

Third, the 4-D energy-momentum vector is

(2.2.28)

Eμ= (E,cP^1 ,cP^2 ,cP^3 ),

E=

mc^2

√

1 −v^2 /c^2

,

Pk=

mvk

√

1 −v^2 /c^2

, vk=

dxk

dt

for 1≤k≤ 3.

Fourth, by (2.2.28), we derive the most important formula in relativistic, called the Ein-
stein energy-momentum relation:

(2.2.29) E^2 =c^2 P^2 +m^2 c^4.

Remark 2.21.In 1905, Albert Einstein conjectured that a static object with massmhave
energyE, and satisfy the relation, called the Einstein formula:

(2.2.30) E=mc^2.

Thus, in a static coordinate system the energy-momentum is in the form

(2.2.31) (E,cP) = (mc^2 , 0 )

Then, it follows from (2.2.31) that the energy-momentum(E,cP)of a moving object with
velocityvis taken in the form of (2.2.28), which implies that the relation (2.2.29) holds true.
Hence, the energy-momentum relation (2.2.29) is based on postulating (2.2.30), which was
verified by numerous experiments.

Fifth, in classical mechanics, the Newtonian second law takes the form

d

dt

P=F, P=mvis the momentum.

In the relativistic mechanics, the 4-D force is

Fμ=

(

dE

ds

,c

dP

ds

)

,

and the relativistic force isF= (F^1 ,F^2 ,F^3 ):

Fk=

1

√

1 −v^2 /c^2

d

dt

Pk for 1≤k≤ 3 ,