Space-Time symmetries and
conservations law
Properties of space
1.
2.
3.
4.
Three dimensionality
Homogeneity
Flatness
Isotropy
Properties of Time
1. One-dimensionality
2. Homogeneity
3. Isotropy
Homogeneity of space and Newton
third law of motion
y
Y’
s
s’
1
a
o
z
o’
z’
x1
2
x2
x’
x
Consider two interacting particles 1 and 2 lying along x-axis of
frame s
Let x1 and x2 are the distance of the particles from o. the
potential energy
of interaction U between the particles in frame s is given by
U=U(x1,x2)
Let s’ be another frame of reference displaced with respect to s
by a distance a along x-axis then oo’= a
The principal of homogeneity demands that
U(x1,x2) = U(x’1,x’2)
Applying Taylor’s theorem, we get
F12 = -F21
This is nothing but Newton’s third law of motion.
Homogeneity of space and law of
conservation of linear momentum
• Consider tow interacting particles 1 and 2 of masses
m1 and m2 then forces between the particles must
satisfy Newton’s third law as required by
homogeneity of space .
• F12 = -F21
• Newton’s 2nd law of motion
• m1dv1/dt = F12 --------- (1)
• m2dv2/dt = F21 ---------(2)
• Adding (1) and (2) and simplifying we get ,
• m1v1 + m2v2 = constant
Isotropy of space and angular
momentum conservation
y’
y
x’
x
dΩ
z
z’
Let U = U(r1,r2) be the P.E. of interaction in frame s
U = U(r1+dr1,r2+dr2) be the potential energy in frame
s’. Then using property of isotropy of space
U(r1,r2) = U(r1+dr1,r2+dr2)
Applying Taylor’s theorem, we get
dL/dt = 0
L = constant
This is just the law of conservation momentum and
is a consequence of space.
Homogeneity of time and energy
conservation
• Consider tow interacting particles 1 and 2 lying
along x-axis of frame s. The P.E. between the two
particles is given by
• U = U(x1,x2)
• If x1 and x2 change w.r.t. time then U will also
change with time but U is an indirect function of
time. The homogeneity of time demands that
result of an experiment should not change with
time
•
𝜕𝑈
𝜕𝑡
=0
Let us assume U = U(x1,x2,t)
dU =
𝜕𝑈
𝜕𝑈
𝑑𝑥1 +
𝑑𝑥2
𝜕𝑥1
𝜕𝑥2
+
𝜕𝑈
dt
𝜕𝑡
Using newton’s 2nd law , we get
2
2
d/dt(1/2m1v 1 + 1/2m2v2 + U)=0
2
2
Or 1/2m1v1 + 1/2m2v2 + U = constant
Which is nothing but law of conservation of total
Energy.