Relativistic Dynamics (II)
and Doppler Shift
Total Energy and Rest Energy
• Kinetic Energy:
K = mc2(γ-1)
• Reordering the expression:
γmc2 = K + mc2
Total Relativistic Energy E
Rest energy E0
Note: as mentioned before, m is defined as the rest mass. If we were to use m0
as the rest mass and the relativistic mass to be m=γm0, the formula would then
read:
E=mc2 = K + m0c2
Energy and Momentum
• Starting from: p=γmu p2=(γmu)2
• Multiplying by c2 (left) and c4/c2 (right):
(pc)2 = (γmc2)2.(u/c)2
• With (u/c)2=1-1/γ2
(pc)2 = (γmc2)2 - m2c4
• E = γmc2 and E0=mc2, we find:
E2 = p2c2 + m2c4 = p2c2 + E02
2
E=mc
• Equivalence Mass-Energy
• An object with no velocity has energy stored
through his mass
• The conservation of Mass (classical) has to be
dropped in favor of an extension of the concept
of the conservation of Energy
• Mass Energy: Nuclear Reactions,…
Energy Mass: Pair production,…
Massless particle ?
• Example: the photon
• Energy: E2 = p2c2 + m2c4 = p2c2 E=pc
A massless particle has an energy originating
only from its momentum.
• One can show that massless particles travel at
the speed of light.
Since the photon is the carrier of the electromagnetic radiation (e.g. light), it
is indeed very reassuring that light travels at the speed of light !!!
Relativistic Doppler Shift (I)
K
v
Light
K’
λK
• Source moving away from the receiver: λK = T0(c+v)
• Frequency of light: ν = c/λ (in all reference frames)
and ν0=1/T0
νK = ν0 ( c / (c+v) )
• Time dilation: T’K’ = γTK ν’K’ = νK / γ
ν’K’ = ν0 . ( c / γ(c+v) )
• Which simplifies into:
ν
ν0
Source and Observer
moving away
Note relative to today’s lecture
• Writing: λ = (c+v)Ts does NOT violate the
special theory of relativity. The wavelength λ
is the distance between two crests. The two
crests are still traveling at c, it is just their
point of origin that changes.
Relativistic Doppler Shift (II)
K
Light
v
K’
λK
• Source approaching the receiver: λK = T0(c-v)
• Frequency of light: ν = c/λ (in all reference frames)
and ν0=1/T0
νK = ν0 ( c / (c-v) )
• Time dilation: T’K’ = γTK ν’K’ = νK / γ
ν’K’ = ν0 . ( c / γ(c-v) )
• Which simplifies into:
ν
ν0
Source and Observer
approaching
Relativistic Doppler Shift (III)
• Assuming β = v/c including sign:
– β = v/c (source and observer approaching)
– β = -v/c (source and observer moving away)
ν
ν0s
More experimental proofs
(with particles & particle accelerators)…
Relativistic Momentum
VS
Classical Momentum
Relativistic Energy: