28
3 The Motion of Particles
This is a good place to mention an arbitrary sign choice we have made in the last
three chapters. The Lorentz metric as defined in (2.3a) contains a single plus sign
and three minus signs. With that choice for the metric the relation between arc length
and proper time for a particle trajectory is c
2 dτ
2
= ds
2 so the proper time interval
is dτ = ds/c; it is positive for a moving particle. Some authors instead choose the
opposite sign for the Lorentz metric since it contains only a single minus sign. With
this choice the relation between proper time and arc length becomes the somewhat
awkward dτ =
√
−ds 2 /c.
One drawback of our sign choice is that the Einstein equations that we will study
in Part III contain a minus sign between the left side describing geometry and right
side describing energy and momentum.
Another way to view the choice of signs is that we might want to think of time
as more “important” than space, or space as more “important” than time; the choice
is obviously one of taste and notational convenience. It is also relevant that during
much of the twentieth century the choice we have made was the dominant one on the
west coast of the US and the other was the dominate one on the east coast! At present
both choices are common; the text of Misner Thorne and Wheeler contains a large
table of sign conventions for the metric and other tensors used in relativity theory
until 1973 (Misner 1973). In particle physics the choice we have made is prevalent
(Bjorken 1963; Griffiths 1987). See Exercise 3.7.
Exercises
3.1. Consider a particle of mass M that decays at rest and turns into two particles
of equal mass m, with 2m < M. What is the energy of each decay particle?
What is the momentum of each? What is the kinetic energy of each, that is the
energy minus the rest energy mc
2 ?
3.2. For a particle of zero rest mass such as a photon the relations for energy and
momentum in (3.10a) are not meaningful, but the relation between energy and
momentum in (3.14) remains reasonable. Using elementary quantum theory
and the Planck and de Broglie relations for energy and momentum show that
(3.14) is indeed correct for a photon, that is E
2
= =
p
2 c
4 .
3.3. Take the case of constant proper acceleration, a = constant, and solve for the
trajectory from (3.25) and (3.29).
3.4. Show that the trajectory is a hyperbola in the variables ct and x, and the
asymptotic velocity is c.
3.5. Draw a nice graph of the hyperbolic motion from Exercise 3.4, and from it
show that a photon sent from x = 0 after time t = c/a will never catch the
rocket.
3.6. There have been many papers and books written on the twin paradox, wherein a
twin who travels at high velocity to a nearby star system and returns is younger
than his twin who remains on earth. Think about this and convince yourself that
there is no contradiction. The problem is discussed in many reputable books,
for example the readable Feynman lectures (Feynman 1963; Schutz 2009).
However it is also a favorite topic in books and articles by people with limited
or incorrect understanding of relativity, so beware.
3 The Motion of Particles
This is a good place to mention an arbitrary sign choice we have made in the last
three chapters. The Lorentz metric as defined in (2.3a) contains a single plus sign
and three minus signs. With that choice for the metric the relation between arc length
and proper time for a particle trajectory is c
2 dτ
2
= ds
2 so the proper time interval
is dτ = ds/c; it is positive for a moving particle. Some authors instead choose the
opposite sign for the Lorentz metric since it contains only a single minus sign. With
this choice the relation between proper time and arc length becomes the somewhat
awkward dτ =
√
−ds 2 /c.
One drawback of our sign choice is that the Einstein equations that we will study
in Part III contain a minus sign between the left side describing geometry and right
side describing energy and momentum.
Another way to view the choice of signs is that we might want to think of time
as more “important” than space, or space as more “important” than time; the choice
is obviously one of taste and notational convenience. It is also relevant that during
much of the twentieth century the choice we have made was the dominant one on the
west coast of the US and the other was the dominate one on the east coast! At present
both choices are common; the text of Misner Thorne and Wheeler contains a large
table of sign conventions for the metric and other tensors used in relativity theory
until 1973 (Misner 1973). In particle physics the choice we have made is prevalent
(Bjorken 1963; Griffiths 1987). See Exercise 3.7.
Exercises
3.1. Consider a particle of mass M that decays at rest and turns into two particles
of equal mass m, with 2m < M. What is the energy of each decay particle?
What is the momentum of each? What is the kinetic energy of each, that is the
energy minus the rest energy mc
2 ?
3.2. For a particle of zero rest mass such as a photon the relations for energy and
momentum in (3.10a) are not meaningful, but the relation between energy and
momentum in (3.14) remains reasonable. Using elementary quantum theory
and the Planck and de Broglie relations for energy and momentum show that
(3.14) is indeed correct for a photon, that is E
2
= =
p
2 c
4 .
3.3. Take the case of constant proper acceleration, a = constant, and solve for the
trajectory from (3.25) and (3.29).
3.4. Show that the trajectory is a hyperbola in the variables ct and x, and the
asymptotic velocity is c.
3.5. Draw a nice graph of the hyperbolic motion from Exercise 3.4, and from it
show that a photon sent from x = 0 after time t = c/a will never catch the
rocket.
3.6. There have been many papers and books written on the twin paradox, wherein a
twin who travels at high velocity to a nearby star system and returns is younger
than his twin who remains on earth. Think about this and convince yourself that
there is no contradiction. The problem is discussed in many reputable books,
for example the readable Feynman lectures (Feynman 1963; Schutz 2009).
However it is also a favorite topic in books and articles by people with limited
or incorrect understanding of relativity, so beware.