Download Astronomy 748 Homework 1: Special Relativity Due Monday, September 28

Astronomy 748 Homework 1: Special Relativity
Due Monday, September 28
1. Relativistic Doppler Shift
You are familiar with the derivation of the Doppler shift using Galilean transformations. Here you will use Lorentz transforms.
a) Consider a source at rest with respect to the observer, i.e. with four-velocity uα =
(1, ~0). Now consider a moving source, i.e. one in a Lorentz boosted frame. What is the
source four-velocity?
b) Energy is a scalar, so the energy of a particle measured by an observer is the
projected four momentum E = −p · u = −pα uα , where pα is the particle four-momentum
in some frame and uα is the four-velocity (boost) of that frame with respect to the observer.
Check this for a frame at rest with respect to the observer by calculating it for a massive
particle 1) at rest, 2) with velocity ~v. The results should look familiar.
c) Now allow the frame to move with velocity ~v and consider a photon of energy Ee
emitted from a source at rest in that frame. Relate Ee in the boosted source frame to
the energy Eo the observer measures. As v → 0 you should find the usual Doppler shift
formula. Define a new parameter z = Ee /Eo − 1. Is z positive or negative for a source
receding (radially) from the observer? approaching (radially) the observer? At what angle
between the source velocity and the line of sight from the observer does z change from
negative to positive? In the Galilean case this angle is 90◦ – there is no transverse Doppler
shift. In the Lorentz transverse case what is the shift to lowest order in v?
Since for a photon the energy is inversely proportional to the wavelength, this effect
(and z) is sometimes called the redshift or blueshift. The scalar product of four-vectors
above will also hold in general relativity and exhibits the equivalence of the photon energy
shift from a moving source, the gravitational redshift of a black hole, and the shift of a time
dependent gravitational lens. (You will soon learn that this is an aspect of the Principle of
Equivalence underlying general relativity).
2. Relativistic Particle Dynamics
The universe is filled with the cosmic background radiation, photons with an average
energy E ≈ 10−3 eV. Energetic processes can also create high energy photons – gamma
rays. Although the cross section is small, interactions do take place, with sufficiently
energetic ones between the two types of photons able to cause, e.g., electron-positron
creation. Since this destroys gamma ray photons above a certain threshold energy, one
expects an observable feature in the gamma ray energy spectrum. To find the threshold it
is convenient to use the four-momenta, allowing you to solve simultaneously the conditions
of energy and momentum conservation.
a) Show that for a massive particle the square of the four-momentum pα pα = −m2 .
b) Find the square of the total four-momentum of the left hand side of the reaction
γγ → e+ e− , i.e. (pα + p′α )(pα + p′ α ) in terms of the photon energies E and E ′ .
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c) Since the total four-momentum in the reaction must be conserved, the result in
b) gives the total amount of mass equivalent available, −m2 from a), for massive particle
production. In order to create an electron and positron, each with mass 511 keV, what is
the minimum gamma ray energy E ′ necessary?
3. Energy-Momentum of the Vacuum
Quantum physics teaches us that the vacuum – the ground state – has energy and
hence momentum. The principle of relativity teaches us that all observers must agree on
the ground state, thus its energy-momentum tensor must be Lorentz invariant (having the
same value for observers moving relative to each other). The only such Lorentz invariant
is the Minkowski tensor η αβ , so T αβ ∼ η αβ .
a) Find the pressure p in terms of the energy density ρ.
b) Use the conservation equations T αβ ,β = 0 for α = i to verify that the pressure is
not only isotropic (as it must be for a perfect fluid) but homogeneous.
c) Use the conservation equation T αβ ,β = 0 for α = 0 to find the evolution of the
energy density with time.
The vacuum energy has to keep these properties even in general relativity, and is what
Einstein called the cosmological constant.
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