Physics 616

• Prof. Andrew W. Steiner
(or Andrew or "Dr. Steiner")
• Office hour: 103 South College, Thursday 11am
• Email: awsteiner@utk.edu
• Homework: Electronically as .pdf
• You may work with each other on the homework, but you must write the solution in your own words

Use down and up arrows to proceed to the next or previous slide.

Outline

• Units
• Metrics
• Intervals
• Space-time diagrams
• Lorentz transformations

Units

• I will (often without warning) change units, sometimes using $\hbar = c = G = k_B = 1$ or combinations thereof
• In some cases, you will be required to provide answers with some unit system I specify
• The neutron mass is $1.7 \times 10^{-24}~\mathrm{g}$, but I often use $c=1$ and specify it's mass as an energy $939~\mathrm{MeV}$ or additionally assume $\hbar=1$ and then convert it's mass to $4.75~\mathrm{fm}^{-1}$. Additionally assuming $G=1$ would give the neutron mass as a totally unitless quantity. Or, alternatively, sometimes I'll use $k_B =1$ and then the neutron mass is $10.9 \times 10^{13}~\mathrm{K}$ .
• In relativity, it's even worse, authors don't even agree on the metric, the inertial reference frame, or on the representation of the group we are using.

Special relativity

• No experiment can measure the velocity of an observer
• The speed of light relative to an unaccelerated observer is constant, regardless of the motion of the source
• "Inertial reference frame": unaccelerated (thus also non-rotating) observer
• One can derive the SR results from these two facts, but we'll present SR from a GR perspective

Indices

• Row vectors (covariant vectors, covectors, one-form), $x_{\alpha}$
• Column vectors (contravariant vectors), $x^{\alpha}$
• Dot product: $x^{\alpha} x_{\alpha}$
• Remember how matrix multiplication works?
• Modified Einstein summation convection: repeated indices are summed, but only when one is covariant and the other is contravariant

Metric

• Metric tells you how to define a distance
• Fundamental element in GR
• This is the "Minkowski" metric (as opposed to the Euclidean metric)
• The distance is $$d^2 = M_{\alpha \beta} x^{\alpha} x^{\beta}$$
• One can think of the metric as "lowering" an index
• For the Minkowski metric (but not for all metrics!) $$M_{\alpha \beta} = M^{\alpha \beta}$$
• Metric is a general term, applying e.g. to function spaces
E.g. used by LIGO

Distance "interval" between two points

• Interval is $$\Delta s^2 = -\left(\Delta t\right)^2 + \left(\Delta x\right)^2 + \left(\Delta y\right)^2 + \left(\Delta z\right)^2$$
• It is hard to over-emphasize the significance of the minus sign!
• Interval can also be computed as $$\Delta s^2 = \sum_{\alpha} \sum_{\beta} M_{\alpha \beta} \Delta x^{\alpha} \Delta x^{\beta}$$ where $M_{00} = -1$, $M_{ij} = 1~\mathrm{iff}~i=j$, and the other entries are zero
• Interval is invariant, i.e. independent of the observer
• Interval is negative (timelike) or positive (spacelike)
• What is true if the interval is zero?
• What is the nature of the interval if two events are causally related?

Space-time diagrams

• Proper time between two events is time ticked off by a clock which passes through both events
• Proper time, $\Delta \tau^2 = - \Delta s^2$
Schutz
Strickland (2014)

Space-time diagrams II

Schutz
• Observer ${\cal \bar{O}}$ moves to the right relative to observer ${\cal O}$ with velocity $v$ $$\phi = \mathrm{arctan}~|v|$$ (signs determined by figure)

Length Contraction

 World path of a rod at rest in $\bar{\cal O}$ Length in $\bar{\cal O}$ is $\Delta s_{AC}^2$ and in ${\cal O}$ is $\Delta s_{AB}^2$ In $\bar{\cal O}$, the coordinates of C are $$\bar{t}_C=0, \quad \bar{x}_C=\ell$$ while in ${\cal O}$, the coordinates of C are $$x_C = \ell/\sqrt{1-v^2} \quad t_C= \ell v / \sqrt{1-v^2}$$ but the interval is the same in both systems so $$x_C^2 - t_C^2 = \ell^2$$ and $$t_C = v x_C$$ Also, $$\frac{x_C-x_B}{t_C-t_B}=v$$ Schutz We want $x_B$ when $t_B =0$ $$x_B = x_C - v t_C = \ell \sqrt{1-v^2}$$ This is length contraction

SR results

• Presuming ${\cal \bar{O}}$ moves with velocity $v$ relative to ${\cal O}$
• Time dilation $$\left( \Delta t\right)_{\cal O} = \left( \Delta t\right)_{\cal \bar{O}} \left( 1-v^2/c^2 \right)^{-1/2}$$
• Lorentz contraction $$\ell_{\cal O} = \ell_{\cal \bar{O}} \left( 1-v^2/c^2 \right)^{1/2}$$
• Relativistic mass $$E_{\cal O} = E_{\cal \bar{O}} \left( 1-v^2/c^2 \right)^{-1/2} \equiv \gamma E_{\cal \bar{O}}$$

Invariant Hyperbolae

 Curves with fixed intervals What might these curves be useful for? Schutz

Group Work

Draw the t and x axes of the spacetime coordinates of an observer ${\cal O}$ and then draw:
• The world line of ${\cal O}$’s clock at x = 1 m.
• The world line of a particle moving with velocity dx/dt = 0.1, and which is at x = 0.5 m when t = 0.
• The $\bar{t}$ and ${\bar x}$ axes of an observer ${\cal \bar{O}}$ who moves with velocity v = 0.5 in the positive x direction relative to ${\cal O}$ and whose origin (${\bar x}$ = $\bar{t}$ = 0) coincides with that of ${\cal O}$.
• The locus of events whose interval $\Delta s^2$ from the origin is −1 $\mathrm{m}^2$ .
• The locus of events whose interval $\Delta s^2$ from the origin is +1 $\mathrm{m}^2$ .
• The calibration ticks (defined as the tick marks on the axis that refer to $\bar{x}=1,2,3,\ldots$ and $\bar{t}=1,2,3,\ldots$ ) at one meter intervals along the ${\bar x}$ and $\bar{t}$ axes.
• The locus of events whose interval $\Delta s^2$ from the origin is 0.
• The locus of events, all of which occur at the time t = 2 m (simultaneous as seen by ${\cal O}$ )
• (For the remaining, presume the observer ${\cal \bar{O}}$ travels with v=0.5 and whose origin coincides with ${\cal O}$ )
• The locus of events, all of which occur at the time $\bar{t}$ = 2 m (simultaneous as seen by ${\cal \bar{O}}$).
• The event which occurs at $\bar{t}$ = 0 and ${\bar x}$ = 0.5 m.
• The locus of events ${\bar x}$ = 1 m.
• The world line of a photon which is emitted from the event t = −1 m, x = 0, travels in the negative x direction, is reflected when it encounters a mirror located at ${\bar x}$ = −1 m, and is absorbed when it encounters a detector located at x = 0.75 m