of Physics, University of Maryland, Prof. T. Jacobson
Physics 675, Fall
Relativity, Gravitation and
were written for Fall 2004,5,6. Please refer to them as well.
The main purpose of these notes as I start
out is just to indicate what was covered in class,
occasionally to include some extra info that was not in the previous
notes or the textbook.
- Hubble's law: z ~ v/c =
(H_0/c) d = d/4000 Mpc. So Slipher's measurements out to
z~ 0.006 corresponded to d ~ 24 Mpc, and 100 Mpc is z ~ 1/40 = 0.025.
- Reshift from expansion: drew
spacetime diagram: "Mercator projection of the history
of the universe", and used diagram to explain two derivations of
1) Coordinate wavelength is preserved in time, thanks to translation
wavelength is a(t) times coordinate wavelength, so 1 + z =
2) The way Hartle derives it: On a light ray we have ds^2 = 0, so dx =
so the coordinate separation traversed in a finite time is the integral
Thus if two light rays travel the same coordinate separation, one
over the interval (t_e, t_o) and the other over (t_e + dt_e, t_o +
it must be that the missing dx from the first part is made up for by
the extra dx
at the end, i.e. dt_e/a(t_e) = dt_o/a(t_o). So if a time interval
from past to future by signals at the speed of light, dt_0/dt_e = a(t_o)/a(t_e).
This applies to all such time intervals, including the period of a
yields the redshift formula 1 + z = a(t_o)/a(t_e). Another example is duration of a
supernova light curve, originating at z = 1. If it lasts for us for 2
months, then at the
source it lasted only 1 month.
- How far back do we see? The CMB is from the surface of last
when the temperature was T ~ 3000 K. The Planck distribution is
and simply redshifts to another Planck distribution, with a temperature
lower by a
factor 1/(1+z). Since the CMB today has T ~ 2.7 K, we have 1 + z ~ 1100.
I did not say this in class, but the oldest quasars are around z ~ 6,
oldest stars might have z ~ 10. Primordial nucleosynthesis of helium
at a redshift around 10^9.
- Relation of cosmic redshift to
Hubble's law: Hubble observed not velocities
but redshifts, and measured distances using the luminosities of Cepheid
For small redshifts like Hubble was working with, one need not think
the effect of expansion or possible spatial curvature on the
luminosity, and the "distance"
can be simply interpreted as the separation distance D between the
source and the
observer, measured on one cosmic time slice. This is a functon of time:
D(t) = a(t) D_x, where D_x is the coordinate separation which is fixed
for a pair of isostropic
worldlines. Taking the time derivative, we get dD/dt = da/dt D_x =
((da/dt)/a) D. Evaluating
this at the present epoch, and interpreting dD/dt as simply the
relative velocity, which is
valid for small redshifts, it reproduces Hubble's law, with H_0 = ((da/dt)/a)_0.
(Hartle uses an alternate analysis: instead of relating the rate of
change of this distance
to the velocity and hence the redshift, one can work directly with the
z = a(t_0)/a(t) - 1 = (a(t_0)
- a(t))/a(t) = ((da/dt)/a)_0 (t_0 - t) + O[(t- t_0)^2]. For small
enough distances t_0 - t = d/c, and the higher order term int he Taylor
negligible, so again we recover Hubble's law.
- By the way, when D_x is large enough, dD/dt is greater than the speed
But so what? Both objects are moving within their future lighcones. No
moving faser than light.
- Dynamics of a(t): apply
Newtonian kinetic + potential energy conservation to a test particle
at the surface of a
small sphere inside FLRW universe, to infer
(da/dt)^2 - (8pi/3) G rho a^2 = constant.
The scale freedom in the scale factor can be exercised to arrange for
the constant to be equal
to +/-1 if it is nonzero. It follows from the Einstein equation (and so
far I don't know how to get this
from a Newtonian argument), that with this scaling a(t) in fact
measures the spatial radius
- discussed varoius aspects of gravitational waves
- started cosmology: parsec, distance scales of the Universe, history
and observational development of cosmology (see previous years notes),
law, interpreted as Doppler effect.
- deep sky survey & CMB evidence for homogeneity & isotropy of
- FLRW (Friedman-Lemaitre-Robertson-Walker) metrics: scale factor a
cosmic time: ds^2 = -dt^2 + a^2(t) dl^2, where dl^2 is the metric of a
and isotropic 3d space, of which there are locally only three
possibilities: R^3, S^3, H^3.
I mentioned in the next class that one can also have "quotients" of
these spaces, such as
the 3-torus for example, that arise by identifying points related
by the action of a symmetry
transformation that leaves no point fixed.
- Unruh effect and its relation to Hawking effect ( = redshifted
- Minkowski vacuum state as superposition of correlated pairs of right
and left Rindler wedge
modes: so that the vacuum, restricted to one side of the Rindler
horizon, is a mixed state. In the
Hawking effect, the picture is the same, and the pairs separate
due to the tidal force.
- Spacelike singularity: inside Schwarzschild black hole: what
1) nothing, or 2) bounce (to either a classical or a nonclassical
spacetime), or 3) something
- Information loss in bh evaporation? Maybe the Hilbert space of a
"baby universe" formed
at the singularity must be accounted for.
- Singuarily of Kerr: timelike, ring shaped, with infinite
circumference. Can go through the
middle and out the other side, but to make a smooth continuation of the
metric there must
analytically extend r to negative values. If you do this, there are
closed timelike loops on the
other side. If you fall freely along the axis of a Kerr black hole, you
bounce at finite r and proceed
to larger r values, but not in the original spacetime. To understand
this, we turned to Carter-Penrose
- Carter-Penrose diagrams in general, and diagram for the axis of Kerr.
Shows the path to another
part of the universe inside the black hole, even with its own
- Something fishy: the inner horizon is probably unstable: when you
fall across it you can receive signals
from the exterior that originated over an INFINITE amount of time.
Studies suggest that in the presence of
any perturbation the inner horizon is unstable and will become
a null or spacelike signularity.
See for example Oscillatory null singularity
inside realistic spinning black holes, by
- The "Clausius relation"
a.k.a. the "first law of black hole
dM - Omega dJ = (kappa/8pi G)dA. This was
first derived by Bekenstein, but
without knowing that kappa is the surface gravity. Instead, he just had
formula for it in terms of M and J. So he could not know it was an
variable, i.e. it is locally defined on the horizon. Moreover, being
defined, it is non-trivial that it is CONSTANT on the horizon. This
called the 0th law of black hole mechanics, and was proved by Bardeen,
and Hawking. They also proved the 1st law in a more geometric fashion.
In fact, it was later proved in the deepest way by Wald, who showed it
a consequence of general covariance of GR. (If more derivatives are
allowed in the
field equations, then the entropy could receive contributions from
addition to area, but these would be suppressed by powers of a small
- The reason for the Omega dJ term (macroscopic energy, not "heat", and
it can be mechanically extracted).
Eg non-relativistic rotational kinetic energy E_rot = J^2/2I has dE_rot
= (J/I) dJ = Omega dJ.
This generalizes to relativistic systems as well. I think this can be
seen in the Hamiltonian
formalism: Omega = d phi/dt = \partial H/partial J, so the change in
the Hamiltonian due to
a change in J, other coordinates and momenta being held fixed, is Omega
- Area of the horizon: integral d theta d phi (g_theta theta g_phi
phi)^1/2. (Exercise: check that
this yields 4\pi(r_+^2 + a^2).)
- Area increase: a) from
future 4-momentum of particles crossing the horizon and b) from
Hawking's area theorem. That theorem is proved roughly as follows:
Suppose at a point of
the horizon the null generators were converging. Then as long as
gravity is not repulsive
for null rays (energy density + pressure not negative) the generators
will cross in a finite
affine parameter (as long as a singularity doesn't get in the way). But
it is a consequence of
the definition of the horizon as the boundary of the causal past of the
future asymptotic region
("future null infinity"), that the generators can never leave the
horizon, nor can they cross on
the horizon. (Roughly, the reason is that if they left or if they
crossed then at the point where
that happed the horizon would be a timelike surface, but being the
causal boundary it must
be null, not timelike.) Thus the theorem is proved under the
assumptions of the "null energy
condition" defined above, and the assumption that there are not
singularities on the horizon.
A stronger version replaces the second assumption with one saying that
lying to the future of some initial spacelike slice are visible from
future null infinity
("Cosmic censorship"). Hawking's
original area theorem paper:
"Gravitational Radiation from Colliding Black
- Information and area: Bekenstein's
derivation of S = eta A/L_P^2 based on the statistical interpretation
of entropy: He argued given the area theorem one is led to guess
entropy of the black hole is a
monotonic function S(A) of area. Additivity of entropy would have
suggested S is proportional to A.
But Bekenstein derived this, as well as the L_P^2 normalization, using
a simple argument:
dS_min = (dS/dA) dA_min, and dS_min = ln 2. The minimal area change is
nonzero since the
uncertainty principle (or more generally, QM) prevents adding an
arbitrarily small energy.
Bekenstein used the "1st law" to infer dA = (8pi G/kappa)(dM - Omega
Note dM - Omega dJ is the Killing energy added, where the
horizon-generating Killing vector is
So the question is what is the minimal Killing energy that can be
considered a mass m with radial spread b in its own rest frame, so QM
implies mb > hbar/c.
(E.g. b can be the Compton wavelength.) To deliver this to the bh with
energy it should be dropped from a position at rest as close as
possible to the horizon,
i.e. with its center at proper length b above the horizon. At that
location the Killing vector has
norm kappa b, so the Killing energy is m(kappa b). Thus dA = 8pi
G mb so dA_min = 8pi hbar G.
Putting this together with the above yields dS/dA = (ln 2)/(8pi L_p^2). Note this is
M and J for the black hole! Integration then yields S_BH ~ (ln 2/8pi) A/L_p^2. As we saw,
Hawking's temperature calculation T_H = hbar kappa/2pi yielded S_BH =
So Bekenstein's estimate was only off by a factor ln2/2pi.
This is a deep argument that area is proportional to entropy, but he
also had the
fact that the 1st law placed kappa dA in the position of T dS. However,
he did not know that
kappa is an intensive variable, because he did not relate it to the
Still, he felt quite confident that black hole are measures entropy.
entropy and generalized second law:
"Black Holes and
- It is rather strange that classical GR gave use the 1st law,
and the area theorem,
very strong indications that area ~ entropy, given that the microstates
counted by the
entropy are presumably quantum meachanical. Note the entropy diverges
in the classical
limit, since it is A/4hbar G.
- What are the microstates? The proposal I find most compelling is that
they are the vacuum
fluctuations of quantum fields outside the horizon entangled with those
within. This makes
perfect sense, since these repesent precisely the "missing information"
that can affect the evolution
of the outside world, which is where the entropy matters. (In
particular, "internal" states of the black hole
should, I think, be irrelevant.) Also, it is the same quantum
fluctuations that give rise to the
Hawking radiation itself.
A problem with this interpretation however is that it yields infinite
entropy in quantum field theory,
due to the infinite number of field modes at short distances. However
that infinity is based on a fixed
horizon, whereas the true horizon must itself have quantum
fluctuations...whatever that means.
It is an outstanding problem to understand how to count these states.
Quantum gravity itself may
already render the number finite, or a more radical modification of the
structure of spacetime may be necessary.
Review of Horizon
Debate on the meaning of black hole entropy http://arxiv.org/abs/hep-th/0501103.
Calculation of cut-off entanglement entropy: http://arxiv.org/abs/hep-th/0703233.
Another apparent problem with the vacuum fluctuations interpretation,
that I did not have time to mention in class,
is the "species problem": the
total entropy in the quantum fluctuations should depend on the number
and nature of
the quantum fields (electromagnetic, electron, neutrino, gluon, etc).
There is a good candidate solution to this
problem however: the value of Newton's constant reflects already the
dependence on the quantum field
- String theorists have
succeeded in computing black hole entropy in terms of microstates for
very special (supersymmetric) black holes by relating these, with the
gravitational coupling tuned to zero,
to flat spacetime configurations called D-branes, and counting the
microstates of those. A supersymmetry
property implies that the counting must agree with that for the black
hole. One still does not
know what are the microstates of the black hole itself, although some
progress towards that has been made,
using string theory, at least for these special black holes.
For further reading:
Introductory lectures on
black hole thermodynamics: http://www.physics.umd.edu/grt/taj/776b/lectures.pdf
Introduction to quantum fields in curved space-time and the Hawking
has a subscription so you can access the linked PRD articles from a
campus computer, or from elsewhere using your
University ID number, at http://www.lib.umd.edu/. Click the Research
- negative Killing energy with future pointing 4-momentum:
4-momentum must have positive
inner product with the spacelike Killing vector (using -+++
- A surface of constant f is spacelike, null or timelike if
the gradient of f is timelike, null, or
spacelike at the surface, i.e. according as g^ab f,a f,b is
negative, zero, or positive.
- Using the previous, a surface of constant r-coordinate is null
if g^rr = 0. For the Kerr
metric in BL coordinates, this is true precisely where Delta=0.
- Maximal efficiency of energy extraction from rotating black hole: dM
- Omega_H dJ = 0.
(This comes from the fact that the future pointing 4-momentum of matter falling into the black
hole must have negative or zero inner product with the
horizon-generating Killing vector.
The extracted angular momentum for a given extracted mass is minimized
when this inner
product is zero, i.e. the matter 4-momentum is null and tangent to the
it enters the black hole on a "skimming" trajectory.)
- Otherwise this quantity is positive, and equal to (kappa/8pi G) dA,
where kappa is
surface gravity. Kappa is uniform over the horizon, and is an intensive
both properties of temperature. A is an extensive quantity, and
additive for multiple
systems, like entropy. So this is like dQ = T dS.
- Wheeler wondered aloud to Bekenstein whether one could hide entropy
by throwing it into a black hole. Bekenstein proposed that no, a black
hole itself has
entropy that would increase in that case. Specifically, he proposed the
to the area A: S_BH = eta A/L_P^2, where eta is a dimensionless
of order unity and L_P^2 = hbar G/c^3 = (10^-33 cm)^2 is the Planck
This means the "temperature" would be T_BH = hbar kappa/8 pi eta.
- Bekenstein proposed the generalized second law (GSL): the sum
of the of entropy outside
the black hole plus S_BH never decreases. He noted that if you
shine a beam
of thermal radiation into the BH at temperature T_rad < T_B
this GSL can be violated,
but he argued that since the wavelength of such radiation is larger
than the BH, it won't
always go in an the process will be dominated by quantum fluctuations.
This was an invalid
escape, since, like the SL, the GSL should have been expected to be
true in the mean,
so if only a fraction of the photons entered the BH, it would still be
a problem. There was only
one correct wasyto save the GSL: to infer that actually T_BH must
represent not just an analogy
but a true physical temperature of the black hole. Since T_BH is
proportional to hbar, this
would be a quantum mechanical effect, an instability of the vacuum
around a black hole.
But Bekenstein missed that opportunity...
- Hawking was a major critic of Bekenstein's proposal that black holes
have real entropy.
However, it turned out to be Hawking himself who discovered that in the
presence of quantum
fields, black holes do emit thermal radiation, at precisely the
termperature T_H = hbar kappa/2pi.
This meant that Bekenstein's eta coefficient is precisely equal to 1/4,
and hence that the
black hole entropy is precisely S_BH = A/4L_P^2.
UMD Astronomy, on Astrophysical Black Holes
- Kerr metric:
+ meaning of total mass in general relativity,
+ expansion in a/r,
+ magnitude of frame dragging effect down by V/c compared to
leading GR effects,
where V = tangential speed of spinning source,
+ maximal spin value a = M, Bardeen spin-up, Thorne limit 0.998 M
(differential photon absorption),
+ effective potential, orbits, ISCO, binding energy
+ non-equatorial orbits integrable: the miraculous Carter
constant: it's quadratic in 4-momentum, and
not associated with an additive conserved
+ gravitational radiation and the Carter constant
+ ergosphere and horizon
+ negative energy states in ergosphere, Penrose process,
extraction of rotational energy
Notes on homework assignment:
- One diagnostic of the timelike/spacelike/null character of a
3d-surface in a 4-d spacetime is the
negative/positive/zero sign of the determinant of the 3d-metric you get
by restricting the 4d metric
to the 3d surface. (This is fine as long as the coordinates are well
behaved on the surface.)
- Note that the Killing vector that is expressed as d/dt in BL
coordinates is expressed as d/dv in EF coordinates.
It's the SAME vector, just expressed in a different coordinate system.
How do I know? Look at the definitions:
BL: hold r,theta,phi fixed, vary t.
EF: hold r, theta,psi fixed, vary v.
These define the same curve, and on that curve t=v, so they are the
The same remark goes for d/dphi and d/dpsi, being the rotation Killing
Hence the horizon generator (15.10) can be directly translated from BL
to EF coordinates.
- redshift of spectral lines from atoms orbiting in accretion disk.
(Went through the calculation of section 11.2 in detail.)
- Kerr metric introduction, black hole uniqueness ("no hair theorem")
- EF coordinates, outgoing and ingoing, past-future asymmetry
- Rindler space, Minkowski analog
- maximal extension of Schwarzschild: Kruskal-Szekeres coordinates
This is all discussed in Chapter 12 of the textbook. For derivations of
various coordinate changes see 2004 notes of
10/19/04 and 10/21/04.
- binary pulsars
- white dwarfs, neutron stars
- black holes, growing horizon
- closed trapped surfaces, Penrose's singularity
- nature of singularity?
- photon orbits: unstable orbit at 3M, impact parameter, black hole
absorption cross-section, deflection angle
lensing: deflection, Einstein ring, multiple images,
lensing (here's a
presentation about weak lensing)
- Shapiro time delay (light arrival time depends on intervening
curvature); the question was asked whether
by this effect one could detect the frame-dragging around
a spinning star. In principle, yes, in practice,
I don't know. I'll look into it...
- orbital precession, the case of Mercury (see 2006
notes for some details and the history of Einstein and Besso's
collaboration on this). Note that
- radial effective potential in Newtonian mechanics, angular
- Schwarzschild line element, uniqueness of spherical solution to
relation of Schwarzschild time to Eddington-Filnkelstein
time: t = v - r - 2M ln(r/2M -1),
Schwarzschild radius r = 2M = 2GM/c^2 = 3km (M/M_sun).
- radial effective potential in GR, bottomless pit in the potential,
- stable and unstable circular orbits, ISCO: innermost stable circular
at r = 6M, with energy Sqrt[8/9] = 0.94 and angular momentum l =
Sqrt M per unit
Hence 6% of rest mass has been extracted as mass worked its way
an accretion disk to the ISCO.
- conserved quantities along geodesics and spacetime symmetries,
examples of "energy" and
- Killing energy vs. locally measured energy, possibility of negative
Killing energy in black hole
spacetime behind the horizon (key to Hawking effect) in nonrotating case, because the
becomes a space translation there. In rotating case this happens
even outside the
horizon (allowing extraction of rotational energy of black hole).
- analogy of "time translation" Killing energy in static black hole
spacetime with "boost energy"
in MInkowski spacetime: conserved quantity conjugate to
hyperbolic rotation angle, i.e. boost
parameter of Lorentz transformation; negative behind horizon. In
Euclidean analog this would be
angular momentum (conjugate to rotation symmetry about a point).
- comparison of Killing energy with energy measured by a static
E_Killing = - p.(d/dt) = -p.u_stat ||d/dt|| = E_stat ||d/dt||,
where ||d/dt|| = Sqrt[-g_tt] = Sqrt[1 - 2M/r] is
the norm of the Killing vector. Expanding ||d/dt|| = 1 - M/r +
... and expanding in slow velocities
E_stat = m + 1/2 mv_stat^2 + ... we get E_Killing = m + 1/2
mv_stat^2 - Mm/r + .... That is, Killing energy
is approximately the Newtonian conserved energy plus the rest
- lower mass to horizon and extract 100% of its rest mass as work
at infinity! (In Newtonain gravity
could extract an infinite energy from a pair of point
- gravitational redshift, derived from conservation law for
- coordinate freedom: dx^m transforms by Jacobian ("contravariant")
, g_mn transforms by one factor of
inverse Jacobian ("covariant") for each of the two indices.
Contraction (summation over) a contravariant-covariant
index pair yields a scalar, since the Jacobian and inverse
- can set g_mn to eta_mn at a point p, provided the signature is
(-+++). The Jacobian has 16 numbers, while
g_mn = eta_mn at p is 10 equations; the remaining 6 degrees of
freedom correspond to Lorentz transformations
at p. Next the first partials g_mn,a are equal in number to the
second partials of the coordinates, so these can be
set to zero at p. Next the second partials g_mn,ab are 10x10=100
in number, while the third partials of the
coordinates are only 4x20=80 in number. So generally 100-80=20
second partials can not be set to zero.
These correspond to curvature.
- conserved momentum conjugate to a coordinate that does not appear in
the Lagrangian (and translation
of which is therefore a symmetry); application to geodesics: if
metric is independent of a coordinate x^1 the
corresponding momentum g_1m dx^u/ds is conserved (this is the
form of the momentum if s is an affine parameter).
- All about geodesic equation:
+ 4 coupled 2nd order nonlinear
+ coordinate independence
+ evaluation in "adapted coordinates at a point p": implies second
derivative of curve coordinates zero at p.
+ lightlike geodesics: same equation, but parameter is "affine
+ ratios of affine parameters are limits of ratios of proper times on
timelike geodesics; affine parameter defined only up to constant
+ Affinely parameterized geodesic equation follows from the "squared"
Lagrangian 1/2 g_mn dx^m/ds dx^n/ds;
+ Note this is like 1/2 mv^2, with m replaced by a position dependent
inertia tensor: 1/2 m_ij(x) v^i v^j.
+ Note the action integral with the squared Lagrangian is not
reparametrization invariant, in agreement with the fact that the
Euler-Lagrange equation only characterizes
the affinely parameterized
+ Note the squared Lagrangian is constant along a solution. I
proved this by a slick argument using the adapted coordinates at a
point. (It could instead be proved by just evaluating the derivative of
the squared Lagrangian in a generic coordinate system and using the
geodesic equation.) This integral of
the motion is always present, and reduces the number of independent
degrees of freedom from 4 to 3, like in Newtonian motion.
limit: v << 1 and g+mn = eta_mn + h_mn, with h_mn << 1 in
coordinates x^m = (t,x^i).
+ To compare with Newton use t for
the path parameter. Since this is not an affine parameter, use square
root Lagrangian, which allows any parameter.
+ Expanded L_1 in powers of v and
h_mn and found that h_tt corresponds to -2 Phi, where Phi is
+ A higher order term is h_ti v^i, which looks like the
velocity-dependent potential of a charged particle coupled to a
magnetic vector potential. This is the gravito-magnetic term.
+ Size of gravito-magnetic term compared to Newtonian potential h_ti
v^i/h_tt. The source that produces h_ti is mass (or energy) current, so
is suppressed compared to h_tt by a factor of the velocity of the
source v_source. So this term is smaller than the Newtonian term by the
factor (v_source v)/c^2.
+ The gravito-magnetic field in the post-Newtonian limit is the curl of
h_ti. How large is this for the earth? It would be of order v_source
Phi/R where Phi is the Newtonian potential and R is the radius of the
earth. So v_source/R is of order 2Pi/day. The Newtonian
potential/c^2 is ~ 10^-9. The angle of precession of a gyro
in one year should thus be of order 2Pi*365*10^-9 radians, i.e.
of order 0.0001 degrees. Actually this estimate was too crude, it turns
out to be about ten times smaller, 0.00001 degrees per year.
+ The website of Gravity Probe
B has lots of information about the experiment to measure this, http://einstein.stanford.edu/
+ While a tiny effect near the earth, gravito-magnetism becomes a
dominant effect near a rapidly spinning black hole.
- equivalence principle, gravity as a pseudoforce, real
gravity as the gradient of the
- geodesics/free-fall/force-free motion:
maximal time with respect to local small variations of the path;
Riemannian analogy; geodesic equation as Euler-Lagrange equation
- example of orbits in neighborhood of earth.
- Doppler effect
- gravity as a pseudoforce
- special relativistic dynamics, energy-momentum conservation,
GZK cutoff, invariant
description of observed frequency.
- Intro to course: syllabus, homework rules (N.B. you may not make
use of previous
solutions), and topics to be covered.
- To be covered: special
relativity, curved spacetime, orbits of particles and light rays,
to motion around nonrotating stars or black holes and then
rotating black holes, black hole
thermodynamics, cosmology, tensor calculus, Einstein field
equation, linearized gravitational
- Review of special relativity.
- Key idea of
relativity: time is arc length along a spacetime curve.
- Comparison of relativistic with
Newtonian spacetime structure:
absolute time function, 2) Euclidean spatial
metric on constant time slices, and
3) a preferred set of worldlines at absolute rest.
Einstein: the "spacetime
takes over the job of all three of these, determines the causal,
temporal, spatial metric and inertial structures of spacetime.
- Proper time, 4-velocity, Minkowski inner product, 4-momentum, rest
mass, E^2 = p^2 + m^2.