PHYM432 Relativity and Cosmology 7. Moving from SR to GR

PHYM432 Relativity and Cosmology
7. Moving from SR to GR
With special relativity, Einstein successfully formulated the physical laws
for inertial frames, i.e frames moving at constant velocity, Newtons 1st
law. However, this did not cover the more general case of observers
moving with a relative acceleration (i.e. an accelerated reference frame),
which meant it could not yet describe gravity.
Newton’s laws were fundamentally flawed, evident by the fact that in
Newtons laws gravity was “instantaneous” (no time dependance) while in
SR specifically stated that the speed of light would be the fastest gravity
could act.
1907
The principle of equivalence - “turning gravity off”
The equivalence principle was the staring point which lead Einstein to a
full theory of GR, it took 10 years for him to complete the theory.
“the gravitational field has only a relative existence, because for an
observer freely falling from the roof of a house there exists (at least to
his immediate surroundings) no gravitational field”
- Einstein
free fall
locally
=
weak of equivalence principle (WEP) - within a sufficiently localised
region of spacetime adjacent to a concentration of mass, the motion of
bodies subject to gravitational effects alone can not be distinguished by
any experiment from the motion of bodies within a region of appropriate
uniform acceleration.
also refereed to the principle of universality of free fall
This is a direct consequence of the inertial mass being equal to the
gravitational mass
mi = F /a
GM mg r̂
=F
2
r
Galileo: inertial mass = gravitational mass
thus the acceleration of any body is independent of the mass, any
difference would show up as a difference in accelerations.
for two masses A and B, a dimensionless parameter summarises the
differences in ratios of gravitational and inertial mass.
⌘=
= ( 0.2 ± 2.8) ⇥ 10
12
the equality of gravity and inertial masses is one of the most accurately
tested principles in physics.
aside: Why might WEP not be satisfied?
various theories of quantum gravity predict a WEP violation, as
other effects might come into play. Extra term summarises
contribution to violation, scale not known
mg = mi +
X
i
2
E/c
i
strong equivalence principle (SEP)
All experiments in a sufficiently small freely falling laboratory, over a
sufficiently short period of time, give results that are indistinguishable from
those of the same experiments in an inertial frame in empty space.
The power of the SEP is the assertion that it applies to ALL laws of
physics.
Example: - If we accept the SEP, then we must conclude that light falls in
a gravitational field. Light is massless (and composed of oscillating E&B
fields) so it it not so straightforward in Newtonian mechanics to calculate
the effect of gravity on light, there is no F=ma.
But with the SEP Gravity attracts light.
gravitational
field
rocket frame
light pulse
=
g
The equivalence principle again raises the issue of coordinate
transformations just as in SR, but the transformations have to be much
more general, and broader than the special case of Lorentz
transformations of SR. Accelerated reference frames would be needed
(non-inertial frames).
Einstein realised that he could formulate a theory for gravity by
transforming the reference frame to a that of a freely falling frame (which
you can always do) where gravity is “turned off”. The freely falling frame
is a local inertial frame, where Special relativity holds (Minkowski
spacetime).
Because of SEP; all forces, all particles, everything is subject to gravity
and behaves universally the same in its presence.
Founding Principles of General Relativity
Einstein was motivated by three basic principles which needed to be met
to successfully formulate a theory of gravity.
1) Principle of Equivalence
2) The principle of covariance
3) The principle of consistency
principle of consistency
A new theory must reduce to Newtonian gravity for weak fields and slow
motions, to account for the success of that theory. In addition, special
relativity must also be satisfied.
A spacetime metric that is locally equivalent to the Minkowski
spacetime (i.e. an inertial frame in free fall) ensures that SR is
satisfied.
Consistency
By reducing to Newtonian mechanics, the equations of GR can be
constrained when trying to formulate the correct equations. There is no
“derivation” of GR, so such constraints are extremely valuable. Einstein
had to essentially guess and check when finding the right form of the
equations.
In special relativity, energy and momentum are conserved, this must also
hold true for GR. While this may seem trivial, Einstein realised that with
E 2 = m2 c4 + p2 c2
ALL forms of energy would be ‘sources’ of gravitational fields, including
both mass and momentum. Thus for GR, it is not sufficient to know the
mass, all forms of energy and momentum must be taken into account as
sources of gravity... including the gravitational field (spacetime curvature)
itself!
Principle of General Covariance
is an extension of the principle of relativity from SR, which stated the laws
of physics should take the same form in all inertial frames. GR extends
the principle of relativity by requiring the equivalence of all frames
including non-inertial frames (i.e accelerated frames). For SR, the
physical laws were form invariant under Lorentz transformations
(Minkowski spacetime) though gravity had to be left out as it did not
transform properly.
In practice, Einstein used Tensors to describe the physical laws, which
ensured that they would be invariant under a broad class of coordinate
transformations.
Many physical quantities are naturally regarded not as vectors themselves,
but as correspondences between one set of vectors and another. Because
they express a relationship between vectors, tensors themselves are
independent of a particular choice of coordinate system.
Geometry of spacetime
Unlike E&M which has positive and negative charges, Gravity can not be
‘switched off’, it’s always attractive, and all forms of energy will produce a
gravitational attraction. Every bit of matter and energy attracts every
other bit of matter throughout the entire universe, and everything is
subject to it’s pull (SEP). This ever-present nature of gravitation played a
key role in Einstein’s GR. He argued that because of its permanence,
gravitation must be related to some intrinsic feature of space and time,
that way all particles (and all forces) would still be subject to its presence.
The genius of Einstein was to relate this feature with the geometry of
space and time. Any effects we ascribe to gravitation actually arise
because the geometry of spacetime is ‘unusual’ or curved.
With this viewpoint, matter (and energy/momentum) bends spacetime. A
particle then responds to the curved geometry, still moving in ‘straight
lines’ with no external forces, but along geodesics, which are the shortest
path between two points in a general multi-dimensional Riemannian
space.
Clocks in a gravitational field
Thought experiment - Alice and Bob are in a rocket ship accelerating.
Alice sends light pulses to Bob, from the tip to the tail of the rocket. Bob
accelerates after the light pulse was sent, and “meets” the light pulses in a
shorter time than Alice sends them.
The acceleration affects time.
Equivalence principle states that this happens in a gravitational field as
well.
t=0
1st pulse
emitted
by A
t = t1
t=
rocket
accelerates
A
1st pulse
received
by B
t = t1 +
2nd pulse
received
by B
2nd pulse
emitted
by B
B
g=a
✓
B = 1
equivalence principle
gh
c2
◆
A
clocks run slower in a gravitational
potential well
gh
is the gravitational potential difference between A and B
A
B
B
=
✓
= gh
1+
⇥B
⇥A
c2
◆
A
Newtonian Gravity in Spacetime terms
dS =
2
✓
2
1+ 2
c
◆
(cdt)
2
✓
1
2
c2
◆
(dx2 + dy 2 + dz 2 )