in 1687 isaac newton changed the way we
humans view the cosmos by connecting a
force
we experience on earth with the force
that controls the movement
of heavenly bodies that force of course
is gravity
and he expressed this analytically in a
remarkable equation that is still used
today
the law of universal gravitation this
equation explained the movement of
planets and it remains very accurate
even today
under most circumstances but it had a
few problems
one of which was the idea of action at a
distance
that even newton had trouble accepting
this was the idea that a massive object
could exert a force instantaneously
to another massive object at a distance
without ever touching it
later in 1859 newton's laws could not
explain the rate of precession of
mercury
the way its elliptical orbit rotates
around the sun
and is not fixed newton explained
gravity
as an innate property of objects a
constant instantaneous force that could
act over long distances
250 years later in 1916 einstein changed
this paradigm
with the publication of the general
theory of relativity
it solved the riddle of mercury's
precession and explained that gravity
was not a mysterious force
acting at a distance in the background
of space and time
but was a result of a bending of the
background itself
the curvature of space-time it was a new
paradigm
that could explain more things today a
hundred years after
einstein's publication we find some
holes in einstein's theory as well
such as its inability to explain the
singularity inside black holes and the
big bang
one of the biggest pursuits in physics
is the attempt to find a better theory
a quantum theory of gravity in order to
understand quantum gravity though you
have to first understand
general relativity however the
mathematics of general relativity
is so difficult that even a genius like
einstein could not
easily derive it and there are still
papers being written today on specific
solutions
to these equations i'm going to try to
explain those equations to you
intuitively and visually to give you an
idea of what the math is trying to say
and i hope
that this will help you better
understand quantum gravity that
explanation
is coming up right now
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if we look at newton's equation for
gravitation and einstein's equation for
general relativity
we see a remarkable correlation for one
thing newton's gravitational constant is
present in both equations
in newton's equation we have a force on
the left side created by a mass
on the right side in einstein's equation
we have the analog of force
the curvature of space time on the left
side it has two components that describe
the curvature and how distances in the
curvature are determined
this is the analog of force described by
newton i want to be careful here because
in general relativity gravity is not a
force within the background of
space-time
like it is in newton's equation but it
is a curvature of the space-time
background itself in other words
gravity does not cause the curvature of
space-time it
is the curvature of space-time the right
side of the general relativity equation
tells us about the mass
energy content this is represented by
something called the mass
energy momentum tensor it's a source of
the curvature
in the context of these equations think
of a tensor as a multi-dimensional array
of mathematical components
this can be an array of vectors scalars
or other tensors
and it's represented by an n-by-n matrix
a vector is a type of tensor
represented by an n by one matrix in
general relativity the dimensions are
four so the vectors are four by one
and the tensors are four by four
matrices
so this equation describes what john
wheeler said so succinctly
about general relativity space-time
tells matter how to move
matter tells space-time how to curve the
right side is the matter and energy
which tells space-time
on the left side how to curve in return
the left side
the curvature resulting in gravity tells
matter
on the right side how to move
essentially
what this equation shows is that matter
and energy results in a curvature of
space-time
which we perceive as gravity although
this looks like
one somewhat simple equation it is
actually ten equations
and is very complicated mathematically
if we wrote down all the ten equations
fully
it would take up more than a page of
text a simplified version is shown here
it is so complicated in fact that
einstein was skeptical about being able
to solve them
there are papers written to this day on
various solutions to these equations
let's take a closer look at what these
equations are trying to say
the equations describe the curvature of
space time by treating it
as being flat at infinitesimally small
distances
so you can consider that general
relativity behaves like special
relativity
at these small distances with no bending
or curvature but
overall curvature is taken into account
r
mu nu is the richie curvature tensor it
tells us
how space time is deviating from flat
it tells you how space time curves at a
given point
the second term on the left side is
composed of r
the scalar curvature it tells you how
much
the space is changed at any given point
such that you know how to correctly
measure distances
little g mu nu is the metric tensor
it tells you the geometry and structure
of space time
together this term defines how distances
are calculated
given a curvature at any point note that
sometimes the third term is added
lambda times g mu nu lambda
is the cosmological constant this term
describes the intrinsic energy density
of the vacuum or empty space
it's the mathematical expression for
what we observe as
dark energy the accelerating expansion
of the universe
it would comprise energy and opposition
to gravity
lambda is a very small number about 1.1
times 10
to the negative 52 inverse meter squared
at small distances this effect is
largely negligible
on the right side t is the stress energy
momentum tensor
which tells us the density of energy and
momentum
at each point in space-time it is the
source
of the curvature there's also a constant
which is equal to
8 pi g divided by the speed of light to
the 4th power
g is newton's gravitational constant but
this whole term is sometimes referred to
as
einstein's gravitational constant it's
really just a conversion factor to make
sure we get the proper units
the way these equations are formulated
is by treating
space time in four dimensions three
spatial dimensions
and one dimension of time this is
incorporated in the mu
and new subscripts the value of mu and
nu can be
0 1 2 or 3. the zero subscript
conventionally represents time and the 1
2 and 3 represent the three spatial
dimensions
this is why we have 10 equations so for
example if we draw a matrix
of all the values that mu and nu could
have
we would see that there are 16 possible
combinations
but this matrix is symmetrical so 0 and
1 for example is equivalent to 1 and 0.
so if we take all the non-symmetrical
combinations
we come up with 10 combinations hence
10 equations let's look at what some of
these combinations mean
so for example one equation is where mu
and nu are zero and zero zero represents
the dimension of time what this would
describe
on the left side is the speeding up or
slowing down of time
at a point in space and the right side
would describe the
energy at that point similarly in
another equation the value of mu and nu
can be zero and one this would represent
time in one direction
and one spatial direction in another so
the left side in this case
represents the combination of the
stretching of time
with one spatial dimension the right
side in this case
would be the momentum or momentum
density in another equation the value of
mu and nu can be 1 and 1.
this would represent spatial dimension
for both mu and nu in this case the left
side is describing
the stretching of space in one of the
dimensions the right side in this case
would describe the pressure
at that point in space the same can be
said if mu and nu are 2 and 2 or 3 and
3.
here's a matrix of what the various
combinations of mu and nu
mean for the stress energy tensor note
that the combination of the curvature in
two different spatial dimensions is
called
shear stress is general relativity true
yes in general for example it predicts
the bending of light around massive
objects this has been observed for our
sun as well as gravitational lensing
around large clusters of galaxies it
predicts the time will run more slowly
on the surface of earth than on a
mountain top
this has been confirmed with atomic
clocks so
why do we think that it's incorrect or
at least incomplete
the problem is that it does not fit with
what is probably an even
more accurate theory and that is quantum
mechanics
there is no theory that describes how
gravity works
at quantum scales or even very small
scales
nobel laureate stephen weinberg of
electroweak theory fame
said that no one should take general
relativity seriously
for distances shorter than about 10 to
the negative 35 meters
so for example let's take a simple
hydrogen atom
which is composed of one electron and
one proton
quantum theory says that the electron is
in a superposed state
meaning it is in multiple positions at
various distances from the nucleus at
the same time
we only know the probability of finding
it at a particular radius
if we were to measure it since the
electron has mass
according to general relativity it must
curve
space-time but it is in multiple
locations
at the same time then the question is
where is the curvature
is it also at multiple locations at the
same time
we don't know there's nothing in general
relativity
akin to superposition like there is in
quantum mechanics
a second problem occurs in the case of
black holes
general relativity predicts matter and
energy being compressed to an
infinitely small point in space with
supposedly
infinite curvature but the theory breaks
down here if you go back to our original
equation of general relativity
it's easy to see why it breaks down
since the equations treat
space-time mathematically as being flat
at infinitesimally small distances
and then calculate the overall curvature
based on this presumption
a problem occurs when space is seemingly
not flat
at infinitesimally small distances this
occurs
at the singularity since it's supposedly
an infinitesimally small point
so how do you calculate space-time
curvature using
flat space-time with an infinitely
curved point
it's a hole in space-time the
mathematics of general relativity
seem to describe singularities in this
way but mathematical infinities like
this are usually wrong
so there's probably something else going
on here that we just haven't figured out
there are some other issues as well when
you try to integrate
general relativity with quantum
mechanics such as the problem
of information loss due to hawking
radiation
the bottom line is we don't really know
how gravity behaves
at quantum scales general relativity has
been tested to scales of about 1
10th of a millimeter at atomic scales
and below
there's probably some better theory that
will describe it
the problem is that quantum mechanics
works fine
with space-time curvature as the
background the problem with gravity is
that it is not
something that works with space-time
curvature is the background
it is the space-time curvature
space-time curvature
is gravity do we have anything that
comes close to describing gravity at
really small scales
there are two theories that seem to be
promising loop quantum gravity
and string theory and these will be the
subject of my next video
so stay tuned you won't want to miss it
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i will see you in the next video my
friend
[Music]
you