Einstein's Field Equations of General Relativity Explained
Physics Videos by Eugene Khutoryansky
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General Relativity & curved space time: Visualization of Christoffel symbols, Riemann curvature tensor, and all the terms in Einstein's Field Equations. My Patreon page is at https://www.patreon.com/EugeneK
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00:01
Einstein’s Field Equations describe how mass and momentum
00:05
create a curvature in space-time, thereby creating the apparent force of gravity.
00:14
Einstein’s Field equations also describe phenomena such as
00:18
gravitational waves, black holes, and the accelerating expansion of the Universe.
00:26
This represents two of the four dimensions.
00:31
Just as the Earth appears flat to us when we are looking at
00:34
only a tiny portion of it, this curved space-time will also
00:39
appear flat if we zoom in and look at only a tiny section.
00:46
If we approximate this region with the mathematics of flat space-time,
00:51
we can define an axis for one of the three spatial dimensions and an axis for time.
01:01
Different observers in this region moving relative to each other will have different
01:06
orientations for these axes, in accordance to the rules of Special Relativity.
01:18
Suppose we move forward along the time axis
01:21
seen by the observer moving in the direction of the red arrow.
01:34
Our approximation for flat space-time is now no longer valid.
01:45
We correct for this by rotating our red arrow
01:48
along the line 90 degrees to the surface.
02:00
We can now create a new flat space-time approximation for this new region.
02:11
We can use this new approximation to again
02:13
move forward in the direction of the red arrow.
02:19
If we repeat this over and over again, using infinitesimally small time steps,
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we get the path that the object will follow through curved space-time.
02:32
The path appears curved to us, thereby creating the illusion of a gravitational force.
02:41
To describe four dimensional space-time,
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we need four coordinates, which we will refer to as X0, X1, X2, and X3.
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The numbers zero through three are index values, not exponents.
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Einstein’s Field equations remain valid no matter what coordinate system we choose.
03:11
None of the coordinates necessarily need to
03:14
correspond to the time coordinate of any observer.
03:22
Curved coordinates can also be used to describe flat space-time.
03:34
In order to better understand the critical concepts,
03:37
let us first focus on a flat two dimensional space.
03:45
The following arrows show the direction in which our position will change
03:50
if we increment one of the coordinates at various points in space by a small amount.
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These arrows are what we refer to as the basis vectors.
04:17
Let’s move one of the basis vectors by exactly one unit as shown.
04:32
The rate at which the basis vector changes is described by the white vector.
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The white vector can be represented as a
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linear combination of the two basis vectors at this point.
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Let’s replace the numbers with the following symbols,
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each of which has three index values.
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The index value at the top indicates which basis vector it is multiplying.
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The two index values at the bottom indicate
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which basis vector is moving, and in what direction.
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These symbols indicate the rate at which the
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basis vectors are changing at each point in space.
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The basis vectors could be changing because
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we are using a curved coordinate system in a flat space-time.
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Or the basis vectors could be changing because we have a curved space-time,
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even though we have coordinates as straight as possible.
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Suppose we have a curved four dimensional space-time
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with coordinates as straight as possible, where X0 represents time,
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and X1 through X3 represent the three spatial dimensions.
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In situations where Newtonian physics gives accurate approximations,
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the acceleration due to gravity is approximately equal to the following
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symbols related to the curvature of space-time.
07:05
In these cases, the gravitational acceleration is approximately equal to
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the rate at which the basis vector for time changes as it moves forward in time.
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This change is due to curvature.
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In order to know how a vector changes due to curvature,
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we need to know how to move it from one point to another.
07:44
Suppose we move a vector by an infinitesimally small
07:47
amount without changing the direction it is pointing.
08:07
Every time we do this, we need to take the curvature into account
08:11
by rotating the vector along the line 90 degrees to the surface.
08:16
If we repeat this over and over again, we can move a vector over long distances,
08:22
which we refer to as parallel transport.
08:30
If we parallel transport a vector in flat space-time,
08:35
when we return the vector to its starting position,
08:38
it will always still be pointing in the same direction as before.
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The same is not necessarily true in curved space-time.
09:27
We can define the curvature of space-time in terms of
09:30
how much the angle of a vector changes when it returns to its original position.
09:38
But, this angle also depends on the size of the area that the vector travels around.
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When the vector travels around this area, it is rotated by 90 degrees.
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If we double the area, we also double the amount by which the vector is rotated.
10:02
We can therefore define the curvature in terms of the amount by which the vector
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is rotated divided by the size of the area around which the vector traveled.
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Note that when the vector travelled around the area in
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the counter clockwise direction, the vector rotated counter clockwise.
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And when the vector travelled around the area in the clockwise direction,
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the vector rotated clockwise.
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If the vector rotates in the same direction in which it is travelling,
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we will refer to this as positive curvature.
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If the vector rotates in the opposite direction in which it is travelling,
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we will refer to it as negative curvature.
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An example of negative curvature is a saddle point,
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such as the region colored in green.
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The sphere had the same curvature at every point.
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On the other hand, this surface has a different curvature at every point.
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We can refine our definition of curvature by defining the curvature at each point
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in terms of the amount by which the vector is rotated, divided by
11:45
the size of the area, as the size of the area approaches zero.
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The curvature at each point is signified by the scalar variable “R”,
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which is one of the terms of Einstein’s Field Equation.
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Though, this should not be confused with
12:25
the tensor variable that has an R with two index values.
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This tensor variable is also related to curvature,
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but it represents sixteen different numbers at each point in space-time.
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The other two tensor variables in Einstein’s Field equations also each represent
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sixteen different numbers at each point in space time.
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At each point in space time, the relationships between all of these different tensors
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must be as is specified by the equation, regardless of what coordinate system we use.
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But, the individual values of each of the tensor components
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strongly depends on what coordinate system we select.
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In order to simplify the following explanation, let us assume that we are
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not dealing with four dimensional space-time, but just with a two dimensional space.
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Suppose we zoom in close enough for us to be able to
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approximate this region of space as flat.
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Keep in mind that the “1” and “2” are index values, not exponents.
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We want to know the length of the white arrow,
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but the yellow and red arrows are not 90 degrees to each other,
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so we can’t use the Pythagorean Theorem.
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Instead, the length squared will be given by the following equation.
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We can extend this concept from two dimensions to higher dimensions.
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When dealing with four dimensional space-time, this does
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not calculate spatial length, but what we call the “space-time interval.”
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In Special Relativity, the space time interval was defined with
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the following equation, which is the quantity all observers can agree on.
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In General Relativity, this equation is no longer necessarily true
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due to the curvature of space-time.
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Instead, we calculate the space-time intervals inside each infinitesimally
15:36
small region using the tensor shown, which we call the “metric tensor.”
15:45
Let’s now focus on the tensor with the “R.”
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We will first show what this tensor means in three dimensions,
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before moving on to four dimensions.
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We have a coordinate system and the basis vectors
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are not necessarily all 90 degrees to one another.
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As always, keep in mind that the following numbers are index values, not exponents.
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We are going to parallel transport the white vector
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around this infinitesimally small parallelogram.
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If we are in a flat three dimensional space, the vector will still be
16:52
pointing in its original direction when it returns to its original position.
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But, if we are in a curved three dimensional space, then this is not necessarily true.
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We want to know how much the vector has changed,
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which means that we want to know “dV1”, “dV2”, and “dV3.”
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These quantities will be proportional to the product of
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the lengths of the two sides of the parallelogram, multiplied by some other terms.
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These other terms will be a linear combination of the components of the white vector.
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For the missing terms in the following equations,
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let us invent a new variable with the letter R,
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but it will be different from the variables with the letter R that we saw before.
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This new variable will have four index values.
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The top index indicates which component of “dV” it is calculating,
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the first bottom index indicates which component of “V” it is multiplying,
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and the last two bottom indexes indicate
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the two dimensions for the lines of the parallelogram.
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This is shown for only three dimensions,
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but let us suppose that we have extended this concept to four dimensions.
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This term in Einstein’s Field Equations can then be computed as follows.
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There is only one term remaining on the left hand
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side of the equation that we have not yet talked about.
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This is the cosmological constant governing the rate
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at which the expansion of the Universe is accelerating.
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This is the phenomena we associate with what we call dark energy.
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Let us now focus on the right side of the equation.
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Capital “G” is the gravitational constant, and it is
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the same gravitational constant that appears in Newton’s equation for gravity.
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This term with the “C” is the speed of light raised to the power of four.
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This last parameter with the “T” describes
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how much energy and momentum we have at each point.
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In classical physics, the momentum is mass multiplied by velocity.
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If the word “mass” always refers to the “rest mass”,
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which does not change with velocity,
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then this equation needs to be modified by adding the following variable,
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which increases to infinity
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as the velocity of the object approaches the speed of light.
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The vector for velocity has a component in each of the spatial directions.
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We can define the velocity vector as follows.
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We can also say that the object is moving through time at the speed of light,
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and extend the velocity vector to four dimensions as shown.
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Using the four dimensional vector for velocity,
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we can also create a four dimensional vector for momentum,
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which we will symbolize with the variable “p.”
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The variable “m” refers to rest mass.
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The term for momentum in the time dimension can be rearranged as shown.
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We can extend this definition of a four dimensional momentum vector
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to phenomena such as electromagnetic radiation,
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which has momentum and energy, but no mass.
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Suppose we multiply one of the components of the momentum vector
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with one of the components of the velocity vector, and we create a matrix
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showing all the different ways in which we can do this.
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If a volume of space-time contains no matter or energy,
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then all of these terms are zero for that volume.
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At each point we can calculate the values of each of these terms per unit volume,
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and signify the result with the following symbol.
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Provided that none of the objects have velocities close to the speed of light,
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then the term in the upper left hand corner will be far larger
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than any of the other terms, and all the other terms will be negligible by comparison.
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This is because the term in the upper left hand corner has
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the square of the speed of light in its equation, which is an extremely large number.
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This is the equation for the energy density.
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These are the values we have defined, but they are not the same as
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the values in Einstein’s Field Equations, which are symbolized as follows.
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These two sets of values can be calculated from each other by using the metric tensor.
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We now have the final component for Einstein’s Field Equations.
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We can call this the stress – energy – momentum tensor,
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and it tells us the density of energy and momentum at each point in space time.
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It includes the energy and momentum of substances such as matter and
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electromagnetic radiation, but it does not include the energy of curved space-time.
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For example, all the components of the stress energy momentum tensor
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could be zero while there are gravitational waves passing through the region.
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Throughout this video, we have visualized curved space-time as a two dimensional
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sheet embedded in a three dimensional space that obeys Euclidean Geometry.
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However, General Relativity does not require four dimensional space-time
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to be embedded inside a space of more than four dimensions
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in order for the mathematics to work.
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It could just be that the Universe is simply not governed by Euclidean Geometry.