This post is the second part of a series where I explain the basic concepts of special and general relativity and finally build up to a quantitative understanding of black holes and Hawking radiation.
These posts have been taken from the chapters of a research paper that I have recently finished which will be uploaded at the end of this series.
The link to the first part is given below:
Let us begin!
Special Relativity (2) – Vector Algebra, Four Velocity, and Momentum:
First, let us start by talking about what vectors are and how they will be represented using different notations in this chapter.
A vector is a quantity that has both magnitude and direction. For example A displacement vector.
In our case, the displacement vector is used to show the coordinate differences between two events in a frame. It is written as:
This is also written as:
Where Δx^α represents (Δx0, Δx1, Δx2, Δx3).
To represent in another frame, say θ’, we can apply the Lorentz transformations we learned in the previous sections to obtain the coordinates in θ’ with respect to θ.
This is a linear transformation.
It can be written as:
Where A0B represents four numbers such that:
This is further reduced the final equation:
Is formed for a vector x in frame θ’. It can easily be formed for frame θ also.
We will now move on to doing some basic vector algebra.
I will now define something called basis vectors.
Basis vectors are mutually independent vectors that are present in a given vector space.
For a frame θ, they can be defined as:
With these as the basis vectors, A vector X can be written as:
Transformations of basis vectors:
We have found how a vector can be expressed in terms of its basis vectors. This vector is the same no matter which frame we view it from. This means that even though the basis vectors of θ’ and θ are different and so are their coordinates, their sum is the same as the original vector.
This is written mathematically as:
From this we can obtain the law of transformation of basis vectors:
We know from the last section what ABa means. This shows that the basis vectors in θ can be written as a linear combination of those in θ’.
(A detailed proof of this theorem along with extra examples is given in Chapter 2, Section 2.2 of )
We will now define a new vector called the ‘Four Velocity’. The four velocity is tangent to the world line of the particle and is of length equal to one unit of time in the particle’s frame.
The inertial frame of the particle is defined as the frame in which the particle appears to be at rest. Thus, in the way we defined four velocity before, we can see that the velocity can be seen as the time basis vector (e0) of this inertial frame. The symbol for four velocity is U .
Therefore, four velocity can be defined as the e0 basis vector of the rest frame of the particle.
In the case of an accelerated particle, there is no frame with reference to which the particle is at rest at all times. Therefore, we use something called the Momentarily Comoving Reference Frame (MCRF). This is a frame defined such that the particle is momentarily at rest with reference to it. There are infinite MCRFs and can be obtained for any point.
The four velocity at that point is defined as the e0 basis vector of the MCRF at that point.
Four momentum is defined as mU where m is the mass and U is the four velocity.
This vector has components:
(p0, p1, p2, p3)
p0 is written as the Energy of the particle (E).
(An example for four velocity and momentum is given in Chapter 2, Section 2.4 of )
Conservation of four momentum:
We can say that the total momentum of the particles before and after the collision in the system is conserved for any inertial frame.
Although we can define two collision events A and B such that they are before and after say, t = 0 in one frame (θ), they may not be in the same orientation as seen by another frame.
This is shown below. A and B are before and after t = 0 in θ, but they are both behind t’ = 0 in θ’.
However, due to the fact that in each collision the momentum is conserved, regardless of the observer the total momentum is the same. This means that any inertial observer can be chosen, and the momentum vector will be the same
We can also define something called a center of momentum frame. In this frame the total momentum becomes zero.
We know that the magnitude of a vector is given as:
The scalar product of 2 vectors A and B is defined as:
The scalar product is also invariant of the inertial frame.
If A.B = 0, the vectors are said to be orthogonal.
For more details and some examples of scalar product the reader can consult Chapter 2, section 2.5 of .
: Bernard Schutz, ‘A First Course in General Relativity’, Cambridge University Press, 1985