In special relativity, a four-vector (also known as a 4-vector) is an object with four components, which transform in a specific way under Lorentz transformation. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the (1/2,1/2) representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts (a change by a constant velocity to another inertial reference frame).Four-vectors describe, for instance, position xμ in spacetime modeled as Minkowski space, a particle's four-momentum pμ, the amplitude of the electromagnetic four-potential Aμ(x) at a point x in spacetime, and the elements of the subspace spanned by the gamma matrices inside the Dirac algebra.
The Lorentz group may be represented by 4×4 matrices Λ. The action of a Lorentz transformation on a general contravariant four-vector X (like the examples above), regarded as a column vector with Cartesian coordinates with respect to an inertial frame in the entries, is given by
X
′
=
Λ
X
,
{\displaystyle X^{\prime }=\Lambda X,}
(matrix multiplication) where the components of the primed object refer to the new frame. Related to the examples above that are given as contravariant vectors, there are also the corresponding covariant vectors xμ, pμ and Aμ(x). These transform according to the rule
X
′
=
(
Λ
−
1
)
T
X
,
{\displaystyle X^{\prime }=\left(\Lambda ^{-1}\right)^{\textrm {T}}X,}
where T denotes the matrix transpose. This rule is different from the above rule. It corresponds to the dual representation of the standard representation. However, for the Lorentz group the dual of any representation is equivalent to the original representation. Thus the objects with covariant indices are four-vectors as well.
For an example of a well-behaved four-component object in special relativity that is not a four-vector, see bispinor. It is similarly defined, the difference being that the transformation rule under Lorentz transformations is given by a representation other than the standard representation. In this case, the rule reads X′ = Π(Λ)X, where Π(Λ) is a 4×4 matrix other than Λ. Similar remarks apply to objects with fewer or more components that are well-behaved under Lorentz transformations. These include scalars, spinors, tensors and spinor-tensors.
The article considers four-vectors in the context of special relativity. Although the concept of four-vectors also extends to general relativity, some of the results stated in this article require modification in general relativity.
In Gravitation by Misner, Thorne and Wheeler (p.139), stress-energy tensor for a single type of particles with uniform mass m and uniform momentum p (and E = p2 +m2) ½ ) can be written as a product of two 4-vectors,T(E,p) = (E,p)×(E,p)/[V(E2 – p2 )½ ]
Since Einstein equation is G = 8πGT, I am...
I am confused. My understanding is that proper time is used in 4 vectors analysis because proper time is frame invariant. Every other inertial frame will agree on the same time increment if they use the proper time of that one reference frame. But when you do the Lorentz transformation, the...
I was trying to show that the field transformation equations do hold when considering electric and magnetic fields as 4-vectors. To start off, I obtained the temporal and spatial components of ##E^{\alpha}## and ##B^{\alpha}##. The expressions are obtained from the following equations...
If anybody has studied the book:
A First course in String Theory - Barton Zweibach - 2nd edition
This statement is present in 6th chapter of book on pg 110
I am having trouble following a step in a book. So we are given that $$\varphi (x) = \int \frac {d^3k}{(2\pi)^3 2\omega} [a(\textbf{k})e^{ikx} + a^*(\textbf{k})e^{-ikx}] $$
where the k in the measure is the spatial (vector) part of the four-momentum k=(##\omega##,##\textbf{k}##) and the k in the...
Homework Statement
I am reading through Griffiths' Electrodynamics, and I have come to the scenario in the Relativity chapter where in an inertial reference frame ##S##, we have a wire, with positive charges (linear density ##\lambda##) moving to the right at speed ##v##, and negative charges...
Are there any theories in physics that allow for a time component of the various vector quantities besides the x,y,z components? For example the velocity of a particle to have a time component ##v_t## besides the x,y,z components ##v_x,v_y,v_z##
I beg your pardon for not writing out the math explicitly in the following. I started to do so, and realized that it would take me hours to debug my attemps at Latex! And I hope that the my explanation is clear enough that it's not needed.
I've been working through R.E. Turner's "Relativity...
Hello! Why do we need to impose a change on the basis vector, when going from a reference frame to another. I understand that the components of the vector and the basis change using inverse matrices (the components use a matrix and the vector basis the inverse). But the transformation condition...
Homework Statement
Given:
An object at rest with respect to an inertial reference frame S.
2 other inertial reference frames S' and S''.
S' has velocity (vx, vy) = (-.6c, 0) with respect to S.
S'' has velocity (vx, vy) = (-.6c, +.6c) with respect to S.
Assumptions:
If I transform my...
I recently had someone ask me why we use 4-vectors in special relativity and what is the motivation for introducing them in the first place. This is the response I gave:
From Einstein's postulates( i.e. 1. the principle of relativity - the laws of physics are identical (invariant) in all...
We know that 4-vectors are invariants, in the sense that they have the same meaning in all reference frames/coordinate systems. We know they transform by the Lorentz transformation in SR, and have an invariant Minkowski norm (let's not bring in GR at this point unless it becomes necessary). It...
Homework Statement
Homework Equations
Relabelling of indeces, 4-vector notation
The Attempt at a Solution
The forth line where I've circled one of the components in red, I am unsure why you can simply let ν=μ and μ=v for the second part of the line only then relate it to the first part and...
Homework Statement
In a particular inertial frame of reference, a particle with 4-velocity V is observed by an
observer moving with 4-velocity U. Derive an expression for the speed of the particle
relative to the observer in terms of the invariant U · V
Homework Equations...
Homework Statement
[/B]
(i) Prove that dL/dt = 0
(ii) Find the relation between space part and 3-angular momentum vector
(iii)Show that 3-angularmomentum vector is independent of pivot
Homework EquationsThe Attempt at a Solution
[/B]
I'm not sure what part (iii) is trying to get at, but I...
Homework Statement
Let A and B be 4-vectors. Show that the dot product of A and B is Lorentz invariant.
The Attempt at a Solution
Should I be trying to show that A.B=\gamma(A.B)?
Thanks
Hi, can someone confirm those or did I not get the meaning of the 4-vectors indices:
\partial^{\mu}x_{\mu}=4;\partial^{\mu}x^{\mu}=2;\partial^{\mu}x_{\nu}= \delta ^{\mu}_{\nu};\partial^{\mu}x^{\nu}=g^{\mu\nu}
How do you take the cross product of two 4-Vectors?
\vec{r} = \left( \begin{array}{ccc}c*t & x & y & z \end{array} \right)
\vec{v} = \left( \begin{array}{ccc}c & vx & vy & vz \end{array} \right)
\vec{v} \times \vec{r} = ?
Homework Statement
I'm confused about the difference between the following two statements:
\mathbf{V_1}\mathbf{V_2}=V_1V_2\cosh (\phi)
and
\mathbf{V_1}\mathbf{V_2}=\gamma c^2
Where \gamma is the Lorentz factor of the relative speed between the two vectors. Both vectors are time-like.
The...
I'm slightly confused by the difference between covariant and contravariant 4-vectors and how they transform under Lorentz boosts. I'm aware that x_{\mu} = (-x^0 ,x^1, x^2, x^3) = (x_0 ,x_1, x_2, x_3), but when I do a Lorentz transform of the covariant vector, it seems to transform exactly like...
A quote from an old thread reads "Energy is the time component of the momentum 4-vector"
That quote came from a Science Advisor.
Does this mean that time can be either a) substituted for energy in the momentum 4-vector, or b) seen as equivalent to energy in the momentum 4-vector? Hmm...
I'm working through the start of the Quantum Field Theory book by Peskin and Schroeder. The first section deals with an electron and positron colliding to give a positive and negative muon traveling along a line at an angle theta to the line of the e,p collision.(This is using center of mass...
I am trying to figure out the exact meaning of the concepts of 4-vector and relativistic tensor in the Minkowski spacetime. In my understanding, a tensor is a map that assigns an array of numbers to each basis in such a way that certain transformation rules apply. A vector can be viewed as a...
Homework Statement
for a compton-scattering-problem, i want to show that:
\delta(p_{10}+k_1-p_0-k)=p_0\delta(\underline{k}_1(\underline{p}+\underline{k})-\underline{k}\underline{p})
Homework Equations
the momentum- and energy-conversion-law for two particle scattering...
I have a question...
I would like to generically describe a 4-vector locally in space-time. Would i go about that by simply taking a 4-vector and multiplying it by a metric? like
u^{\alpha}=u_{\beta}g^{\alpha\beta}
with u^{\alpha} the new 4-vector in the space specified by the metric?
Homework Statement
If a and b are 4-vectors give the definition of the scalar product a.b and demonstrate its Lorentz invariance
Homework Equations
The Attempt at a Solution
So (with 4-vectors double underlined!)
a.b = a0b0-a1b1-a2b2-a3b3
a' = (a0*gamma - beta*gamma a1 ...
Homework Statement
I am aving a bit of trouble understanding how to show something is a 4 vector. For example K = (v/c, 1/lamdba, 0, 0 ) Show it is a 4-vector. I am not quite sure how to start this.
Similarly I have the amplitude of a wave in frame S described as A=cos[2PI(vt-x/lambda)] and...
Homework Statement
Here is my problem:
DOPPLER EFFECT:
Consider a photon traveling in the x direction. Ignoring the y and z components, and setting c=1, the 4-momentum is (p,p). In matrix notation, what are the Lorentz transformations for the frames traveling to the left and to the right at...
My question deals not with the Lorentz Tranformation itself, but the matrix representation of it:
I see readily how the space-time 4-vector: x^{\mu}=\left( c \ast t, x, y, z\right) transforms approptiately so that x^{\acute{\mu}}=\Lambda_{v}^{\acute{\mu}} \ast x^{\mu}=\left( \gamma \ast...
Homework Statement
Given the binomial expansions for sinh and cosh, so that the acceleration and position 4-vectors reduce to x=1/2 a t^2 for small t.
Homework Equations
Okay, so we have cosh (x) = 1 + x^2/2 + ... and
sinh(x) = x + x^3/3 ...
We have the...
Homework Statement
Hello everyone, thanks for reading.
This might be a little more math than physics (don't run away though!), but it's an excercise on my General relativity text :)
I need to prove that there exists an analog formula, like a*b = abcos(theta) for 3-vectors, only for...
If a and b are four-vectors then are ka and a+b also four-vectors?
My question arises because of the four-velocity, which always has magnitude c. So the sum or difference of two four-velocities will not generally be a four-velocity, but will it be a Lorentz invariant four-vector? If so...
...for a point particle is a 4-vector. Consequence : E^2-c^2(\vec{p})^2 is an invariant
Nevertheless, for a system of particles, the energy momentum is not a 4-vector. See here.
Hence (\Sigma E)^2-c^2(\Sigma \vec{p})^2 is not an invariant. See here
Hi all,
I had a rather poor introduction to special relativity and right now I'm refreshing myself in order to study quantum field theory.
In particular, I've always found the concept of four-vectors confusing. The problem is that from the mathematical point of view 4-vectors are nothing other...
(See thread "Relativistic velocity in the time dimension", post 11)
I delved into this some time ago. An interesting source I found in Feynman's Lectures on Physics, volume II, in particular because of his non-standard way of approaching these things. Do you have suggestions for additional...