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.
So the line element is given by $$ ds^2 = (1- \frac{R_s}{r})dt^2 - (1- \frac{R_s}{r})^{-1}dr^2 - r^2d\Omega ^2$$
The object is orbiting at constant radius ##r## in the plane ## \theta = \frac{\pi}{2}##. I am supposed to find the values of ##a## and ##b## in the 4-velocity given by: $$U =...
I see this question in PSE and it seemed interesting. The Question is like this,
Consider a semi-Riemannian manifold which of these statements is false:
1) All vectors on the light-cone are light-like, all vectors in the interior of the light-cone are time-like and all vectors in the exterior...
I used the two equations above to solve for u_x and u_y and got u = 0.987c, where u_parallel = u_x and u_perpen = u_y. I wonder if I can use velocity four-vectors to solve this problem. Modify η'μ = Λμνην so we can use it for velocity addition?
Hello! I am reading some notes on Lorentz group and at a point it is said that the irreducible representations (IR) of the proper orthochronous Lorentz group are labeled by 2 numbers (as it has rank 2). They describe the 4-vector representation ##D^{(\frac{1}{2},\frac{1}{2})}## and initially I...
A stationary Monopole exist at the Origin. I am trying to get an understanding of the time derivative of a Four-Vector of ##\vec{B}## and ##\vec{E}##
##\vec{B} = B_r \hat r + B_\theta \hat \theta + B_\phi \hat \phi + \frac{1}{c}B_t \hat t##
##\vec{E} = E_r \hat r + E_\theta \hat \theta +...
Could someone tell me if this 4-Vector cross product is correct:
i j k t
dx dy dz 1/c*dt
Ex Ey Ez Et
=[(dy(Ez)-dz(Ey))-(dy(Et)-1/c*dt(Ey))+(dz(Et)-1/c*dt(Ez))]*i
-[(d(E)-d(E))-(d(E)-d(E))+(d(E)-d(E))]*j...
I hope I'm not violating Forum protocol, again.
I tried posting this question in the Homework section but it got locked for violating homework protocol.
My understanding for the relativistic transformation of a velocity u to u' is given by
$$
\begin{bmatrix}
\gamma_{u'} \\
\gamma_{u'} u'_x...
Homework Statement
Given
Space Time Coordinate of object <t, x, y, z> = <0, 0, 0, 0>
Velocity of object as Vector <betaX, betaY, betaZ> = <.866, 0 ,0>
Velocity of target reference frame as Vector <betaX, betaY, betaZ> = <-.866, 0 ,0>
Transform velocity of object to the...
How can a curl of 4-vector or 6-vector be writen? Let's say that we have a 4-vector A4=(a1,a2,a3,a4)
how can we write in details the ∇×A4
Can we follow the same procedure for 6-vector?
Homework Statement
I am meant to show that the following equation is manifestly Lorentz invariant:
$$\frac{dp^{\mu}}{d\tau}=\frac{q}{mc}F^{\mu\nu}p_{\nu}$$
Homework Equations
I am given that ##F^{\mu\nu}## is a tensor of rank two.
The Attempt at a Solution
I was thinking about doing a Lorents...
Assuming a four velocity
##u_{\nu}=(1,0,0,0)##
we can use the Maxwell Energy-Momentum Tensor to build a 4-vector in the following way
##P^{\mu}=u_{\nu}T^{\mu\nu}=\left(\frac{E^{2}+B^{2}}{2},\mathbf{E}\times\mathbf{B}\right)##
So, we have a vector whose time component is the energy density of...
In our particle physics lecture this term comes up often, it doesn't look right to me but the lecturer uses it so it must be:
##{\partial }^{2}A^{\mu} = - {\partial }_{\mu}{\partial }^{\mu}A^{\mu}+ {\partial }_{\mu}{\partial }^{2}A^{\mu}##
I understand if you have:
##F^{\mu v} = {\partial...
Homework Statement
Homework Equations
KG Equation
##\left ( \delta ^{2} + \frac{m^{2}c^{2}}{\hbar^{2}}\right )\Psi = 0##
The Attempt at a Solution
To solve this problem I'm not sure whether to use the KG equation OR apply the operator given in the question (I don't know how to do this)
I...
Homework Statement
In a particular frame of reference a particle with 4-momentum ##P_p## is observed by an observer moving with 4-momentum ##P_o##. Derive an expression for the speed of the particle relative to the observer in terms of the invariant ##P_p.P_o##
I am completely stuck on this...
Some subtleties of the metric tensor are just becoming clear to me now. If I take ##g_{\mu\nu}=diag(+1,-1,-1,-1)##
and want to write ##\partial_\mu\phi^\mu##, it would be ##\partial_0\phi^0 -\partial_i\phi^i##, correct? ##\phi## is a 4-vector.
Hey everyone,
So I've come across something in my notes where it says that these two Lagrangian densities are equal:
\mathcal{L}_{1}=(\partial_{mu}\phi)^{\dagger}(\partial^{\mu}\phi)-m^{2}\phi^{\dagger}\phi
\mathcal{L}_{2}=-\phi^{\dagger}\Box\phi - m^{2}\phi^{\dagger}\phi
where \Box =...
Hello there,
Given any 4-vector ##x = (x_0,x_1,x_2,x_3)##, hvor do I compute its magnitude? I couldn't find a simple explanation online.
For example I want to compute the magnitute of the momenergy 4-vector of a photon traveling in the positive x-direction where ##E## is the energy of the...
For a spin 1 field described by a 4-vector, the condition m \partial_\mu A^\mu=0 can be derived from the equations of motion. This condition reduces the degrees of freedom from 4 to 3 for a massive particle (when m is not equal to zero).
However, for the photon, \partial_\mu A^\mu=0 has to be...
Hello everyone, I have a problem with deriving following equality:
\begin{equation}
u^{s}(p) \bar{u}^{s}(p) = 1/2 ((\slashed{p} + m)(1+\gamma^5 \slashed{s}))
\end{equation}
where s is spin 4-vector. I know how to calculate this tensor product when there is spin sum in front of it, but without...
We can write the geodesic condition as
\frac{d^2 x^\alpha}{d\tau^2}={-\Gamma^\alpha}_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}
On the RHS we have two contravariant vectors ( or (0,1) tensors ) contracted with the connection. But the connection is NOT a tensor. So is the LHS a...
Several questions in a kind of stream of consciousness format...
I was going over some special relativity notes and I'm not sure why I should buy this. The statement was saying that we can introduce a four-force by differentiating the four-momentum with respect to proper time. That is,
K^μ...
Assume that there is a dielectric material with a mass density of ρ0 observed in the dielectric-rest frame. And further, it is assumed that observed in the lab frame, v(x,y,z,t) is the velocity distribution, β=v/c is the normalized velocity, and γ=(1-β2)-1/2 is the relativistic factor, with c...
I have a few basic questions about the Pauli-Lubanski spin 4-vector S.
1. I've used it in quantum mechanical calculations as an operator, that is to say each of the components of S is a matrix operator that operates on an eigenvector or eigenspinor. But my question is about the utility of S...
Hello,
I have a particular derivation of a four-vector integration measure, basically changing the measure to some related more useful measure - but I'd like to do this in 3-vector notation. Here it is, from the integral...
Meaning of ct in Lorentz transformation -
In Lorentz transformation matrix, the first column is defined as - ct, not t itself. Is it because ct satisfies the units of x, y, z? Or, simpler Lorentz transformation matrix will be derived? The idea of 'ct', instead of t, is quite abstract for me...
I have a problem understanding the Lorentz transformation of the spin. The spin 4-vector is defined in the rest frame of the particle as
s^{\mu} = (0, \vec{s})
and then boosted in any other frame according to
s'^{\mu} = (\gamma \vec{\beta} \cdot \vec{s}, \vec{s} +...
Newton's second law of motion is given in Minkowski space by
\bar{F}=m(c\gamma\dot{\gamma}, \gamma\dot{\gamma}\tilde{v}+\gamma^{2}\tilde{a})
where \dot{\gamma}=\frac{d\gamma}{dt}=\frac{\gamma^{3}}{c^{2}}\tilde{v}\cdot\tilde{a} and \tilde{v}(t) and \tilde{a}(t) the 3-velocity and...
Uh, the title pretty much says it: I'm wondering what the 4-vector analog to the classical 3-angular momentum is. Also, is the definition
L = r \times p
still valid for the 3-angular momentum in special relativity?
Since the "tensors" of relativity are defined with respect to the tangent spaces of a pseudo-Riemannian manifold, which include velocity vectors (i.e. timelike tangent vectors), this might be taken to suggest (carpet-from-under-feet-ingly) that none of the objects called tensors in relativity...
Suppose you're given a 4 tuple and told that its scalar product with any 4-vector is a lorentz scalar. How do I show that this implies the 4-tuple is a 4-vector?
Thanks
Suppose you're given a 4 tuple and told that its scalar product with any 4-vector is a lorentz scalar. How do I show that this implies the 4-tuple is a 4-vector?
I'm checking how k_y in the wave 4-vector transforms, but not getting what I expect:
The wave 4-vector is defined as (\omega/c,\ \textbf{k} ) where \textbf{k} = 2\pi/ \boldsymbol{\lambda},\ \textbf{u} is the velocity of propagation of the plane wave
Let s' travel, as usual, along the...
I've been studying relativity and standard model physics, and I don't understand how it is determined what 'things' go together to form a 4-vector. For example, there is the familiar energy momentum 4-vector, the charge-current density four vector, the phi-A (scalar/vector potential) 4-vector...
Homework Statement
OK - the problem is thus:
In an inertial frame two observers (called a & b) travel along the positive x-axis with velocities Va and Vb. They encounter a photon traveling in the opposite x-direction. Without using the Lorentz transformations, show that the ratio of the...
Hi all,
Long time no see! Had an interesting (non-hw question lol) posed to me as to what a general transform would be to turn a right-handed system to a left handed system in 4-space. I realize that there is no analogy of a vector cross product to use for 4-vectors (which is what i'd...
I was summarizing for myself the various four-vectors of mechanics:
\begin{align*}
x &= ct + \mathbf{x} \\
V &= \frac{dx}{d\tau} = \gamma(c + \mathbf{v}) \\
P &= m V = E/c + \gamma\mathbf{p} \\
f &= m\frac{d^2 x}{d\tau^2} = m\frac{d V}{d\tau} \\
\end{align*}
where...
What is the rationale for the sign convention in the space-time 4-vector? How is it related to the sign convention in the energy-momentum 4-vector, if at all?
Can someone explain to me why the spin polarizations of a particle can be represented by the four unit 4-vectors, ie partial derivative vector fields with respect to each coordinate function?
I also do not understand why the probability of a particle to be created or absorbed with spin...
I'm having a problem understanding this:
P^2=P_{\mu}P^\mu=m^2
If we take c=1.
Here is what bothers me:
P(E, \vec{p})=E^2-(\vec{p})^2
Now, I assume that E=mc^2, and for c=1, E^2=m^2? Is that correct?
And I don't know what p^2 is, I look at it as:
(\vec{p})^2=m^2(\vec{v})^2...
Hi,
I just want to share my curiosity
in the definition of 4-vector quantities such as world line 4-vector x^alpha, 4-velocity vect, gauge potential etc. the ones with subscript for indices usually have the first component with negative sign and the ones with superscript for indices have all...
Are there names for the Lorentz invariant norm of the four-potential and four-current? I assume that they are invariant under the transformations. Also, is it true that any physical quantities which form a four-vector have an invariant quantity associated with them (i.e. the norm of the...
I made a new web page to describe/define the electric field 4-vector to someone. I thought I'd post a link to that page here. Some of you might find it interesting since most people only think of the electric field as being only the components of a 2-tensor. See...
My direct question is at the bottom of this post, but I thought I would set the scene. Skip the first part if you wish.
The magnitude of any 4-vector is scalar and therefore the same in all frames. For the velocity vector
u = (\gamma c, \gamma v)
where v is a 3-vector velocity term, we...