What is Four vectors: Definition and 32 Discussions

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



{\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









{\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.

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  1. K

    I Understand 4-Vectors & Spacetime: Hartle Gravity Chapter 5

    Hartle, gravity. Chapter 5 "A four-vector is defined as a directed line segment in four-dimensional flat spacetime in the same way as a three-dimensional vector (to be called a three-vector in this chapter) can be defined as a direcied line segment in three-dimensional Euclidean Space"For...
  2. patric44

    Confusion about four vector notation

    hi guys I am trying to learn special relativity and relativistic quantum mechanics on my own and just very confused about the different conventions used for the notation!?, e.g: the four position 4-vector some times denoted as $$ x_{\mu}=(ct,-\vec{r})\;\;or\;as\;x_{\mu}=(ict,\vec{r}) $$ or...
  3. dkhurana

    I Four Velocity Vector: why divide by time according to the particle?

    So I understand that time is now part of the four vector, and so dividing delta X by delta t (time according to me), would produce just c as the first dimension of the vector, which gives us no intuition as to how fast time is moving for the observer, so is not useful. I understand why we...
  4. sophiatev

    I Four-Vector in Spacetime: Invariance Across All Reference Frames?

    I know that the mathematical form of the line element of spacetime is invariant in all inertial reference frames, namely $$ds^2 = -(cdt^2) + dx^2 + dy^2 + dz^2$$ From what I understand, the actual spacetime distance between two events is the same numerical quantity in all reference frames...
  5. S

    How Does the Relativistic Rocket Equation Describe Velocity?

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  6. C

    I Levi-Civita Contraction Meaning: Undergrad Research

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  7. P

    Dust in special relativity - conservation of particle number

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  8. S

    Contravariant Four-gradient ESN in Wikipedia appears wrong

    Homework Statement I am self studying relativity. In Wikipedia under the four-gradient section, the contravariant four-vector looks wrong from an Einstein summation notation point of view. https://en.wikipedia.org/wiki/Four-vector Homework Equations It states: E0∂0-E1∂1-E2∂2-E3∂3 = Eα∂α...
  9. F

    I Why is energy not Lorentz invariant?

    As I understand it, since space-time is modeled as a four dimensional manifold it is natural to consider 4 vectors to describe physical quantities that have a direction associated with them, since we require that physics should be independent of inertial frame and so we should describe it in...
  10. F

    I Motivation for the usage of 4-vectors in special relativity

    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...
  11. F

    I Zeroth component of 4-momentum & energy-momentum relation

    As I understand it one is forced to use 4-vectors since we require objects that transform as vectors under application of Lorentz transformations and 3-vectors do not (technically they do under rotations, but not under boosts). Equivalenty, if one starts off with Minkowski spacetime from the...
  12. J

    Four momentum proton-proton scattering question

    Homework Statement In a fixed target experiment a particle of mass M and kinetic energy T strikes a stationary particle of mass M. By evaluating s, t and u in the laboratory frame and using the above relation, or otherwise, show that the kinetic energy T' of the particle scattered elastically...
  13. tomdodd4598

    Can the Dot Product of Four Vectors Maintain Positive Component Signs?

    Hi there, I understand that taking the dot product of two four vectors automatically applies the metric tensor to the second vector. Is there a way to take write the dot product, using vector notation, in a way which keeps the signs of all of the components positive? Thanks in advance.
  14. kontejnjer

    Timelike and spacelike vectors

    Homework Statement Show that if x^{\mu} is timelike and x^{\mu}y_{\mu}=0, y^{\mu}\neq 0, then y^\mu is spacelike. Homework Equations ds^2=\\>0\hspace{0.5cm}\text{timelike}\\<0\hspace{0.5cm}\text{spacelike}\\0\hspace{0.5cm}\text{lightlike} metric is diag (+---) The Attempt at a Solution Don't...
  15. U

    How Does a Relativistic Photon Rocket Calculate Acceleration and Speed?

    Homework Statement (a) Show ##a = \frac{a_0}{\gamma^3}##. (b) Find proper acceleration of rocket (c) Find speed as a function of time. (d) Find acceleration of second rocket. Homework EquationsThe Attempt at a Solution Part(a) 4-vector acceleration is given by ##\gamma^2 \left[...
  16. S

    Subscripts and superscripts in four vectors

    Homework Statement I'm having trouble with understanding four vectors in particle physics. I'm reading this wikipedia page,http://en.wikipedia.org/wiki/Einstein_notation, and its telling me that ## v^\mu= \begin{pmatrix} \mu_0 \\ \mu_1 \\ \mu_2 \\ \mu_3 \end{pmatrix} ## and ## v_\mu=...
  17. U

    Four-potential of moving charge

    Homework Statement (a) Find the four-vector potential of a moving charge (b) Find source time and z-component of electric field (c) Find electric potential to first order of x and hence electric field[/B] Homework EquationsThe Attempt at a Solution Part(a) [/B] \phi = \frac{q}{4\pi...
  18. U

    Deriving Equation 4.8 in Special Relativity: Tips and Tricks

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  19. S

    Four vectors and lorentz transformations

    1. Consider a four vector x^{\mu}, that is timelike (i.e x^{2}>0. show that it is always possible to find a frame where the coordinates of x are of the form (x^{0'},0). Determine the lorentz transformation relating the initial frame to this particular frame 3. I figured that assuming that the...
  20. S

    Using Four vectors & Special Relativity

    Hey, My question concerns parts (a), (b) and (c) in the below. In part (a) we're asked to find the energy, velocity and momentum of the incoming particle AND then to find gamma, however I know the energy of the incoming particle is simply 3MeV but I'm not sure how to find the velocity...
  21. T

    Angles in momentum four vectors

    Homework Statement A particle decays to two photons. In the rest frame the two photons are emitted on the x-y plane, in opposite directions along a line that forms an angle alpha with the the x axis. Derive the momentum four vector of the two photons in the lab frame. Homework Equations...
  22. G

    Transforming Four Velocities with the Lambda Matrix

    In SR, I know we transform three velocities using the einstein velocity addition law. However, in the case of four velocities, do they simply transform the same way as four positions? With the lambda matrix?
  23. E

    Proca Lagrangian (Math troubles with four vectors)

    I'm reading Griffith's Elementary particles and I'm stuck on the math for one of the examples, could anyone show me what I'm missing or point me in the right direction? I attached a pdf (of the word doc I was using) that shows what I did so far since I'm really bad with LaTeX and it would've...
  24. N

    Four vectors in complex notation

    Hi, I recently completed a course on Electrodynamics, with a short introduction to special relativity (and E/B fields in special relativity). Both in the book we were asked to study, and the final exam, the four vectors we used used real components, with the time coordinate being the...
  25. Z

    Particle Dacay with four vectors (not too hard)

    Problem: Given that the \Lambda has Q=0, B=1, S=-1 and the K(+) has Q=1, B=0, S=1 and the K− is its antiparticle, show that the following reactions are suitable for making strange particles. Calculate the threshold kinetic energy of the beam for producing these reactions in a ‘fixed target’...
  26. P

    How to Calculate the Limit of Fraction Involving Four Vectors with q --> 0

    I should calculate the limit of the following fraction \dfrac{- (pq) p^{2}}{-pq \pm \sqrt{(pq)^{2} - p^{2} q^{2}}} . with q --> 0, but I don't know how to do that. p and q are two four-vectors, so we have: pq = p_{\mu} q^{\mu} and so on. Does anyone have an idea? Or at least: Is...
  27. S

    How can the orthonormal basis of four vectors be found?

    Homework Statement How can I find the orthonormal basis of four vectors? The vectors are: (0, 3, 0, 4), (4, 0, 3, 0), (4, 3, 3, 4) and (4, -3, 3, -4). The Attempt at a Solution I am not sure, whether I should use Gram-Schmidt process or the process of finding eigenvalues, eigenvectors and...
  28. A

    What is the magnitude of the vector sum of the four vectors

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  29. J

    Studying special relativity and using the idea of four vectors

    We are currently studying special relativity and using the idea of four vectors. The position 4-vector has been defined in class as \vec{X} = (ct, \vec{r}) \ \ \ \ \ \ \ where \ \vec{r} = (x, y, z) \ \ (ie: the usual 3 space position vector) and the velocity 4-vector is then...
  30. S

    Four Vector Analysis: Calculating Particle Velocity and Ratio in Particle Decay

    A particle of rest mass Mo is at rest in the laboratory when it decays into three identical particles, each of rest mass mo. Two of the particles (1 and 2) have velocities and directions as shown. [1 is 3c/5 at -90degrees, 2 4c/5 at 180degrees] (a) Calculate the direction and the speed of...
  31. E

    Four vectors and Lorentz invariance

    Does anyone know where I can find a mathematical proof that the norm of any four-vector is Lorentz invaraint?