# What is 4-vector: Definition and 47 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

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.

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9. ### A How can curl of 4-vector or 6-vector be writen?

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10. ### Prove Lorentz invariance for momentum 4-vector

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12. ### Is this a correct 4-vector identity?

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13. ### Klien-Gordon equation to solve 4-vector problem? (Particle)

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14. ### 4-momentum and relative velocity

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16. ### Difference between these 4-vector derivatives?

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17. ### Compute Magnitude of 4-Vector: A Step-by-Step Guide

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18. ### Describing 3-component spin 1 with a 4-Vector

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19. ### Deriving Equality with Spin 4-Vector: Help Needed

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20. ### Zeroth component of Spin 4-Vector

The spin 4-vector is defined in the rest frame of the particle as s^{\mu}= (0, \vec{s}). Why is the zeroth component of the same zero in this frame?
21. ### What are the Lorentz transformation tensors used for?

Hi all, I got a 3 part Qs: γ=1/√1-v^2-c^2 Part A Homework Statement Consider the Lorentz transformation tensor Matrix Row 1: [ γ 0 0 -vγ/c] Row 2: [ 0 1 0 0 ] Row 3: [ 0 0 1 0 ] Row 4:-[vγ/c 0 0 γ ] for transforming 4-vectors from frame S...
22. ### What is the correct transformation for a 4-vector in special relativity?

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23. ### Is acceleration due to gravity a 4-vector?

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24. ### Why is it obvious that the derivative of a 4-vector is also a 4-vector

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25. ### What is the dielectric momentum density 4-vector?

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26. ### Classical vs. Quantum interpretation of spin 4-vector

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27. ### Transforming a 4-vector integration measure

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28. ### Meaning and derivation of 4-vector

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29. ### Understanding Lorentz Transformation of Spin 4-Vector

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30. ### 4-vector law of motion in different inertial frames

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31. ### What is the angular-momentum 4-vector?

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?
32. ### Nonlinear Addition of Four-Vectors in Relativity: Exploring the Inconsistencies

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33. ### Determining 4-vector character of a 4-tuple

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
34. ### Determining 4-vector character of a 4-tuple

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?
35. ### Transformation of k_y in the wave 4-vector

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36. ### How do you know what goes together to form a 4-vector?

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...
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38. ### Transformation to flip handedness of a 4-vector

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39. ### Is the 4-velocity always a space-tipe 4-vector?

Is the 4-velocity always a space-tipe 4-vector?
40. ### Invarients for 4-vector quantites.

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...
41. ### Sign convention in the space-time 4-vector

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?
42. ### Spin polarization 4-vector

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...
43. ### Understanding 4-Vector Momentum^2

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44. ### Definition of 4-vector quantities

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...
45. ### Names of 4-Vector Norms & Physical Quantities

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...
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47. ### What is the significance of a 4-vector's magnitude being c?

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