# 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

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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Is it generally the case even with light like paths that ##\dot t>0##?
2. ### I Understand 4-Vectors & Spacetime: Hartle Gravity Chapter 5

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3. ### Confusion about four vector notation

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

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

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12. ### I Zeroth component of 4-momentum & energy-momentum relation

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13. ### Four momentum proton-proton scattering question

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14. ### 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.
15. ### Timelike and spacelike vectors

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16. ### How Does a Relativistic Photon Rocket Calculate Acceleration and Speed?

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18. ### Four-potential of moving charge

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20. ### Four vectors and lorentz transformations

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21. ### Using Four vectors & Special Relativity

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22. ### Angles in momentum four vectors

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23. ### Transforming Four Velocities with the Lambda Matrix

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24. ### Proca Lagrangian (Math troubles with four vectors)

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25. ### Four vectors in complex notation

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28. ### How can the orthonormal basis of four vectors be found?

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30. ### Studying special relativity and using the idea of four vectors

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31. ### Four Vector Analysis: Calculating Particle Velocity and Ratio in Particle Decay

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32. ### 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?