What is 4-vectors: Definition and 35 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. E

    A Can Einstein Tensor be the Product of Two 4-Vectors?

    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...
  2. e2m2a

    I Proper Time & 4-vectors: Clarification Needed

    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...
  3. U

    Trying to understand electric and magnetic fields as 4-vectors

    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...
  4. AhmadKhaqan

    A Constant Scalings of 4-Vectors" - Zweibach, 2nd Ed.

    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
  5. A

    I Invert a 3D Fourier transform when dealing with 4-vectors

    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...
  6. CDL

    Einstein Velocity Addition for a Moving Charge in a Wire

    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...
  7. Delta2

    I Does Physics Allow for a Time Component in Vector Quantities?

    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##
  8. K

    I Compton Scattering w/Moving Electron: Turner's Eq 5.29

    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...
  9. S

    I Transform Bases for 4-Vectors in Ref. Frames

    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...
  10. M

    Trouble with 2 step velocity transformation in SR

    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...
  11. 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...
  12. Ken G

    A Measuring 4-Vectors: Is It Possible?

    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...
  13. R

    Symmetric rank-2 tensor, relabelling of indices? (4-vectors)

    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...
  14. S

    Derive Particle Speed in Terms of Invariant U.V | Relative 4-velocities Homework

    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...
  15. U

    3-angular momentum : independent of pivot?

    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...
  16. C

    Proving the dor product of 4-vectors is Lorentz invariant

    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
  17. N

    Contractions of indices of the 4-vectors

    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}
  18. Philosophaie

    Cross product of two 4-Vectors

    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} = ?
  19. K

    The scalar product of 4-vectors in special relativity

    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...
  20. E

    Transforming co- and contravariant 4-vectors

    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...
  21. jaketodd

    Time in Physics 4-Vectors: Is Time Included in the Position Four-Vector?

    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...
  22. J

    Polarization 4-vectors to get matrix element in QFT

    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...
  23. S

    What Are 4-vectors and Relativistic Tensors?

    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...
  24. U

    Deltafunktion of 4-vectors for energy and momentum coversation

    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...
  25. jfy4

    Describing 4-vectors in space-time

    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?
  26. C

    How Is the Scalar Product of 4-Vectors Defined and Proven Lorentz Invariant?

    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 ...
  27. C

    Solving 4-Vectors and Wave Amplitudes in Different Frames

    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...
  28. K

    Doppler Effect: Solving 4-Vectors Problem

    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...
  29. Y

    Question about applying the Lorentz Transformation to velocity 4-vectors

    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...
  30. J

    Relativity: position/acceleration 4-vectors to Newton's laws

    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...
  31. C

    Proving 4-vector Analog Formula for Lorentz Boost

    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...
  32. Dale

    Are Scaled and Summed Four-Vectors Still Four-Vectors?

    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...
  33. 1

    Energy-momentum for a point particle and 4-vectors

    ...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
  34. J

    Understanding the Concept of 4-Vectors in Physics: A Mathematical Perspective

    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...
  35. M

    What Are the Differences Between Electric and Magnetic Field 4-Vectors?

    (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...