**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

Concept# Four-velocity

Summary

In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetime that represents the relativistic counterpart of velocity, which is a three-dimensional vector in space.
Physical events correspond to mathematical points in time and space, the set of all of them together forming a mathematical model of physical four-dimensional spacetime. The history of an object traces a curve in spacetime, called its world line. If the object has mass, so that its speed is necessarily less than the speed of light, the world line may be parametrized by the proper time of the object. The four-velocity is the rate of change of four-position with respect to the proper time along the curve. The velocity, in contrast, is the rate of change of the position in (three-dimensional) space of the object, as seen by an observer, with respect to the observer's time.
The value of the magnitude of an object's four-velocity, i.e. the quantity obtained by applying the metric tensor g to the four-velocity U, that is 2 = U ⋅ U = gμνUνUμ, is always equal to ±c2, where c is the speed of light. Whether the plus or minus sign applies depends on the choice of metric signature. For an object at rest its four-velocity is parallel to the direction of the time coordinate with U0 = c. A four-velocity is thus the normalized future-directed timelike tangent vector to a world line, and is a contravariant vector. Though it is a vector, addition of two four-velocities does not yield a four-velocity: the space of four-velocities is not itself a vector space.
The path of an object in three-dimensional space (in an inertial frame) may be expressed in terms of three spatial coordinate functions xi(t) of time t, where i is an index which takes values 1, 2, 3.
The three coordinates form the 3d position vector, written as a column vector
The components of the velocity (tangent to the curve) at any point on the world line are
Each component is simply written
In Einstein's theory of relativity, the path of an object moving relative to a particular frame of reference is defined by four coordinate functions xμ(τ), where μ is a spacetime index which takes the value 0 for the timelike component, and 1, 2, 3 for the spacelike coordinates.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related courses (30)

Related units (8)

Related concepts (21)

Related people (31)

Ontological neighbourhood

Students will learn the principles of mechanics to enable a better understanding of physical phenomena, such as the kinematics and dyamics of point masses and solid bodies. Students will acquire the c

Continuum conservation laws (e.g. mass, momentum and energy) will be introduced. Mathematical tools, including basic algebra and calculus of vectors and Cartesian tensors will be taught. Stress and de

The course presents the design, control, and applications of legged robots. It gives a review of different types of legged robots (including two-, four- and multi-legged robots), and an analysis of di

In the theory of relativity, four-acceleration is a four-vector (vector in four-dimensional spacetime) that is analogous to classical acceleration (a three-dimensional vector, see three-acceleration in special relativity). Four-acceleration has applications in areas such as the annihilation of antiprotons, resonance of strange particles and radiation of an accelerated charge. In inertial coordinates in special relativity, four-acceleration is defined as the rate of change in four-velocity with respect to the particle's proper time along its worldline.

In the special theory of relativity, four-force is a four-vector that replaces the classical force. The four-force is defined as the rate of change in the four-momentum of a particle with respect to the particle's proper time: For a particle of constant invariant mass , where is the four-velocity, so we can relate the four-force with the four-acceleration as in Newton's second law: Here and where , and are 3-space vectors describing the velocity, the momentum of the particle and the force acting on it respectively.

In differential geometry, the four-gradient (or 4-gradient) is the four-vector analogue of the gradient from vector calculus. In special relativity and in quantum mechanics, the four-gradient is used to define the properties and relations between the various physical four-vectors and tensors. This article uses the (+ − − −) metric signature. SR and GR are abbreviations for special relativity and general relativity respectively. indicates the speed of light in vacuum. is the flat spacetime metric of SR.

, , , , , , , , ,

Related publications (113)

Related lectures (129)

The non-dimensional energy of starting vortex rings typically converges to values around 0.33 when they are created by a piston-cylinder or a bluff body translating at a constant speed. To explore the limits of the universality of this value and to analyse ...

2022The perpendicular propagation velocity of turbulent density fluctuations is an important parameter in fusion plasmas, since sheared plasma flows are crucial for reducing turbulence, and thus an essential input parameter for turbulent transport simulations. ...

Paolo Ricci, Baptiste Jimmy Frei, Antoine Cyril David Hoffmann

We present a convergence study of the gyromoment (GM) approach, which is based on projecting the gyrokinetic distribution function onto a Hermite–Laguerre polynomial basis, focused on the cyclone base case (CBC) (Lin et al., Phys. Rev. Lett., vol. 83, no. ...

2023Interplanetary Trajectories: Concepts and Strategies

Explores interplanetary trajectories, including departure strategies, sphere of influence, and orbit insertion around the destination planet.

Linear Momentum Balance

Analyzes linear momentum balance equations and explores the consequences of continuity and divergence in angular momentum conservation.

Design Discussions, Documentation

Explores design discussions and documentation in software development, emphasizing scientific programming and code documentation tools like Doxygen and Sphinx.