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Let one also introduce the four-dimensional spacetime metric , where Greek indices span the usual four dimensions of space and time; a 4-vector identified with the electromagnetic vector potential; and a scalar field . Then decompose the 5D metric so that the 4D metric is framed by the electromagnetic vector potential, with the scalar field at ...
Many distinct vector fields can give rise to the same congruence of curves, since if is a nowhere vanishing scalar function, then and = give rise to the same congruence. However, in a Lorentzian manifold, we have a metric tensor , which picks out a preferred vector field among the vector fields which are everywhere parallel to a given timelike ...
Frame fields of a Lorentzian manifold always correspond to a family of ideal observers immersed in the given spacetime; the integral curves of the timelike unit vector field are the worldlines of these observers, and at each event along a given worldline, the three spacelike unit vector fields specify the spatial triad carried by the observer.
A vector field V defined on an open set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that = = (,,, …,). The associated flow is called the gradient flow , and is used in the method of gradient descent .
A field can be classified as a scalar field, a vector field, a spinor field or a tensor field according to whether the represented physical quantity is a scalar, a vector, a spinor, or a tensor, respectively. A field has a consistent tensorial character wherever it is defined: i.e. a field cannot be a scalar field somewhere and a vector field ...
In a classical field theory, the physical states are sections of a Poincaré-equivariant vector bundle over Minkowski space. The equivariance condition means that the group acts on the total space of the vector bundle, and the projection to Minkowski space is an equivariant map. Therefore, the Poincaré group also acts on the space of sections.
Archaeologists believe they may have discovered the final location of Noah’s Ark on Turkey’s Mount Ararat. Soil samples from atop the highest peaks in Turkey reveal human activity and marine ...
One can express the complex scalar field theory in terms of two real fields, φ 1 = Re φ and φ 2 = Im φ, which transform in the vector representation of the U(1) = O(2) internal symmetry. Although such fields transform as a vector under the internal symmetry , they are still Lorentz scalars.