What is metric tensor in special relativity?
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation.
Does the metric tensor commute?
As general tensors, metric tensors are not commutative in general (try in dimension 2 for example to construct two symmetric matrices that do not commute).
Is the metric tensor constant?
The co-variant derivative of the metric tensor is always zero, no matter the coordinate system, that is the definition of a tensor. In euclidean coordinates the metric tensor does change when you move around.
How does the metric tensor transform?
A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor.
How does the metric transform?
Metric transforms give new distances as a functions of given metrics (or given distances) on the same set X. A metric so obtained is called a transform metric.
Are metric tensors invariant?
It is a basic result of special relativity that the Minkowski metric tensor is invariant under the Lorentz group.
How are tensors used in general relativity?
Tensor fields in general relativity The notion of a tensor field is of major importance in GR. For example, the geometry around a star is described by a metric tensor at each point, so at each point of the spacetime the value of the metric should be given to solve for the paths of material particles.
Why is metric tensor covariant?
Thus the metric tensor gives the infinitesimal distance on the manifold. The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor.
Is the metric tensor unique?
There is a unique metric tensor φ∗g on V that makes φ an isometry, i.e. φ is a function that preserves distance.
Do we write a metric tensor as a matrix?
Metric tensor of spacetime in general relativity written as a matrix In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation.
What is the unit of the metric tensor?
It has no units. A metric tensor in any coordinate can have no units, just look at how the components of a diagonal metric tensor fits into the distance formula, if it had units the units on the left would not match the units on teh right.
What is the Riemann curvature tensor?
Riemann curvature tensor. In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common method used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e.,…
Is current a tensor?
The stress tensor: If you have a scalar conserved quantity, the current density of the charge is a vector. If you have a vector conserved quantity (like momentum), the current density of momentum is a tensor, called the stress tensor.