How do you find the metric of a sphere?
It is the first equation on the page D s squared equals D R squared plus R squared D theta squared plus R squared sine squared theta DV squared.
What is metric tensor of spherical coordinates?
From the coordinate-independent point of view, a metric tensor field is defined to be a nondegenerate symmetric bilinear form on each tangent space that varies smoothly from point to point.
How do you calculate metric tensors?
And then the third term should be an R squared sine theta. But let’s let’s pretend that we don’t know that and see how do we calculate the components of the metric to begin with.
How do you find the Ricci tensor?
The general steps for calculating the Ricci tensor are as follows: Specify a metric tensor (either in matrix form or the line element of the metric). Calculate the Christoffel symbols from the metric. Calculate the components of the Ricci tensor from the Christoffel symbols.
What is spherical metric?
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal …
What do you mean by metric tensor?
Roughly speaking, the metric tensor is a function which tells how to compute the distance between any two points in a given space. Its components can be viewed as multiplication factors which must be placed in front of the differential displacements.
How do you convert to metric?
To convert from one unit to another within the metric system usually means moving a decimal point. If you can remember what the prefixes mean, you can convert within the metric system relatively easily by simply multiplying or dividing the number by the value of the prefix.
Does the metric tensor have to be symmetric?
The metric tensor is an example of a tensor field. coordinate system. Thus a metric tensor is a covariant symmetric tensor. From the coordinate-independent point of view, a metric tensor field is defined to be a nondegenerate symmetric bilinear form on each tangent space that varies smoothly from point to point.
What is Ricci identity?
The Riemann curvature tensor is also the commutator of the covariant derivative of an arbitrary covector with itself: since the connection is torsionless, which means that the torsion tensor. vanishes. This formula is often called the Ricci identity.
What is the order of Ricci tensor?
The Ricci tensor is a second order tensor about curvature while the stress- energy tensor is a second order tensor about the source of gravity (energy density).
What is Contravariant metric tensor?
The contravariant metric tensor is defined as. (1.18) where Gik is the algebraic adjunct to the element gik, and g is the determinant consisting of the elements gik. With the tensors gik and gik available, we obtain. (1.19)
What is phi in a sphere?
Relationship between spherical and Cartesian coordinates
Lastly, ϕ is the angle between the positive z-axis and the line segment from the origin to P. We can calculate the relationship between the Cartesian coordinates (x,y,z) of the point P and its spherical coordinates (ρ,θ,ϕ) using trigonometry.
Are metric tensors invariant?
The metric in general relativity is a tensor. It is covariant. It can be represented as a a matrix whose values do change with coordinate transformation. However, physical quantities (like the mass, or the length of a stick) calculated from the metric are invariant.
Why 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.
What is an example of a metric?
There are various metric units used for measuring length, mass, area, and capacity. For example, millimeters, centimeters, meters, and kilometers are the metric units of the measurement of length. Grams and kilograms are the units for measuring weight.
What does Khdmdcm stand for?
Acronym. Definition. KHDMDCM. King Henry Danced Merrily Down Center Main (mnemonic for metric units of measure)
Is spacetime a metric space?
It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past.
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).
What rank is Ricci tensor?
Answers and Replies. The rank of a tensor can be thought of as the number of distinct indices that the tensor has. Thus is a fourth-rank tensor, while the Ricci tensor is a second-rank tensor. On the other hand, the Ricci scalar is a scalar quantity and hence a zero-rank tensor.
Who invented the Ricci tensor?
Gregorio Ricci-Curbastro
0. Born on 12 January 1853 in Lugo in what is now Italy, Gregorio Ricci-Curbastro was a mathematician best known as the inventor of tensor calculus.
Is Ricci tensor symmetric?
so the covariant derivative of the Ricci tensor IS NOT ZERO! But this funny combination of the ricci tensor and curvature scalar IS. So we have the Ricci Tensor, which is a symmetric second order tensor, but its divergance IS NOT zero.
What is the meaning of contravariant?
Adjective. contravariant (not comparable) (category theory, of a functor) which reverses composition.
What is the difference between covariant and contravariant tensors?
In differential geometry, the components of a vector relative to a basis of the tangent bundle are covariant if they change with the same linear transformation as a change of basis. They are contravariant if they change by the inverse transformation.
Is azimuth theta or phi?
Matlab convention
Here theta is the azimuth angle, as for the mathematics convention, but phi is the angle between the reference plane and OP. This implies different formulae for the conversions between Cartesian and spherical coordinates that are easy to derive.
What is dA for a sphere?
where dA is an area element taken on the surface of a sphere of radius, r, centered at the origin. We have just shown that the solid angle associated with a sphere is 4π steradians (just as the circle is associated with 2π radians).