Einstein Relatively Easy

Tensors

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Category: Dictionary

Last Updated: 17 December 2022

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           Let us consider as we did in our article Introduction to Tensors the transformation of a vector A under a rotation θ of the coordinate system.

The components A'^x and A'^y of the vector A in the primed/rotated coordinate system relative to the components A^x and A^y in the unprimed/untransformed coordinate system can be defined as follows:

More precisely, whe have shown that the vector components in the new primed coordinate system could be written as below, and represent thus weighted linear combinations of the original components.

Using matrix representation and Einstein summation convention, we can equivalently write:

Tensors as a generalisation of vectors

Now let us generalize the concept of a vector by considering objects carrying more indices.

Imagine a collection of mathematical objects T^ij carrying two indices, with i,j = 1,2,..,N in N-dimensional space.

If the T^ij , which represent a collection of N² mathematical entities transform into linear combinations of one another (exactly as  the vector components do), then T^ij is a tensor.

We thus can generalize the tensor components transformation as per below

That's all we need to know ;-)

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Geodesics from covariant derivative

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Category: General Relativity

Last Updated: 24 October 2020

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                       A geodesic of spacetime is a curve that is straight and uniformly parametrized, as measured in each local Lorentz frame along its way.

If the geosidesic is timelike, then it is a possible wordline for a freely falling particle, and its uniformly ticking parameter λ (called affine parameter) is a multiple of the particule's proper time, λ = κτ + μ.

This definition of geodesic translates into the abstract and coordinate-free language: a geodesic is a curve P(λ) that parallel-transports its tangent vector u = dP/dλ along itself.

Now defining a coordinate system {x^α(P)), along with basis vectors e_α = ∂/ ∂x^α, we can define the tangent vector u and its components

and thus finally  the component version of the abstract geodesic equation definition becomes

CQFD

This geodesic equation can be solved (in principle) when both initial data x^α and dx^α / dλ have been specified.