X_iu^i: Physical Significance & Conservation

[c299792458]

Let us denote by $X^i=(1,\vec 0)$ the Killing vector and by $u^i(s)$ a tangent vector of a geodesic, where $s$ is some affine parameter.

What physical significance do the scalar quantity $X_iu^i$ and its conservation hold? If any...? I have seen this in may books and exam questions. I wonder what it means...

 

[WannabeNewton]

Hi there! The point is that the scalar quantity you formed is constant along the geodesic! Using your notation, $\triangledown _{U}(X_{i}U^{i}) = U^{j}\triangledown _{j}(X_{i}U^{i}) = U^{j}U^{i}\triangledown _{j}X_{i} + X_{i}U^{j}\triangledown _{j}U^{i}$. Note that $U^{j}U^{i}\triangledown _{j}X_{i}$ vanishes because $U^{j}U^{i}$ is symmetric in the two indices whereas, by definition of a killing vector, $\triangledown _{j}X_{i}$ is anti - symmetric in the two indices and it is very easy to show that the contraction of a symmetric tensor with an anti - symmetric one will vanish. The second term $X_{i}U^{j}\triangledown _{j}U^{i}$ vanishes simply because U is the tangent vector to a geodesic thus we have that $\triangledown _{U}(X_{i}U^{i}) = 0$. In particular note that if this geodesic is the worldline of some freely falling massive particle then its 4 - velocity is the tangent vector to the worldline and we can re - express the condition for the worldline being a geodesic in terms of the 4 - momentum of the particle (and for photons just define the geodesic condition like this) and we can have that if $X^{i}$ is a killing field on the space - time then $X_{i}P^{i}$ will be constant along this geodesic. It is a geometric way of expressing local conservation of components of the 4 - momentum; these killing fields are differentiable symmetries of the space - time and you might be able to see that more clearly by the fact that the lie derivative of the metric tensor along the killing field will vanish.

 

[jfy4]

dat signature...

 

[WannabeNewton]

dat signature...

:[ don't judge me T_T

 

[PJC01]

The scalar quantity X_iu^i represents the contraction of the Killing vector X^i with the tangent vector u^i. In general relativity, the Killing vector is a special type of vector field that generates a symmetry of the spacetime metric. This means that the spacetime metric remains unchanged under infinitesimal transformations generated by the Killing vector.

The tangent vector u^i represents the direction and speed of a geodesic, which is the path followed by a free-falling particle in a curved spacetime. The scalar quantity X_iu^i represents the rate of change of the Killing vector along the geodesic, which has physical significance in terms of the symmetries of the spacetime.

The conservation of X_iu^i means that this scalar quantity remains constant along the geodesic, regardless of the affine parameter s. This can be understood as a conservation of the symmetry of the spacetime along the geodesic. In other words, the spacetime remains unchanged as the particle moves along its geodesic path.

This conservation has important implications in terms of the laws of physics. It means that certain physical quantities, such as energy and momentum, are conserved along the geodesic. This is a consequence of the symmetries of the spacetime and is a fundamental principle in general relativity.

In summary, the scalar quantity X_iu^i has physical significance in terms of the symmetries of the spacetime and its conservation holds as a fundamental principle in general relativity.