- #1
jk22
- 729
- 24
Does there exist a form of the geodesic equation which is invariant under coordinates change ?
The geodesic equation cannot be invariant because it has one free (vector) index. It is, therefore, covariant (i.e., form invariant) with respect to an arbitrary change of coordinates. Indeed, it is a good exercise to show that the left-hand-side of the geodesic equation transforms as a vector:jk22 said:Does there exist a form of the geodesic equation which is invariant under coordinates change ?
samalkhaiat said:The geodesic equation cannot be invariant because it has one free (vector) index. It is, therefore, covariant (i.e., form invariant) with respect to an arbitrary change of coordinates. Indeed, it is a good exercise to show that the left-hand-side of the geodesic equation transforms as a vector:
[tex]\left(\frac{d^{2}\bar{x}^{\mu}}{d\tau^{2}} + \bar{\Gamma}^{\mu}_{\rho \sigma}(\bar{x}) \frac{d\bar{x}^{\rho}}{d\tau}\frac{d\bar{x}^{\sigma}}{d\tau}\right) = \frac{\partial \bar{x}^{\mu}}{\partial x^{\nu}} \left( \frac{d^{2} x^{\nu}}{d\tau^{2}} + \Gamma^{\nu}_{\rho \sigma}(x) \frac{d x^{\rho}}{d\tau}\frac{d x^{\sigma}}{d\tau}\right).[/tex]
If you are using the geodesic equation in a space that is rotationally symmetric about a point and you rotate solutions about that point, they will still be solutions.jk22 said:I wanted to say rotationally invariant in the 3d space. I ask this because of the following : there are circular uniform orbits by fixing theta at a certain value, but if one rotate this trajectory around the origin then it is no more solution of the 4 geodesic equation.
You haven't stated what spacetime you are working in, so once again we need to guess part of your problem. Please try to state a complete problem.jk22 said:$$\ddot{\phi}=2\cot\theta\dot{\theta}\dot{\phi}$$
/* Maxima batch file to calculate geodesic equations in Schwarzschild */
/* spacetime, simplifying for the case of a circular orbit (dr/dtau=0, */
/* dt/dtau=const) */
/* Load ctensor and set up for Schwarzschild spacetime */
load(ctensor);
ct_coordsys(exteriorschwarzschild);
depends(r,tau);
depends(t,tau);
depends(theta,tau);
depends(phi,tau);
/* Calculate inverse metric and Christoffel symbols */
ug:invert(lg);
christof(mcs);
/* Function to generate the ith geodesic equation. Note that Maxima */
/* numbers arrays from 1 and the Christoffel symbol \Gamma^i_{jk} is */
/* stored in array element mcs[j,k,i]. */
geodesic(i):=block(
[j,k,geo],
geo:diff(ct_coords[i],tau,2),
for j:1 thru 4 do block (
for k:1 thru 4 do block (
geo:geo+mcs[j,k,i]*diff(ct_coords[j],tau)*diff(ct_coords[k],tau)
)
),
return(ratsimp(geo)=0)
);
/* Generate the geodesic equations and simplify */
[geodesic(1),geodesic(2),geodesic(3),geodesic(4)];
substitute(0,diff(r,tau,2),%);
substitute(0,diff(r,tau),%);
substitute(R,r,%); /* R is a constant */
substitute(0,diff(t,tau,2),%);
substitute(K,diff(t,tau),%); /* K is a constant */
ratsimp(%);
expand(%);
jk22 said:For a circular motion ##r## is constant and time flows linearly. The geodesic equation reads
$$\ddot{\theta}=\sin\theta\cos\theta\dot{\phi}^2$$
$$\ddot{\phi}=-2\cot\theta\dot{\theta}\dot{\phi}$$
$$\dot{\theta}^2+\sin^2\theta\dot{\phi}^2=C(onstant)$$
The second equation is divided by ##\dot{\phi}## giving by integration :
$$\dot{\phi}=a/\sin^2\theta$$
Substituting in the first and multiplying by ##\dot{\theta}## gives
$$\dot{\theta}^2=-a^2/\sin^2\theta+d$$
Giving :
$$ (\sin(x) \sqrt(d - a \csc^2(x)) (\sqrt(d) \log(\sqrt(2 a + d \cos(2 x) - d) +\sqrt(2) \sqrt(d)\cos(x)) - \sqrt(a) \tanh^(-1)((\sqrt(2) \sqrt(a) \cos(x))/\sqrt(2 a + d \cos(2 x) - d))))/\sqrt(a + 1/2 d \cos(2 x) - d/2)=t $$(##x=\theta##, by wolframalpha)The last equation becomes
$$D=C$$
So we see it is a solution.
The "Exists ? : Invariant geodesic equation" is a mathematical equation used in the field of general relativity to describe the motion of particles in curved spacetime. It is based on the principle of least action, which states that the path a particle takes between two points is the one that minimizes the action (a measure of the energy) along that path.
The "Exists ? : Invariant geodesic equation" is important because it allows us to understand the behavior of particles in the presence of massive objects, such as planets or stars. It is crucial for understanding the effects of gravity on the motion of objects and has been used to make predictions about the behavior of light and other particles in the universe.
The "Exists ? : Invariant geodesic equation" is derived from the Einstein field equations, which describe the relationship between the curvature of spacetime and the distribution of matter and energy within it. By solving these equations, we can determine the geodesic equation, which describes the motion of particles in curved spacetime.
The word "invariant" in the "Exists ? : Invariant geodesic equation" refers to the fact that the equation remains the same regardless of the coordinate system used to describe spacetime. This means that the equation is valid in any frame of reference and is not affected by changes in perspective or measurement.
The "Exists ? : Invariant geodesic equation" is a fundamental part of the theory of general relativity, which was developed by Albert Einstein in the early 20th century. It is one of the key equations used to describe the effects of gravity on the motion of objects in the universe and has been extensively tested and confirmed through observations and experiments.