Questions tagged [geodesics]
For questions involving consideration of the shortest (or longest) path between two points in a curved space (e.g. a straight line between two points on the surface of a sphere such as the earth).
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Understanding classically equivalent actions of the same physical theory - what went wrong as they produce different E.O.M? [duplicate]
I am working on a specific example where the metric I am using is the $AdS_4$ metric whose ricci scalar $R=-12/l^2$ for some characteristic scale $l$: $$ds^2=-\cosh^2\left(\frac{\rho}{l}\right)dt^2+d\...
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How to transform a partial derivative to a directional derivative with respect to some affine parameter?
Suppose an affine parameter $\lambda$ is defined along a null geodesic with $dx^\mu/d\lambda=k^\mu$. How could I write the partial derivative $\partial f/\partial x^\mu$ by using $df/d\lambda$? If $k^\...
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How to get rid of the affine parameter in geodesic equation?
I encounter a problem that require me to calculate the geodesic of
$$ds^2=\frac{dx^2+dz^2}{z^2}$$ with the endpoint $(x_L,0),(x_R,0)$. I get the answer $\ddot{x}-\frac{\dot{x}\dot{z}}{z}=0$ and $\...
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What did happened with the 2 in equation (3.82) to (3.83)?
I have tried to do the same computing by the definition of the Christoffel symbols by the metric and the result is the same that the image, but i don understand what occur with the number 2. Why did ...
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Derivation of Proper Time of Fall in Schwarzschild Metric
Some time ago, in this question (Proper Time of Fall in Schwarschild Metric), I asked how to find the proper time of fall of an observer in the Schwarzschild metric because I had found many different ...
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On finding geodesics in general relativity
In the following, as usual, a spacetime is a pseudo-Riemannian manifold $M$ with metric $g$ compatible with the Levi-Civita connection $\nabla$. If an affinely-parametrized curve is a geodesic, then ...
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Geodesics: Energy functional vs length functional
The Wikipedia article about geodesics talks about the equivalence of obtaining the geodesic by either minimizing the length functional $L$, or by minimizing the energy functional $L^2/2$, cf. the Phys....
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Photon Sphere Planck Recalibration
Say a black hole's Schwarzchild radius is equal to the Planck length then a horizontal formula can be established as
$A = 4\pi \ell^2$
[1]. I've tried to find an analogue of this set up online but can'...
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Different relativistic actions [duplicate]
I am slightly confused about different action integrals in relativity. When you work through some introductions to general relativity, you usually get in contact with the relativistic action integral
\...
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Affine connection on a manifold induced by acceleration field
Suppose I have a classical force field that accelerates all particles, such that the acceleration is a function $A(p, v)$ of position and 4-velocity alone. E.g., Newtonian gravity with a possible &...
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Is there a connection between geodesics and the Euler-Lagrange equation for mechanical systems? [duplicate]
Given an affine manifold $(M,\nabla)$, the geodesic equation $\ddot{x}^j+\dot{x}^k \dot{x}^l\Gamma_{kl}^j=0$ completely characterizes the geodesics on the manifold. This is often called the Euler-...
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How flat is water?
For the purpose of this question, let us define “flat” as meaning “having a surface which is a plane”.
Clearly, the Earth being round, water is not flat. If you take a sheet of water of length $2l$, ...
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Why is light not affected by gravity?
If a rocket needs to go to mars, it needs to go through a specific trajectory. But i can see the sun straight where it is, so light does not obey gravity?
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What would the Raychaudhuri Equation be for accelerated geodesics?
What would the Raychaudhuri Equation be for accelerated geodesics? Suppose we are not able to assume the geodesic equation but rather have to assume for some tangent vector $u^\alpha$
$$u^\alpha\...
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In and out an (extremal) Reissner-Nordström black hole
The Reissner-Nordström metric is a charged black hole solution of the Einstein equations. When $M=|Q|$, the Reissner-Nordström metric is written
$$ds^2= -\left(1-\dfrac{M}{r} \right)^2 dt^2 + \left(1-\...
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Why does the $\phi$-component of the Schwarzschild geodesic represent specific angular momentum?
The $\phi$-component of a geodesic in Schwarzschild spacetime is: $$0=\frac{d}{d\tau}\left(r^2 \sin^2\theta\frac{\partial\phi}{\partial\tau}\right),$$
which can be integrated to get:
$$r^2\sin^2\theta\...
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Integration constants in geodesic equation for wave equation
I am stuck at a following "hello world problem". Let us consider the most common (d'Alembertian) wave equation:
$$
\frac{\partial^2 \psi}{\partial^2 x} - \frac{1}{c_0^2}\frac{\partial^2 \psi}...
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Need help with the proof to Proposition 4.5.12 in Hawking & Ellis
Proposition 4.5.12 in Large Scale Structure of Space-time by Hawking & Ellis is a proposition about varying a null geodesic with conjugate points to obtain a timelike curve. It states that
If ...
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Schwarzschild Metric - Why only four Geodesic equations considering nine non zero Christoffel symbols?
When employing the Schwarzschild metric, I understand there are nine non-vanishing Christoffel symbols:
$ \Gamma^t_{rt} = -\Gamma^r_{rr} = \frac{r_{\rm s}}{2r(r - r_{\rm s})} \\[3pt]
...
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Analytical Integration of Wormhole Orbits
Is there any way lightlike or timelike orbits in this metric can be found analytically?
$$ds^2=-dt^2+dp^2+\frac{1}{5p^2+4t^2}$$
The $1$ in the numerator is the squared angular momentum.
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What do Hawking/Ellis mean exactly by "non-rotating families of geodesics"?
In The Large Scale Structure of Space-Time, Hawking and Ellis refer twice (page 4, page 78) to non-rotating families of geodesics.
I don't know what that means. Is a rotating geodesic one that ...
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Which curve has the maximum proper length? [closed]
Below is a spacetime diagram in the rest frame of a lab on Earth (with a gravitational field). Which of the worldlines shown below has the greatest proper time?
My attempt: B, because $d\tau=\sqrt{1-...
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What will be the lengthdifference in the paths through spacetime of a simultaneously shot bullet and dropped mass when the mass hits the floor?
A bullet is shot down vertically from the same spacetime point from which we drop a mass from rest. The bullet arrives firstly on the ground and when the mass hits at the same place the clocks of both ...
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FLRW Geodesics with Nonzero $dr$ and $d\theta$
Given the default form of the (spatially flat) FLRW metric, $ds^2=-dt^2+a^2(t)(dr^2+r^2d\theta^2)$, I can't see a straightforward way to integrate Geodesics that have both nonzero $dr$ and $d\theta$. ...
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Implications of Parameter Choice in Geodesic Equation
Is there a difference, conceptually speaking, between solving the geodesic equations using $\lambda$ as an arbitrary parameter vs substituting a coordinate from the metric in it's place?
For instance, ...
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Principle null congruences of Kerr metric
I am trying to derive the principle null congruences of Kerr metric in Boyer-Lindquist coordinate, which is
$$t=r+\left( m+\frac{m^2}{(m^2-a^2)^{\frac{1}{2}}}\right)ln|r-r_+|+\left( m-\frac{m^2}{(m^2-...
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Solving for Geodesics in Dynamic Ellis Wormhole
Is it possible to make a coordinate transformation to the metric $$ds^2=-dt^2+dp^2+(5p^2+4t^2)d\phi^2$$ that would reveal constants of motion that can be used to solve for geodesics analytically? ...
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Are photons affected by "temporal gravity?"
Since objects follow geodesics in spacetime, that is the locally shortest path, it would seem to me that unless objects move, they do not trace any path at all. In other words, if I'm stationary on ...
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"In spacetime, a straight path yields the longest elapsed time between two events". Could someone explain this please?
I know this may appear to be a duplicate question but the other question Straight lines and longest distance doesn't seem to explain in laymans terms. So...
I'm trying to understand this but have read ...
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Newtonian Gravity from curved space?
Imagine you have the arc-length of a curve, in spherical, coordinates:
$$
s = \int_{\mathcal C}{d\tau \; \sqrt{f(r)^2 \left (\frac{dr}{d \tau} \right )^2 + r^2 \left (\frac{d \theta}{d \tau} \right )^...
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How can we describe the class of trajectories around a point mass in general relativity?
As per the answers to this post, a Newtonian gravitational trajectory of a test particle about an ideal isolated point mass is always a conic section.
An ideal point mass in GR is a black hole, either ...
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Peculiar velocities, density gradients and geodesic motion [closed]
I am studying cosmology and I have learnt that the galaxies in the universe can be represented by a pressureless fluid. Therefore, they have a geodesic motion (the so called matter frame). When ...
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Integrating the deviation vector around in a loop?
Consider a $1$-parameter family of timelike geodesics $x_s(\tau)$, where $s$ labels each geodesic in the family whilst $\tau$ is an affine parameter along each $x(\tau)$. Then the vector field $\xi≡\...
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Physical interpretation of radial null geodesics in Schwarzschild geometry
(Note: $c =1$ throughout)
The Schwarzschild metric is $$ds^2 = (1- \frac{2m}{r})dt^2 - \frac{1}{1-\frac{2m}{r}}dr^2 - r^2 d\Omega ^2,$$ with $d\Omega^2$ being the square of the solid angle element and ...
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Relativity and kinetic energy of a mass falling into a black hole
If a mass is accelerated by the application of a force, the mass/kinetic energy of the object approaches infinity as it approaches the speed of light.
Now let's consider the same mass falling toward a ...
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What gives us the equations of motion in GR?
Maybe stupid question, but to my understanding, the Einstein equation tells us the differential equation governing the Geometry of spacetime. That's all good and fine , but suppose I had an actual ...
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Equivalence Principle: Uniform to Non-uniform gravitational fields
2Einstein in his 1916 GR paper describes the equivalence principle and makes a case for general relativity i.e a person in a non-inertial frame is equivalent to a person in a uniform gravitational ...
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Duration of a projectile shot up the ZARM tower, from entering the drop tube until exiting again, vs. duration of the tube being occupied by this proj
In the basement of the ZARM Drop Tower in Bremen, Germany, there's a high-tech catapult! ...
This catapult can be used to launch a payload (projectile $P$) flying (freely, initially upwards) into the ...
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Connecting vectors for two different congruences?
So I was reading this thesis. It presents this equation I haven't seen (Page $2$ equation $6$) before and have no idea how to derive:
The general solution to this second-order differential equation ...
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Point Particle Energy Momentum Tensor from Lagrangian Density
Given the Lagrangian density of a point particle
$$\mathscr{L} = - m \int \sqrt{-g_{\alpha \beta}\dot{z}^{\alpha}\dot{z}^{\beta}} \delta^{(4)}(x,z) d\lambda$$
we could derive the energy momentum ...
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Why does a piece of thread form a straight line when we pull it?
Experience tells that if we pull a piece of thread, it forms a straight line, a geodesic in the Euclidean space. If we perform a similar experiment on the surface of a sphere, we will get an arc of a ...
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Sum of geodesic deviation around a triangle in curved spacetime?
So I was pondering about geodesic deviations and I'm confused about the following. Let's say I have $3$ geodesics $\gamma_1(t)$ , $\gamma_2(t)$ and $\gamma_3(t)$. I introduce a parameter $s$ such that ...
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A problem about relativistic particle actions [closed]
I have a problem of a course in General Relativity which I don’t know to start solving, and I was wondering if someone could give me some indications or ideas on how to figure it out. It is the ...
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Relative velocity and proper time derivative of geodesic deviation?
From wiki
To quantify geodesic deviation, one begins by setting up a family of closely spaced geodesics indexed by a continuous variable s and parametrized by an affine parameter $\tau$. That is, for ...
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What is timelike geodesic?
I have searched the internet for the definition of timelike geodesic curves. But I am not getting a consistent definition. In some places I saw the geodesic maximises the proper time and in some ...
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Confusion in Gravitation Foundation and Frontiers?
Taken from Gravitation: Foundations and Frontiers by T. Padmanabhan Pg $197$
Consider a bunch of particles moving along geodesics in a spacetime. Let $x_i = x_i
(\tau,v)$ denote the
family of ...
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How do tidal forces on incomplete geodesics determine extendability?
Why can we be sure that the manifold with the metric $(M,g)$ does not have a geodesically complete extension if it has an incomplete timelike geodesic along which the tidal force blows up? Does this ...
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Hamiltonian for the time-like particle on the geodesic
I am trying to reproduce the results from this paper. On page 2 of the paper, they have an equation: $$2 H=-\frac{\dot{r}^2}{g(r)}-L \dot{\phi }+E \dot{t}=\epsilon\tag{9}$$
where they make a comment ...
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How much longer is the path through spacetime of a mass that falls freely compared to a resting mass?
A mass that falls to Earth follows a shortest path through spacetime. If a mass falls from a 1km high building, how much longer will its path be compared to a mass resting on a table?
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Geodesic: maximal aging versus extremal aging
From Exploring Black Holes, by Taylor and Wheeler, page 1-7:
Purists insist that we say not maximum reading but rather extremal reading: either maximum or minimum. This book contains only examples of ...
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