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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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General relativity: congruence and integral curves

I've been doing some research on GR for a assignment on the Raychaudhuri equation. To do so, I've had to pick up some math I had not learnt earlier. My university offers a really poor choice of ...
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Doubt about energy conditions: the Time-like Convergence Condition

First of all, consider a congruence of smooth time-like geodesics parametrized by proper time $\tau$. So, a tangent vector to a time-like geodesic is indeed a four-velocity up to a factor constant; ...
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Geodesic curves representation

If there is some 2 dimensional metric given for some space, lets say the metric is diagonal with both elements 1/t^2 with the time being with negative sign. If we now try to find what is the shape of ...
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Are the horizon generators radial null geodesics also?

What I am going to ask is probably a result of unrigorous treatment of the submanifold in question. Radial Null Geodesics of Schwarzschild So start with Schwarzschild spacetime. The metric tensor is ...
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Geodesics - Reparameterization

I am reading Wald's textbook Chapter 3. I am struggling with Section 3.3 and problem 5. It states that any curve that satisfies the weaker condition $T^{a}\nabla_{b}T^{b} = \alpha T^{a}$ is $eq.(3.3....
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Unique null geodesic between two points

Given two points in Lorentzian spacetime $p,q\in M$, is it true that there is only a unique null geodesic (up to affine reparametrization) that connects that the two points? On the one hand, it seems ...
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Spacetime curvature and measurements

From a programming perspective, I've always thought of gravitational influence as a kind of vector field, (crudely drawn) which seems to attribute to the motions of bodies through the field ...
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Trouble in proving geodesic action invariance under diffeomorphism

In this post Diffeomorphism invariance and geodesic action it is said: You found (by computing in local coordinates) that this is invariant under a diffeomorphism $\phi: M \to M$. This statement ...
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1answer
102 views

How to calculate initial conditions to integrate a null geodesic

Suppose, this is the line element of a FLRW metric, $$ ds^2 = -[1 + 2ψ(t,x_i)]dt^2 + a^2(t) [1 - 2ϕ(t,x_i)]dx_i^2 $$ and the geodesic equation is, $$ \frac{d^2x^α}{dλ^2} = - Γ_{βγ}^α \frac{dx^β}{dλ} \...
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Euler-Lagrange Equation, Shortest Path On Sphere

The equations for spherical polar coordinates are $$x = r \sin(\theta) \cos(\phi) \\ y = r \sin(\theta) \sin(\phi) \\ z = r \cos(\theta)$$ Now, consider a path expressed as $\phi = \phi(\theta)$...
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Does a geodesic always extremize its path length? [duplicate]

I've learned that a geodesic maximizes its proper time in Minkowski spacetime. Is this still true in general curved spacetime? In other words, does the geodesic equation give the globally extremal ...
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Does the worldline of light depend on the frequency?

My Question is: If 2 rays of light are emitted with different frequencies from the same spacetime point, does an observer see them in the same worldline? I know that the worldline of light behaves ...
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4answers
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Are there null geodesics inside null infinity?

Looking at a Penrose diagram for Minkowski space, you would think that you could draw a null geodesic running from $i^0$, along $\mathscr{I}^+$, and ending up on $i^+$. In fact there would be many ...
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1answer
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Free particle in Rindler metric

I have to discuss the motion of a free particle in Minkowski using the Rindler metric $$ds^2=e^{2ap}(-d\tau^2+dp^2)$$ So it has to satisfy the geodesic condition $\frac{d^2x^\mu}{d\tau^2}+\Gamma_{\rho\...
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1answer
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Differentiating Scalar along a geodesic

I have been studying GR for sometime and doing exercises from Schutz and I have a question about differentiating along a geodesic. Here is what I know. The equation of geodesic in terms of four ...
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1answer
57 views

Energy in spherically symmetric space times

In deriving the equations of motion for geodesics in spherically symmetric spacetimes through Hamiltonian formalism, we can find some constants of motion, namely, $E$ and $L$, the energy per unit of ...
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1answer
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Direction of gravity

General Relativity explains the path a falling body makes (ex. An apple falling toward the center of the Earth) as a geodesic in curved spacetime. What explains the direction the apple falls? In other ...
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1answer
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Derivation of equation for geodesic deviation

I am trying to figure out the calculation which leads to the geodesic deviation on this site. So far I understood all steps until (14.7) and managed to show that (14.6) = (14.7), namely $$ \ddot\xi^\...
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2answers
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I'm travelling near the speed of light. Do I need to brake before using a super-massive black hole to turn around?

I'm taking one of those new fusion drives for a trip to nearby supermassive black hole. At a comfortable 1 G, it'll take me about 7 months of proper time to accelerate to 95% the speed of light. I'm ...
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2answers
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The time component of the geodesic equation for Newtonian gravity

I am working on a simple and popular GR textbook exercise. In Dodelson's Modern Cosmology (p. 54), it is stated thus: The metric for a particle traveling in the presence of a gravitational field ...
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1answer
55 views

Question about Lagrange method and line element

Consider the following line element: $$ds^{2} = K(x,y,z,t)(-dt^2+dx^2)+M(x,y,z,t)dxdt+dy^2+dz^2$$ Then the lagragian method give to us the lagrangian from line element: $$\mathcal{L}^2 = K(x,y,z,t)(...
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Geodesic curve definition [duplicate]

Do we have a choice in defining the covariant derivative by the use of a set of coefficient functions(Christoffel gammas)? If so, could we then say that these coefficient functions need not to ...
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2answers
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Geodesic equations from action with auxiliary field

A textbook says that the geodesic equations (for both massive and massless) can be derived from the following action: $$ S = -\frac{1}{2} \int d\tau \:\eta \: (\eta^{-2} \dot{x}^\mu \dot{x}^\nu g_{\...
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1answer
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Calculus of Variations help [closed]

I've been studying Chapter 6 in Taylor's Mechanics book. And am working through the odd-numbered problems. I am struggling with 6.13, which reads: In relativity theory, velocities can be ...
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1answer
103 views

Closed form expression for position as function of time of object falling directly into black hole from infinity

Given a Schwarzschild radius $r_s=2 G M/c^2$, the escape velocity (equal to speed if falling from infinity) will be $\sqrt{2 G M/r}=\sqrt{r_s c^2/r}$ where the radial distance "r" is the point at ...
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Tidal acceleration of a body at rest in a Schwarzchild geometry using geodesic equations

I'm self-studying the properties of a Schwarzchild geometry, with line element $$ds^2 = \left(1-\frac{2m}{r}\right)dt^2-\left(1-\frac{2m}{r}\right)^{-1}dr^2-r^2\left(d\theta^2 + \sin^2\theta d\phi^2\...
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3answers
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What is the length of null geodesic?

There are many questions about this but none of them adresses my concrete question. If it is indeed true that for light we have $ds^2 = 0$ does that mean that in 4d spacetime total "distance" is zero ...
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2answers
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Relationship between freefall velocity time dilation and gravitational time dilation in a Schwarzschild metric

If you drop an object into a gravitational field, is its final velocity equal to what it would have to be in flat space in order to generate the same time dilation that you get at a given radius for ...
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Periodic motion(s) on a torus

I recently came across the Lyusternik-Fet theorem concerning closed geodesics on a compact manifold. For simplicity of description, take the 2-torus, and imagine it represents the configuration space ...
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2answers
189 views

Geodesics of anti-de Sitter space

It is said that (p. 9), given the anti-de Sitter space $\text{AdS}_2$, let's say in the static coordinates $$ds^2 = -(1 + x^2) dt^2 + \frac{1}{(1+x^2)} dx^2$$ Every timelike geodesic will cross the ...
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Significance of geodesics of a Hamiltonian surface in condensed matter physics?

Many Hamiltonians in 2D quantum systems can be parameterized as a surface (such as the Bloch sphere) by their k-space coordinates. Another example is given by the (kx,ky) points of the Brillouin torus ...
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Is the geodesic equation valid for a motion of an object with an arbitrary initial condition?

As far as I know, the geodesic equation of motion can be directly derived from the equivalent principle. For instance, as shown by Steven Weinberg, the geodesic equation can be obtained by ...
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Is the geodesic equation independent of an initial condition?

The following argument is used to determine the unknown factors (e.g., $A(r)$ and $B(r)$) in the Schwarzschild metric. $$ \lim_{r \to ∞}A(r) = \lim_{r \to ∞}B(r) = 1 \space\space\space\space\space\...
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What is the meaning of the Equation of Geodesic Deviation?

I've seen the Equation of Geodesic Deviation stated in several text as:$$\frac{D\xi^2}{dt^2}+Riemann(\textbf U,\xi,\textbf U)=0$$ but I haven't seen a real good explanation of why it works or the ...
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Why can we choose affine parameterization?

In general relativity when deriving the geodesic equation $$\ddot{x}^\mu + \Gamma^\mu_{\alpha\beta}\dot{x}^\alpha\dot{x}^\beta = 0\tag{1}$$ from the action $$S = \int d\tau \sqrt{|g_{\mu\nu} \dot{x}...
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According to general relativity, why are two objects at rest attracted to each other? [duplicate]

I'm trying to understand gravity in General Relativity and I'm having some questions. I can understand that an object in orbit around another more massive object is free falling and simply following a ...
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Hamiltonian formulation of the geodesic equation [duplicate]

I am using the Hamilton's formulation of the geodesic equation in order to obtain geodesic equations in terms of the particle's coordinates and the conjugate momenta. Now, I am figuring out a way to ...
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1answer
103 views

Deriving the geodesic equation using a Lagrange multiplier to fix affine parametrisation

The geodesic equation can be derived using the action $$S_0 ~=~ \int d\tau \sqrt{-\dot{x}_\mu\cdot \dot{x}^\mu}\tag{1}$$ (I am using the (-+++) convention and $\dot{x} = \frac{dx}{d\tau}$). To ...
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Doubt in Functional Derivative of Lagrangian

Lecture XXXIII: Lagrangian formulation of GR by Christopher M. Hirata NON-INTERACTING DUST Consider a system with a suite of particles {A} each of mass $\mu_{A}$ following some set of trajectories $...
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Question on affine parameter of null world-lines versus light-like world-lines

I consider Minkowski space $M$ in this question. My question is about the following. For lightlike worldlines we can define the geodesic equation as follows: $$\nabla_TT=0,$$ where $$T = \gamma^\...
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Why is the right hand side of the Equation of Geodesic Deviation 0?

I'm reading through several primers on GR and I keep seeing the same thing over and over again with no explanation: why is the right hand side of this equation zero: $$\frac {D^2\xi^a}{Du^2}+R^{\alpha}...
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Conjugate points in spacetime [Proposition 4.4.2 in Hawking & Ellis]

In Hawking & Ellis The Large Scale Structure of Space-time page 98, they proved a proposition 4.4.2 which is show in the attached figures and . I do not understand the statement underlined in ...
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initial directions in geodesic equations

in solving geodesic equations, how do you find components of initial tangent vectors (or initial direction) given initial values for t, r, theta and phi? im reading this article on using the ...
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What would the Second Law of Motion be in this universe?

I have a universe described by the equation: $$ds^2=c^2d\tau^2=-c^2dt^2+[dr+a_0 td\tau]^2+r^2\Omega^2$$ What would be the second law of motion in this universe?
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Intuitively, why do attempts to delay hitting a black hole singularity cause you to reach it faster?

In general relativity, proper time is maximized along geodesics. Inside of a black hole, all future-oriented timelike trajectories end at the singularity. Putting these two facts together, we find ...
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Can somebody explain why the action in the picture is true?

I can provide the resource for where this is from. Can somebody explain how to get this expression?
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Physical example of geodesic mapping without metric

I am working on my master thesis about harmonic and geodesic mappings and I am looking for some examples with physical meaning. I want to find some geodesic mapping, so a map between manifolds endowed ...
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375 views

Does an electromagnetic field affect neutral particles via the metric because of the EM stress-energy tensor?

I'm just starting to learn general relativity (GR), and I'm a beginner, but I came out with this situation which is unclear to me: The trajectory of a charged particle in GR is given from the equation:...
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1answer
220 views

Conserved quantity along geodesic and metric

I’m studying General Relativity on Schutz’s book. On Chapter 7 he talks about conserved quantities along geodesics, with the equation \begin{equation} m\frac{dp_{\beta}}{d\lambda}=\frac{1}{2}g_{\nu\...
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Conserved quantity in Noethers theorem with the use of a Killing vector

Consider a system given by the action: $$S = \int \sum_{i,j} G_{ij}(q) \dot{q}^i \dot{q}^j dt$$ Now, consider the quantity $Q_v = \sum_{i,j} G_{ij} v^i \dot{q}^j $ with $v^i$ the Killing vector. ...