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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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Alternative covariant form of geodesic equation

I am required to show that : $$\frac{d^2x_\mu}{d\tau^2} = \frac{1}{2}\frac{g_{\rho\nu}}{dx^{\mu}}\frac{dx^{\nu}}{d\tau}\frac{dx^{\rho}}{d\tau}$$ I was thinking that I could perhaps contract the ...
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1answer
55 views

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 ...
3
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1answer
36 views

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
47 views

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
55 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
62 views

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
57 views

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
164 views

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
93 views

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
53 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
55 views

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
65 views

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 ...
4
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1answer
102 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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0answers
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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
109 views

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
67 views

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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0answers
31 views

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
126 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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2answers
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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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2answers
119 views

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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2answers
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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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2answers
97 views

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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3answers
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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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0answers
30 views

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 ...
2
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1answer
79 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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1answer
79 views

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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2answers
129 views

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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0answers
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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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1answer
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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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2answers
109 views

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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2answers
72 views

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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1answer
40 views

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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2answers
372 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:...
2
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1answer
146 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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1answer
99 views

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. ...
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1answer
109 views

Euler-Lagrange equation in General Relativity

In Relativity the Lagrangian of a free particle is \begin{align} \mathcal L=\sqrt{g_{ab}\frac{dx^a}{d\tau}\frac{dx^b}{d\tau}}\end{align} Since $\mathcal L$ is parameterization invariant we can always ...
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0answers
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Trying to reproduce curves with angle of CMB anisotropies as a function of distance and curvature parameter

I am looking for a way to get, by a simple numerical computation, the 3 curves on the following figure: For this, I don't know what considering as abcissa (comoving distance ?, i.e $$D_{comoving} = ...
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2answers
80 views

What is $\tau$ and $\sigma$ in a geodesic $ψ(x) = (\tau(x),\sigma(x))$?

I'm trying to replicate the results of this paper (Eternity in 6 hours, written by Stuart Armstrong and Anders Sandberg). On pages 14-15, it discusses the path of an intergalactic probe that is ...
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1answer
92 views

Geodesic of a massive particle in Schwarzschild metric

what happens to particle in unstable circular orbit in schwarzschild metic when it is pushed outwards? Does its geodesic change into an ellipse with around stable radius corresponding to its angular ...
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57 views

Radial orbits in the Kerr metric

I have a question concerning radial orbits in the Kerr metric. The equations for planar geodesics $\theta=\pi/2$ in Boyer-Lindquist coordinates are well known: $E = -g_{t\mu}u^{\mu} = \left( 1 - \...
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1answer
88 views

Spatial dimensions inside the event horizon

Consider a test particle falling radially to a black hole. The radial direction toward the singularity inside the black hole becomes a direction in time while the dimension that used to be time ...
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1answer
129 views

Hamilton or Hamilton-Jacobi formalism with Hamiltonian equal to zero

I have the Lagrange function: $$L=\sqrt{\frac{\dot{x}^2+\dot{y}^2}{-y}}.\tag{1}$$ The energy is then: $$H=\dot{x}\frac{\partial L}{\partial \dot{x}}+\dot{y}\frac{\partial L}{\partial \dot{y}}-L=0.\...
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3answers
99 views

Is the path that a planet takes orbiting the sun a centripetal one or one that follows a geodesic path?

Is the path that a planet takes orbiting the sun a centripetal one or one that follows a geodesic path? My point is that if the planet follows a centripetal path all objects on the planet will ...
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1answer
78 views

Is spacetime in an ellipse around a massive object, or does it just slope down towards the massive object?

In some discussions, people seem to imply that spacetime is in an actual ellipse around a massive object so that, for example, the planet orbiting a star is actually traveling in a straight line in ...
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Calculating the Gaussian Curvature of Cylinder

In the book "Gravitation" by Misner, Thorne and Wheeler, exercise 1.1 on page 44 reads: Show that the Gaussian curvature $R$ of the surface of a cylinder is zero by showing that geodesics on that ...