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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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Sean Carroll GR - Ex.3.6 (b) & (c) [on hold]

I'm working in the newtonian limit of GR with the metric $$ ds^2 = -(1+2\Phi)dt^2 + (1-2\Phi)dr^2 +r^2d\theta^2+r^2sin^2\theta\;d\phi^2 $$ where $$\Phi = -\frac{GM}{r}.$$ We are first asked to ...
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2answers
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Geodesics from variational principle with respect to coordinate?

I know you can find geodesic equations with respect to proper time $\tau $ using the variational principle, i.e. using Euler-Lagrange equations $$ \frac{\partial}{\partial x^{\mu}}L-\frac{d}{d\tau}\...
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Rotationally invariant metrics and conservation of angular momentum

This was prompted by an exam question, though the questions are more general: A 2D Riemannian space has the metric: $ds^2=dr^2 + \gamma^2 r^4 d\phi^2$ State what conserved quantity ...
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1answer
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Proving that test particles in GR, follow spacetime geodesics

My question is pretty much in the title. According to this paper, this is not exactly proven rigorously yet. What I dont understand is what exactly is not proven. If I'm not too wrong, a test particle ...
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Radial infall in Schwarzschild

In Straumann's book on general relativity, one finds the following solution to the question of geodesic radial infall into the black hole: $$d\tau=(\frac{2m}{r}-\frac{2m}{R})^{-\frac{1}{2}}dr$$ To ...
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Why can we parameterize a null geodesic such that its velocity is four-momentum? [duplicate]

One principle in general relativity is that the wordlines of massless particles are null geodesics. It also seem to be a commonly stated fact (for instance see eq. (3.62) in Section 3.4 of Carroll's ...
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1answer
19 views

Fundamental Principle of Dynamics and equations of geodesics with proper time

I just wanted to have a little precision. In the expression below translating the PFD (Fundamental Principle of Dynamics) in tensor calculus (or more precisely the inertial principle) : $$a^{\nu}=\...
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1answer
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Equality between derivatives of the metric

In one of my lecture, it is said: Let us use the freedom of the choice of parametrization to demand that the variation of $\lambda$ after a small displacement along the curve is proportional to the ...
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2answers
81 views

How does General Relativity explain escape velocities?

In general relativity, objects follow the shortest possible path through curved space-time called a geodesic and that there exists no such force of gravity which pulls objects, it is just because ...
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2answers
48 views

Is the determinant of metric tensor stationary wrt. proper time for a particle moving along its world line?

While writing the expression for stress energy tensor of a free massive particle moving along its world-line some authors take out of the integral sign, the $\sqrt{-g}$ where $g$ is the metric tensor ...
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1answer
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Momentum of a moving object in FRW metric according to an observer comoving with cosmic expansion

I would like to show that in an FRW metric the momentum of a freely falling object decays as the inverse of the scale factor. I know there are many proofs and arguments for this but I am trying to get ...
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2answers
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Geodesics for FRW metric using variational principle

I am trying to find geodesics for the FRW metric, $$ d\tau^2 = dt^2 - a(t)^2 \left(d\mathbf{x}^2 + K \frac{(\mathbf{x}\cdot d\mathbf{x})^2}{1-K\mathbf{x}^2} \right), $$ where $\mathbf{x}$ is 3-...
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1answer
58 views

De Sitter spacetime affine parameter

I am reading Chapter 8 in Carroll's "Spacetime and geometry " textbook and I was lead to exercise 8.2, given as: Consider de Sitter space in coordinates where the metric takes the form $$ds^{2} =...
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Light path in a Schwarzschild spacetime

This might be a silly question for the physicist as I'm not one but I've have watched this YouTube video which shows the steps needed to simulate light path in a Schwarzschild spacetime, what I failed ...
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38 views

Lorentz-like equation from Geodesic equation

In Hobson's General Relativity, it explains the following (p.173) Consider the limit of a weak gravitational field in a coordinate system in which $g _ { \mu \nu } = \eta _ { \mu \nu } + h _ { \mu \...
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General Relativity Lorentz-like equation

In the literature, it says that, in the weak-field, $$g_{µν} = η_{µν} +h_{µν},$$ slow-motion limit, the Geodesic equation reduces to the Lorentz-like equation. Can anyone explain this?
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Why a straight line is the shortest path between two points? [closed]

There is a geometric proof, the triangle inequality. But is there a way to prove this by means of the first law of Newton?
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Geodesic expansion

I have some questions regarding the geodesic expansion: Is the expansion of ingoing congruences always required to be negative? For both ingoing and outgoing geodesics, is $\frac{d\Theta}{d\lambda}$ ...
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1answer
55 views

Why is the Lagrangian for space-like geodesics equal to 1?

In Schwarzschild spacetime, the Lagrangian can be defined as $$ L = -\left( 1 - \frac{2M}{r} \right) \dot{t}^2 + \left( 1- \frac{2M}{r} \right)^{-1} \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2\theta ...
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ingoing and outgoing radial geodesics

Again, playing with a metric of the form $ds^2 = (-\Phi + H^{2}R^{2}\Phi^{-1})dt^2 -2HR\Phi^{-1}dRdt+\Phi^{-1}dR^2+R^2d\Omega^2$ where $\Phi = 1- \frac{2me^{Ht}}{R}$, and $H$ is a positive constant....
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Coordinates in studying geodesic congruences

Will it matter what coordinate I use in studying geodesic congruences? For example, if i want to calculate the expansion scalar of null radial geodesic congruences and from here determine the ...
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1answer
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Interpretation of Normal Coordinates

In my lecture notes, normal coordinates are defined as the following: Def.: Let $\left(\mathrm{e}_\mu\right)$ be a basis of $\mathcal{T}_p\left(\mathcal{M}\right)$. Normal coordinates in a ...
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Existence of a strictly a timelike curve on Lorentzian manifold

I encountered online the following exercise: Let $M$ be a Lorentz manifold of $\dim(M)=n$, and let $\psi:\Sigma\to M$ be a spacelike submanifold of dimension $n-2$ embedded into $M$. We will ...
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1answer
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Effective potential in a time-dependent spacetime

My question is regarding an arbitrary time-dependent spherically symmetric spacetime with line-element, in co-moving coordinates, to be $$ds^2 = -f(R) dt^2 + a(t)\bigg\lbrace\frac{dR^2}{f(R)} +R^2d\...
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2answers
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How does $r$ depend on $\varphi$ in the Schwarzschild metric?

I am confused about the Wikipedia derivation of the equation for geodesic motion in the Schwarzschild spacetime. The derivation of this equation involves variation with respect to the longitude $\...
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1answer
51 views

Geodesic equation in different coordinate systems

Suppose I have a Schwarzschild metric in Schwarzschild-coordinates and I obtain the geodesic equations for this metric. Suppose I transform the Schwarzschild metric into a different coordinate system,...
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0answers
49 views

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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0answers
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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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34 views

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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1answer
95 views

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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3answers
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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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3answers
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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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0answers
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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
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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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1answer
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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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1answer
96 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 ...
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4answers
298 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
58 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
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
72 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
88 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
178 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
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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
59 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
104 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 ...
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1answer
112 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 ...