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).
937
questions
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Some aspect of covariant derivative of point particle energy-momentum tensor
My question is related to Derivation of the geodesic equation from the continuity equation for the energy momentum tensor
I need to understand one step in derivation.
Let's consider the Energy-...
4
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2
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320
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Is there a known mechanism for mass-energy distorting spacetime?
I’ve been really interested in learning about the mechanisms behind physical phenomena that go beyond just learning to manipulate the equations and give a physical intuition about HOW something ...
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1
answer
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How to compute Kerr geodesics?
How would I start to numerically compute trajectories of Kerr geodesics with constants of motion like in this wikipedia page. I want to recreate trajectories like in this picture in Matlab.
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Derivative with respect to a coordiante differential (geodesic equation)
If the arc length is chosen to be the action integral, that is $$ S=\int \sqrt {g_{kn}\frac{dx^k}{ds} \frac{dx^n}{ds}} dx \tag{11.13} $$
Then Lagrangian is given by $$L=\sqrt {g_{kn}\frac{dx^k}{ds}...
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1
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Sean Carroll GR - Ex.3.6 (b) & (c) [closed]
I'm working in the newtonian limit of GR with the metric
$$
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}.$$
We are first asked to ...
2
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2
answers
631
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Geodesics from variational principle with respect to coordinate?
I know you can find geodesic equations with respect to proper time $\tau $ using the variational principle, i.e. using Euler-Lagrange equations
$$ \frac{\partial}{\partial x^{\mu}}L-\frac{d}{d\tau}\...
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0
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58
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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 ...
5
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1
answer
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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 ...
2
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0
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58
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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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1
answer
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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) :
$$a^{\nu}=\...
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1
answer
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Equality between derivatives of the metric
In one of my lecture, it is said:
Let us use the freedom of the choice of parametrization to demand that the variation of $\lambda$ after a small displacement along the curve is proportional to the ...
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3
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How does General Relativity explain escape velocities?
In general relativity, objects follow the shortest possible path through curved space-time called a geodesic and that there exists no such force of gravity which pulls objects, it is just because ...
2
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2
answers
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Is the determinant of metric tensor stationary wrt. proper time for a particle moving along its world line?
While writing the expression for stress energy tensor of a free massive particle moving along its world-line some authors take out of the integral sign, the $\sqrt{-g}$ where $g$ is the metric tensor ...
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1
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Momentum of a moving object in FRW metric according to an observer comoving with cosmic expansion
I would like to show that in an FRW metric the momentum of a freely falling object decays as the inverse of the scale factor. I know there are many proofs and arguments for this but I am trying to get ...
2
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2
answers
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Geodesics for FRW metric using variational principle
I am trying to find geodesics for the FRW metric,
$$
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),
$$
where $\mathbf{x}$ is 3-...
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1
answer
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De Sitter spacetime affine parameter
I am reading Chapter 8 in Carroll's "Spacetime and geometry " textbook and I was lead to exercise 8.2, given as:
Consider de Sitter space in coordinates where the metric takes the form
$$ds^{2} =...
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0
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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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0
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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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1
answer
128
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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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0
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201
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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 $\frac{d\Theta}{d\lambda}$ ...
2
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1
answer
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Why is the Lagrangian for space-like geodesics equal to 1?
In Schwarzschild spacetime, the Lagrangian can be defined as
$$ L = -\left( 1 - \frac{2M}{r} \right) \dot{t}^2 + \left( 1- \frac{2M}{r} \right)^{-1} \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2\theta ...
2
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1
answer
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Interpretation of Normal Coordinates
In my lecture notes, normal coordinates are defined as the following:
Def.: Let $\left(\mathrm{e}_\mu\right)$ be a basis of $\mathcal{T}_p\left(\mathcal{M}\right)$. Normal coordinates in a ...
2
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1
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Effective potential in a time-dependent spacetime
My question is regarding an arbitrary time-dependent spherically symmetric spacetime with line-element, in co-moving coordinates, to be
$$ds^2 = -f(R) dt^2 + a(t)\bigg\lbrace\frac{dR^2}{f(R)} +R^2d\...
3
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2
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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 $\...
3
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1
answer
378
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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,...
3
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0
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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; ...
2
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1
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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 ...
1
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3
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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....
3
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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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0
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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 ...
2
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1
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532
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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,
$$ \frac{d^2x^α}{dλ^2} = - Γ_{βγ}^α \frac{dx^β}{dλ} \...
3
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1
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Does a geodesic always extremize its path length? [duplicate]
I've learned that a geodesic maximizes its proper time in Minkowski spacetime.
Is this still true in general curved spacetime?
In other words, does the geodesic equation give the globally extremal ...
0
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1
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Does the worldline of light depend on the frequency?
My Question is:
If 2 rays of light are emitted with different frequencies from the same spacetime point, does an observer see them in the same worldline?
I know that the worldline of light behaves ...
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4
answers
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Are there null geodesics inside null infinity?
Looking at a Penrose diagram for Minkowski space, you would think that you could draw a null geodesic running from $i^0$, along $\mathscr{I}^+$, and ending up on $i^+$. In fact there would be many ...
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1
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Free particle in Rindler metric
I have to discuss the motion of a free particle in Minkowski using the Rindler metric
$$ds^2=e^{2ap}(-d\tau^2+dp^2)$$
So it has to satisfy the geodesic condition $\frac{d^2x^\mu}{d\tau^2}+\Gamma_{\rho\...
2
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1
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Differentiating Scalar along a geodesic
I have been studying GR for sometime and doing exercises from Schutz and I have a question about differentiating along a geodesic. Here is what I know. The equation of geodesic in terms of four ...
0
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1
answer
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Energy in spherically symmetric space times
In deriving the equations of motion for geodesics in spherically symmetric spacetimes through Hamiltonian formalism, we can find some constants of motion, namely, $E$ and $L$, the energy per unit of ...
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1
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Direction of gravity
General Relativity explains the path a falling body makes (ex. An apple falling toward the center of the Earth) as a geodesic in curved spacetime. What explains the direction the apple falls? In other ...
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1
answer
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Derivation of equation for geodesic deviation
I am trying to figure out the calculation which leads to the geodesic deviation on this site. So far I understood all steps until (14.7) and managed to show that (14.6) = (14.7), namely
$$ \ddot\xi^\...
2
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2
answers
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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 ...
2
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2
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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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1
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679
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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:
$$\mathcal{L}^2 = K(x,y,z,t)(...
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2
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Geodesic equations from action with auxiliary field
A textbook says that the geodesic equations (for both massive and massless) can be derived from the following action:
$$
S = -\frac{1}{2} \int d\tau \:\eta \: (\eta^{-2} \dot{x}^\mu \dot{x}^\nu g_{\...
1
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1
answer
517
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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 ...
5
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1
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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 ...
2
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0
answers
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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 $$ds^2 = \left(1-\frac{2m}{r}\right)dt^2-\left(1-\frac{2m}{r}\right)^{-1}dr^2-r^2\left(d\theta^2 + \sin^2\theta d\phi^2\...
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3
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What is the length of null geodesic?
There are many questions about this but none of them adresses my concrete question. If it is indeed true that for light we have $ds^2 = 0$ does that mean that in 4d spacetime total "distance" is zero ...
2
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2
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Relationship between freefall velocity time dilation and gravitational time dilation in a Schwarzschild metric
If you drop an object into a gravitational field, is its final velocity equal to what it would have to be in flat space in order to generate the same time dilation that you get at a given radius for ...
2
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0
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Periodic motion(s) on a torus
I recently came across the Lyusternik-Fet theorem concerning closed geodesics on a compact manifold.
For simplicity of description, take the 2-torus, and imagine it represents the configuration space ...
8
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2
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Geodesics of anti-de Sitter space
It is said that (p. 9), given the anti-de Sitter space $\text{AdS}_2$, let's say in the static coordinates
$$ds^2 = -(1 + x^2) dt^2 + \frac{1}{(1+x^2)} dx^2$$
Every timelike geodesic will cross the ...