A theory that describes how matter interacts dynamically with the geometry of space and time. It was first published by Einstein in 1915 and is currently used to study the structure and evolution of the universe, as well as having practical applications like GPS.

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How do we expect distance measurements to compare inside and outside the event horizon of a black hole?

I've read that as one approaches the event horizon of a black hole, time is dilated relative to time measured farther away from the event horizon (clocks tick slower near the event horizon). I've ...
4
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3answers
149 views

Is a metric tensor field the same thing as $ds² = -dt² + dx²+ dy² + dz²$?

I am having trouble understanding the nature of the metric tensor field on spacetime manifolds. In particular, a Riemannian manifold $(M,g)$ is defined as a real smooth manifold $M$ equipped with an ...
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2answers
151 views

Tensors as multilinear maps

Sean Carrol's in his book on GR introduces tensors as a multilinear map of a set of dual vectors and vectors onto R. I usually think of tensors as a multidimensional array of numbers with fixed ...
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1answer
76 views

Where is the mass energy of potential energy (such as for the universe?)

As you know, energy has mass via $E=mc^2$. If I understand right, mass must be inside of a particle, and can not just be "free floating." Gravitational potential energy (or what ever its equivalent ...
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1answer
224 views

Geodesic equation from the proper time integral

This is something that has been bothering me for a little while. The usual procedure that I've seen is to write the proper time as the line integral $$\tau=\int_\gamma d\tau$$ along some curve $\gamma$...
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2answers
126 views

Are time and gravity affected when at rest compared to free fall?

A falling object moves along a geodesic path ('straight path') in spacetime. When it comes to rest it now follows a 'curved path' through spacetime. Is the passage of time and force of gravity ...
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1answer
142 views

Curved paths through spacetime when standing still?

I have heard that falling objects fall at the same rate irrespective of their mass. They are 'following straight line paths through curved spacetime'. Does this mean that objects that accelerate in ...
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0answers
25 views

Einstein's equation: good, intuitive introduction anywhere? [duplicate]

Baez - like most others - gives (IMHO) a quite unsatisfactory introduction to Einstein's equation. They all start with the equivalence principle or similarly unintuitive reasoning, relating it to ...
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1answer
152 views

General relativity: is curvature of spacetime really required or just a convenient representation?

I'm not really far into the general theory of relativity but already have an important question: are there formulations that can do without spacetime curvature and describe the general theory of ...
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1answer
309 views

Gravitational wave solutions to the Einstein field equations

It is well known that general relativity predicts gravitational waves, but I would like to know how. What solution(s) to the Einstein field equations yield something which can be interpreted as a wave(...
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3answers
207 views

Expanding universe as predicted by the Einstein Field Equation

Without the cosmological constant, the Einstein field equation predicts the universe is expanding. Why is that? It is counter-intuitive because generally gravity should pull things closer and shrink ...
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2answers
838 views

Books on cosmology

I am a 14 year old who is independently studying physics. I finished the book: Spacetime and Geometry: An Introduction to General Relativity by Sean Carroll. I am specifically interested in cosmology, ...
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1answer
75 views

How to demonstrate frame dragging through the Kerr metric?

I derived the Kerr metric, but in a form which doesn't seem to relate to frame dragging. I have been trying this for some time, so how do we relate the Kerr metric to frame dragging?
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1answer
112 views

Schwarzschild Solution

I'm able to derive the Schwarzschild solution under the assumptions that the metric is (1) static (2) spherically symmetric and that the space is the vacuum. However, I have read that the ...
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2answers
99 views

How much additional light does Earth receive from the Sun due to Earth's gravitational field?

I was reading about how gravity affects light, and that got me wondering how much additional light is collected by the Sun due to the Earth's gravitational field. Is it a significant amount of light (...
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2answers
298 views

Difference between the metric tensor in general relativity and the metric tensor in mathematics?

Is the metric tensor in general relativity the same as the metric tensor in maths, or is there a difference?
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2answers
141 views

Why is the cosmological constant a scalar?

Maybe my understanding is just off, but the cosmological constant is just a scalar, right? What are it's units? Why a scalar? - was a tensor 'cosmological constant' ever considered or is it just not ...
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1answer
458 views

Galactic Rotation Speeds - Ehrenfest Paradox, Gravitational time dilation, Dark Matter - all of the above?

The observed paths and speeds of objects, part of some distant galaxy, do not match up with speed vs distance curves it seems - the observed speeds are not falling off in fact they're trending as ...
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0answers
64 views

Does relativistic glider violate principle of equivalence?

The relativistic glider proposed can slow down the fall of an object in gravitational field. Will this violate the principle of equivalence which says that one cannot distinguish between free falling ...
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1answer
188 views

How did Einstein arrive at the right hand side of his general relativity tensor equation?

It seems Einstein postulated the right hand side of his field equations… I see in books that tell the story of how he arrived at what to put on the right hand side of his tensor equations, the physics ...
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2answers
870 views

Curvature of spacetime as a real thing?

I get the curvature tensor in General Relativity, it is “just” math. Does space-time REALLY curves as a tangible thing, or is Einstein proposing a mathematical abstraction? More naively, please allow,...
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3answers
196 views

What is the evidence of interpreting $g_{\mu\nu}$ as the metric of space-time?

I think if we don't mention the meaning of $g_{\mu\nu}$ as the metric of space-time, we can still construct the equation of motion and Einstein field equation in a way such that $g_{\mu\nu}$ is just a ...
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1answer
71 views

Flatness and Kinetic Energy

Why the curvature parameter can be interpreted as the difference between the average potential energy and the average kinetic energy of a region of space? Curvature parameter:$$\Omega_{k}\equiv\Omega-...
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3answers
167 views

Does gravitational time dilation affect apparent mass?

Suppose I'm on a highly elliptical orbit around some massive body. As I get closer, I start to experience time dilation due to the object's gravitational field - time starts passing slower for me than ...
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2answers
123 views

Time Dilation Properties

I've read up on time dilation and how gravitational/kinematic time dilation work but I have not received a clear answer on how the two work together. If you are observing something traveling at a ...
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2answers
1k views

Spacetime geometry around two black holes

For space-time geometry we all know images like he one below. But if I were to take a neutron star and put it right next to but not touching a black hole what effect would that have on the geometry of ...
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2answers
700 views

Feynman's statement of the Einstein Field Equations

In Feynman's Lectures on Physics (Volume 2, chapter 42) he states that Einstein's field equation is equivalent to the statement that in any local inertial coordinate system the scalar curvature of ...
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1answer
103 views

Post-Newtonian acceleration formula

First of all, I have to say that I know very little about tensors and partial differential equations, as I've never done them at school. I'm developing an application which simulates the motion of ...
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4answers
100 views

Effect of expansion on time?

We know that for moving body times moves at a slower rate, but can this happen for Universe itself whose rate of expansion is accelerating?
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37 views

Where does the dark energy of space expansion come from? [duplicate]

Space has a constant density of dark energy, that means dark energy does not dilute with space expansion. By consequence, any space generated by space expansion does generate new dark energy. ...
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3answers
332 views

why do x Schwarzschild radii equal time dilation effects of speed of light going y times faster than an object^2?

let me walk you through the math. First you start with the gravitational time dilation formula where: $$ T_1=T\sqrt{1-\frac{2GM}{rc^2}} $$ and rather than entering $r$ for the radius we replace $r$ ...
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2answers
440 views

Is space stretched with no limits by a black hole?

Some depictions of black holes show space being warped into a singularity, with no end, e.g. as pictured below. Moreover, in Cosmos, Neil Tyson speculates with the possibility that Black Holes contain ...
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2answers
134 views

Is the surface of a heavy sphere bigger than $4 \pi r^2$ due to general relativity?

I am unfortunately not familiar with the mathematics behind general relativity. However, on a heavy planet (say a sphere) gravity will bend space-time in a way that an object initially in rest, will ...
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3answers
2k views

Event Horizon of Supermassive Black Holes

I'm going to ask/explain this as best I can; I'm sure I have some fundamentals wrong here. Spaghettification is a phenomenon which occurs only in stellar-mass black holes owing to the immense gravity ...
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2answers
238 views

Is gravity a force? [duplicate]

If gravity just emerges from the curvature of spacetime, is it actually a force? Why is it one of the 4 fundamental forces of nature?
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1answer
387 views

Time coordinate inside black hole horizon [duplicate]

I am new to physics and was trying to learn more especially about general relativity. The Schwarzschild metric, changes the sign of the time and radial parts of the metric once we cross the event ...
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5answers
873 views

Does coordinate time have physical meaning?

I have always been a little confused by the meaning of the "$t$" which appears in spacetime intervals or metrics in general relativity. I concluded that $t$ was just a mathematical thing which allow ...
6
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1answer
242 views

Null Coordinates

I have a very basic question: what are the advantages of writing a metric in the null coordinates? Which extra insight do they provide? I've looked in Caroll's "Spacetime and Geometry" and Wald's "...
2
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0answers
148 views

Energy-momentum tensor

I need to show that: \begin{align} \mathcal h_i^a \, T_{ab} \, h_i^b=(\nabla_i \phi)^2-\frac{h_{ii}}{2}[\dot{\phi}^2-(\nabla \phi)^2-m^2 \phi^2] \end{align} where i) $T_{ab}=\nabla_a \phi \...
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1answer
194 views

Are a black hole really connecting 2 parallel universe?

What is beyond the point of singularity in a black hole and universe expands in all the 3 $x$, $y$ and $z$ axis? So how can one thing be on top of another when we don't know where the end point of ...
5
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1answer
118 views

How is $\Omega_0 = 1$ when the characteristic “teardrop” past light cone seems to admit curvature?

Introduction: The top graphic is just one I pulled from a page describing the process of detecting cosmic curvature. The second graphic is one I drew up to illustrate my misunderstanding. My ...
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2answers
315 views

Curved space-time VS change of coordinates in Minkowski space

I'm looking for a rather intuitive explanation (or some references) of the difference between the metric of a curved space-time and the metric of non-inertial frames. Consider an inertial reference ...
5
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2answers
237 views

Can we test QFT on a curved spacetime?

It is possible to extend a quantum field theory to a curved spacetime. But does this lead to predictions that can be tested and measured? Had it been confirmed? The underlying reason I am asking this ...
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2answers
299 views

Particles inside black hole does not hit singularity

We say that if any particles from outside fall into the black hole, they eventually hit singularity. Then why not particles already inside the black hole are at singularity? Or are they? If yes, then ...
5
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2answers
158 views

How to prove explicitly that by including Dirac fermions into the Einstein-Hilbert action we make torsion to be non-zero?

Recently I've heard the statement that by including Dirac fermions into the Einstein-Hilbert action we make torsion be non-zero, so that is one of problem of quantum gravity. How to prove that ...
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1answer
316 views

Relationship between mass and the radius of curvature of space and time

What is the relationship between mass and the radius of curvature of space and time created due to the presence of the mass? please give the mathematical relation if there is any?
2
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1answer
140 views

Why gamma-matrices are associated with tetrads Lorentz rotation?

In Zee's "QFT in nutshell" in a paragraph "Differential geometry of Riemann manifold" he states that Dirac gamma-matrices are associated with tetrads Lorentz rotation, so Dirac lagrangian in curved ...
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2answers
252 views

Raising and Lowering indices of tensor

Why we use metric tensors $g$ to raise or lower indices of tensors, why not using other (invertible) order-2 tensors to do the job?
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1answer
106 views

Proof of expression to combine different redshifts

I am trying to arrive to the following expression: $1+z = (1+z_C)(1+z_G)(1+z_D)$ so the total redshift is the product of the cosmological redshift, the gravitational redshift and the Doppler ...
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818 views

How does a rotating object cause frame dragging?

Frame dragging is a consequence of general relativity. But I don't really understand it. Of course I can find metaphors like the "honey metaphor" where stirring a honey can move the specks even if ...