## Spacetime curvature

at the center of a planet/star/etc

Study matter and its motion through spacetime...

### Re: Spacetime curvature

twistor59 wrote:
CdesignProponentsist wrote:
So in effect that would mean that the acceleration that you feel (how many Gs) is the rate of change in spacetime distortion and not the overall distortion relative to an outside inertial frame.

Is that correct?

By the acceleration you mean the force ? That's related to the connection coefficients, which are the rates of change of the spacetime metric.

Define connection coefficient (in layman's terms please )

And what is the difference between gravitational acceleration and gravitational force?
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### Re: Spacetime curvature

CdesignProponentsist wrote:Define connection coefficient (in layman's terms please )

They're the things that allow you to do parallel transport of a vector. I'll let John Baez describe parallel transport to you because he's good at making things visual:

http://math.ucr.edu/home/baez/gr/parallel.transport.html

A geodesic is a curve that parallel transports it's own tangent vector. It's thus the "straightest" curve possible between two points on a manifold (like spacetime) that may be curved. It's the curve that a freely falling body will follow.

CdesignProponentsist wrote:
And what is the difference between the acceleration and gravitational force?

only a factor of m.
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### Re: Spacetime curvature

It think I got it but I don't have the vocabulary to express it.

The rate of change in the spacetime metric increases as you reach the surface, but will start to decrease at some point as you approach the center of mass which results in a decrease in gravitational force. Correct?

Gravitational acceleration which distorts time and space always increases down to the center of mass. Correct?
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### Re: Spacetime curvature

twistor59 wrote:
CdesignProponentsist wrote:Define connection coefficient (in layman's terms please )

They're the things that allow you to do parallel transport of a vector. I'll let John Baez describe parallel transport to you because he's good at making things visual:

http://math.ucr.edu/home/baez/gr/parallel.transport.html

ETA by the way a picture of the scenario Baez is describing is in the wikipedia article

A geodesic is a curve that parallel transports it's own tangent vector. It's thus the "straightest" curve possible between two points on a manifold (like spacetime) that may be curved. It's the curve that a freely falling body will follow.

CdesignProponentsist wrote:
And what is the difference between the acceleration and gravitational force?

only a factor of m.
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### Re: Spacetime curvature

Bollocks that was meant to be an edit not a new post
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### Re: Spacetime curvature

CdesignProponentsist wrote:It think I got it but I don't have the vocabulary to express it.

The rate of change in the spacetime metric increases as you reach the surface, but will start to decrease at some point as you approach the center of mass which results in a decrease in gravitational force. Correct?

CdesignProponentsist wrote:
Gravitational acceleration which distorts time and space always increases down to the center of mass. Correct?

The thing that distorts time and space is the energy/momentum of the matter (in this case the planet). This is the source of the distortion.
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### Re: Spacetime curvature

Okay, getting closer

twistor59 wrote:

CdesignProponentsist wrote:
Gravitational acceleration which distorts time and space always increases down to the center of mass. Correct?

The thing that distorts time and space is the energy/momentum of the matter (in this case the planet). This is the source of the distortion.

How is gravitational acceleration come into play then?

And what is the difference in gravitational acceleration between an object falling and an object standing on the surface?
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### Re: Spacetime curvature

I don't mean to throw a monkey wrench into the discussion, but I noticed this:

Xaihe wrote:Since gravity is stronger at the surface of a planet than at the center of a planet, does that also mean spacetime is curved less at the center of the planet, or just that the gravitational effect is canceled?

If I remember correctly, gravity is stronger at the center of a planet, not at its surface. This is why gravity is slight stronger at the poles of the Earth than at the Equator; the poles are closer to the core.

Am I wrong, or am I just reading the comment incorrectly?

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### Re: Spacetime curvature

Nautilidae wrote:

If I remember correctly, gravity is stronger at the center of a planet, not at its surface.

No. At the center you have the entire mass distributed equally around you. So the force is canceled like a marble at the bottom of a bowl.

...gravity is slight stronger at the poles of the Earth than at the Equator; the poles are closer to the core.

Yes. You are slightly closer to the core which is much more dense than the mantle, but it is still completely distributed below you whereas if you were IN the core, you would have it distributed all around you, again canceling out for a sum total of 0 gravitational force.

But it also is due to the rotation of the earth as well to a slightly lesser degree.

http://en.wikipedia.org/wiki/Gravity_of_Earth#Latitude
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### Re: Spacetime curvature

CdesignProponentsist wrote:
Nautilidae wrote:

If I remember correctly, gravity is stronger at the center of a planet, not at its surface.

No. At the center you have the entire mass distributed equally around you. So the force is canceled like a marble at the bottom of a bowl.

...gravity is slight stronger at the poles of the Earth than at the Equator; the poles are closer to the core.

Yes. You are slightly closer to the core which is much more dense than the mantle, but it is still completely distributed below you whereas if you were IN the core, you would have it distributed all around you, again canceling out for a sum total of 0 gravitational force.

But it also is due to the rotation of the earth as well to a slightly lesser degree.

http://en.wikipedia.org/wiki/Gravity_of_Earth#Latitude

Ah, alright. Thanks for the information.

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### Re: Spacetime curvature

CdesignProponentsist wrote:Okay, getting closer

How is gravitational acceleration come into play then?

And what is the difference in gravitational acceleration between an object falling and an object standing on the surface?

Need to be precise about terminology here. In GR people don't usually talk about "gravitational acceleration", but rather have two concepts:

Coordinate Acceleration
This would be the rate of change of a space coordinate with respect to the time coordinate for an object in free fall. So for example in the Schwarzschild case, where the coordinates are (t, r, θ, φ), the radial coordinate acceleration component of a radially free-falling object would be dr/dt. You would compute this by working out the geodesic equation.

Proper Acceleration

To get the proper acceleration experienced by an object, you first define an inertial observer who's going to be the reference for the proper acceleration. This reference observer should be initially stationary relative to the object. So for example, if I have an object sitting at a point at distance r from the centre, I take an observer, sit him right by the object, then let the observer drop in free fall. The proper acceleration is the relative acceleration (derivative of four-velocity) between the two. In Newtonian mechanics the proper acceleration would be GM/r2, but in the Schwarzschild solution in GR, there will be a small correction to this involving some more G's, M's and an few c's
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### Re: Spacetime curvature

twistor59 wrote:
hackenslash wrote:
If anybody wants to see this explained in greater detail, I recommend Why Doe E=mc2, by Brian Cox and Jeff Forshaw, from which this example is borrowed, as are the diagrams.

Edit: Tags

Nice explanation
I also (after your suggestion) read Cox & Forshaw's book, and can heartily recommend it to anybody who's interested in relativity.

I just got that book today. Going to start reading it after dins.
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### Re: Spacetime curvature

Animavore wrote:
twistor59 wrote:
hackenslash wrote:
If anybody wants to see this explained in greater detail, I recommend Why Doe E=mc2, by Brian Cox and Jeff Forshaw, from which this example is borrowed, as are the diagrams.

Edit: Tags

Nice explanation
I also (after your suggestion) read Cox & Forshaw's book, and can heartily recommend it to anybody who's interested in relativity.

I just got that book today. Going to start reading it after dins.

We see yo' on here tonight, we know yo' ain't readin'. Yo' in trouble.
Enjoy
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### Re: Spacetime curvature

twistor59 wrote:
Animavore wrote:
twistor59 wrote:

Nice explanation
I also (after your suggestion) read Cox & Forshaw's book, and can heartily recommend it to anybody who's interested in relativity.

I just got that book today. Going to start reading it after dins.

We see yo' on here tonight, we know yo' ain't readin'. Yo' in trouble.
Enjoy

Just read the first 2 chapters. I like that they skimmed over scientific history. Too many books spend a long time going through this. Haven't come across anything new, yet. Just taking a brain break with a go of Killzone 3 then back to read chapter 3.
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### Re: Spacetime curvature

Geronwivit!

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### Re: Spacetime curvature

This book is making my head hurt
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### Re: Spacetime curvature

Animavore wrote:This book is making my head hurt

Well a few of us have copies, so will be happy to try to help with bits you're not happy with...
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### Re: Spacetime curvature

twistor59 wrote:
Animavore wrote:This book is making my head hurt

Well a few of us have copies, so will be happy to try to help with bits you're not happy with...

Oh I will. Especially how he made the leap to a minus figure for Pytagoras' therom. I don't know if my mind wandered and I missed a bit or if this book is a mess.

But I'll come back after I've finishd it so don't answer now, it'll only confuse the issue more.
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### Re: Spacetime curvature

Animavore wrote:Oh I will. Especially how he made the leap to a minus figure for Pytagoras' therom. I don't know if my mind wandered and I missed a bit or if this book is a mess.

He does address it, specifically with the example I gave above. Using the plus version violates causality, because it gives a situation in which two events in spacetime the same distance s apart can happen in reverse, even if one of the events is the cause of the other. Think of a bullet fired from a gun and hitting a target. In the plus sign version of the equation, the bullet can hit its target before it's fired from the gun. The jump to the minus sign version protects causality, because it generates a non-Euclidean spacetime, known as Minkowski spacetime.

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### Re: Spacetime curvature

hackenslash wrote:
Animavore wrote:Oh I will. Especially how he made the leap to a minus figure for Pytagoras' therom. I don't know if my mind wandered and I missed a bit or if this book is a mess.

He does address it, specifically with the example I gave above. Using the plus version violates causality, because it gives a situation in which two events in spacetime the same distance s apart can happen in reverse, even if one of the events is the cause of the other. Think of a bullet fired from a gun and hitting a target. In the plus sign version of the equation, the bullet can hit its target before it's fired from the gun. The jump to the minus sign version protects causality, because it generates a non-Euclidean spacetime, known as Minkowski spacetime.

I got that bit. He doesn't explain why it does or how he gets a circle or why using a minus gives you a hyperbola in the top only that it does.
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