## Spacetime curvature

at the center of a planet/star/etc

Study matter and its motion through spacetime...

### Re: Spacetime curvature

We might as well include a Lorentz transformation for completeness:

From Wikimedia, animated by John Doolan

As you can see, while the distance through space and the distance through time can vary, the distance through spacetime is invariant. This is how we derive that moving through space slows motion through time or, in other words, why time slows down as you move through space.

hackenslash

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### Re: Spacetime curvature

Thanks Hack. I understood that. I think that may be the necessary stepping stool for understanding the maths presented by Twist as well.
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### Re: Spacetime curvature

Thanks everyone. So does this mean that the equivalence principle can't be applied to the center of a gravitational well?

Wikipedia wrote:In the physics of general relativity, the equivalence principle refers to several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's assertion that the gravitational "force" as experienced locally while standing on a massive body (such as the Earth) is actually the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference.

http://en.wikipedia.org/wiki/Equivalence_principle

It makes me wonder about assumptions though. Like the equivalence principle rests on assumptions for simplification, does the math include assumptions (most models have assumptions, after all) that are still valid when applied to the center of gravitational wells? If spacetime is still heavily curved near the center, does this represent another singularity at the center (point) or is it my own failings of imagination that I can't imagine a curved point?

I googled a bit (which I probably should've done more thoroughly before I started the thread) and found that this discussion has been had on other forums as well, and someone suggested experiments to verify.
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### Re: Spacetime curvature

Xaihe wrote:Thanks everyone. So does this mean that the equivalence principle can't be applied to the center of a gravitational well?

Wikipedia wrote:In the physics of general relativity, the equivalence principle refers to several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's assertion that the gravitational "force" as experienced locally while standing on a massive body (such as the Earth) is actually the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference.

http://en.wikipedia.org/wiki/Equivalence_principle

I'm not quite with you. Why do you say that it doesn't apply there ?
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### Re: Spacetime curvature

twistor59 wrote:
Xaihe wrote:Thanks everyone. So does this mean that the equivalence principle can't be applied to the center of a gravitational well?

Wikipedia wrote:In the physics of general relativity, the equivalence principle refers to several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's assertion that the gravitational "force" as experienced locally while standing on a massive body (such as the Earth) is actually the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference.

http://en.wikipedia.org/wiki/Equivalence_principle

I'm not quite with you. Why do you say that it doesn't apply there ?

What I mean is that if time goes slower closer to the center of a gravitational well, then the time dilation must be disconnected from the gravitational acceleration. In other words, the time dilation isn't the same anymore as in a similar situation with the same acceleration (outside a gravity well).
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### Re: Spacetime curvature

It is, because you're still experiencing gravitational acceleration, you're just experiencing it in all directions, rather than just a single direction. In other words, you're still immersed in a gravity well, or rather many gravity wells, all adding to the time dilation.

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### Re: Spacetime curvature

Yes, but isn't that the same as accelerating inside an atmosphere inside a very long box that is accelerating the other way? In effect, stationary to an outside observer but at the same time double the acceleration? It doesn't work like that, does it?
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### Re: Spacetime curvature

Let me try to explain one more time:

In the situation when you're on the surface of the planet, all the gravitational attraction is coming from one direction, loosely speaking, like this:

There is some gravitational contribution from off to the sides, but most of the contribution is directly down. In the case of being at the centre of the gravity well, the contribution is from all directions, like this:

In fact, the gravity is slightly stronger, because the attraction is from the entire planet, not just the portion of it that's below you. In other words, the gravitational contribution is total, which accounts for the additional dilation highlighted above by Twistor. Bear in mind that these are two-dimensional representations of three-dimensional phenomena, and that this contribution lies in all directions.

The actual effect that the body would feel would be exactly akin to no gravitational effect whatsoever, because the contribution from all directions means that it's effectively cancelled out, but the effect on time is not cancelled out, because the gravitational acceleration is still present.

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### Re: Spacetime curvature

I know what you're trying to explain, but I don't think you understand my point exactly. What I'm questioning is whether there is an assumption here and whether that assumption is valid.
First, my own assumption is that mass curves spacetime and that this accounts for gravity and time dilation. Then, the assumption necessary for the picture you laid out is that spacetime at a particular point can be curved in multiple (possibly opposing) directions at once. And this opposite curvature would then account for the apparent decreased acceleration and increased time dilation. I'm just having some trouble wrapping my head around this idea (which is no argument).
If my interpretation is wrong, please let me know where (and if you would, why).
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### Re: Spacetime curvature

Xaihe wrote:I know what you're trying to explain, but I don't think you understand my point exactly. What I'm questioning is whether there is an assumption here and whether that assumption is valid.
First, my own assumption is that mass curves spacetime and that this accounts for gravity and time dilation. Then, the assumption necessary for the picture you laid out is that spacetime at a particular point can be curved in multiple (possibly opposing) directions at once. And this opposite curvature would then account for the apparent decreased acceleration and increased time dilation. I'm just having some trouble wrapping my head around this idea (which is no argument).
If my interpretation is wrong, please let me know where (and if you would, why).

I don't know how to answer your question, but just to clarify what seems to be a misconception: time dilation is a result of Special Relativity, which has nothing to do with gravity. And curvature is a result of General Relativity, which has everything to do with gravity.

Now, it's true that a difference in gravitational potential will affect clocks -- for instance, this is taken into account for GPS operation, but this gravitational correction is not referred to as time dilation.

Neeedless to say, time dilation and gravitational correction are calculated using entirely different equations.

Time dilation: dτ ≈ dt ( 1 + v2/2c2)
Gravitational correction: dτ ≈ dt( 1 + GM/Rc2)

Hope this helps

zaybu

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### Re: Spacetime curvature

There is some truth to that, but the equivalence principle tells us that time dilation also occurs when immersed in a gravitational field, meaning that there is indeed a GR solution for time dilation. Twistor's above calculation gives the solution employing the Schwarzschild interior metric, and it is indeed referred to as gravitational time dilation.

Google scholar results for 'gravitational time dilation.

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### Re: Spacetime curvature

Xaihe wrote:I know what you're trying to explain, but I don't think you understand my point exactly. What I'm questioning is whether there is an assumption here and whether that assumption is valid.
First, my own assumption is that mass curves spacetime and that this accounts for gravity and time dilation. Then, the assumption necessary for the picture you laid out is that spacetime at a particular point can be curved in multiple (possibly opposing) directions at once. And this opposite curvature would then account for the apparent decreased acceleration and increased time dilation. I'm just having some trouble wrapping my head around this idea (which is no argument).
If my interpretation is wrong, please let me know where (and if you would, why).

It's a bit difficult to pick the bones out of this, but I'll try. The best analogy I can come up with is to think of a gravity well as being somewhat like a tornado, but one that lies in all directions. The classical image of the gravity well is this:

Which is not very accurate, not least because the bottom of the well would be centred on the centre of mass of the planet. If you think of that same gravity well existing in every direction, you begin to see a picture that looks a bit more like this:

Although even this isn't very accurate, not least because it's almost impossible to represent this phenomenon in a two-dimensional image, due to the inherent limitations in such a representation.

Now, if you take the image of the gravity well as a tornado, you can think of the centre of that gravity well that exists in all spacetime dimensions as being somewhat like the eye of the tornado, where everything is calm. Indeed, even given a one-dimensional well, it would still be relatively calm, because there's no further to fall when you're at the bottom of the well, but the curvature of spacetime still exists there, and it is maximal for the appropriate mass. Thus, time dilation is still in effect, because the warping of spacetime is still in effect, and it can indeed be curved in lots of different directions at once, because it's curved in all directions.

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### Re: Spacetime curvature

hackenslash wrote:There is some truth to that, but the equivalence principle tells us that time dilation also occurs when immersed in a gravitational field, meaning that there is indeed a GR solution for time dilation. Twistor's above calculation gives the solution employing the Schwarzschild interior metric, and it is indeed referred to as gravitational time dilation.

Google scholar results for 'gravitational time dilation.

Thanks for the clarification

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### Re: Spacetime curvature

Xaihe wrote:I know what you're trying to explain, but I don't think you understand my point exactly. What I'm questioning is whether there is an assumption here and whether that assumption is valid.
First, my own assumption is that mass curves spacetime and that this accounts for gravity and time dilation. Then, the assumption necessary for the picture you laid out is that spacetime at a particular point can be curved in multiple (possibly opposing) directions at once. And this opposite curvature would then account for the apparent decreased acceleration and increased time dilation. I'm just having some trouble wrapping my head around this idea (which is no argument).
If my interpretation is wrong, please let me know where (and if you would, why).

Is it something like this that's troubling you:

"There has to be a gravitational field to produce the time dilation effect, but at the centre of the planet, there is clearly no gravitational field, because an object there feels no force at all, so how the fuck can we see time dilation ?"

One issue is that thinking in terms of a "gravitational field" gets confusing in relativity unless you're very specific about what you mean by it. The object in relativity that is most analogous to the Newtonian "gravitational field" is the connection, (also called the Christoffel symbols). In Newtonian theory the thing you differentiate to get the gravitational field is the potential, and the relatvitistic analog of the potential is the metric. The first derivatives of the metric give the connection coefficients.

Newton
.............................................Einstein

Potential...........................................Metric
....|.....................................................|......
.differentiate....................................differentiate
....|.....................................................|......
....V.....................................................V.....
..Field...........................................Connection

Now, the reason why the field (connection) can be a bit misleading is that at any point, you can find a coordinate system which makes it vanish at that point. Effectively, you choose a local freely falling frame (geodesic normal coordinates), and hey presto, what you thought was a gravitational force vanishes there (if you have a tidal component to your field you can detect it by noting how the paths of freely falling particles diverge or converge there).

However, note that there's more to physics than just the first derivative of the metric at a point. There's also the second derivatives, from which you can construct the curvature. I haven't tried computing it for the Schwarzschild interior metric, but I'm sure the curvature doesn't vanish at that point.

Anyway, for the purposes of analysing the time dilation in the OP, we want to compare two identical clocks, one at the centre and one at the surface. The thing is, to get from the centre to the surface, we have to travel a fair old distance, climbing out of the gravity well in the process. So the gravitational field (connection) value at a single point (in this case the earth's centre) is NOT determining the dilation effect. It's the comparison of the metric (potential) at the two points that determines this.
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### Re: Spacetime curvature

Man! He looks happy to be in the center of Jupiter...
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### Re: Spacetime curvature

CdesignProponentsist wrote:Man! He looks happy to be in the center of Jupiter...

Don't take the piss. Hack's art teacher gave him a gold star for that.
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### Re: Spacetime curvature

I think it may be better to think of it this way. The distortion of spacetime increases as you approach the surface, but as you approach the center of mass the spacetime distortion increases at a lesser and lesser rate. The distortion is still there and at maximum at the center of mass, but never gets any less until you pass it.

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?
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### Re: Spacetime curvature

Interesting stuff here. To make sure I understand the explanations offered, I had a simple follow up question:

Let's say there is a hole running from one end of the earth to the other through the center of the earth. As we know, an object dropped into such a hole (ignoring the effects of rotation), will execute simple harmonic motion with a period of 84 minutes and 23 seconds. However (per my understanding), this period for the SHM is from the reference frame of an observer on the earth's surface. Now, if the object dropped into the hole is me, and I'm wearing a watch, how will my watch record the period of the oscillation?
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### Re: Spacetime curvature

CdesignProponentsist wrote:I think it may be better to think of it this way. The distortion of spacetime increases as you approach the surface, but as you approach the center of mass the spacetime distortion increases at a lesser and lesser rate. The distortion is still there and at maximum at the center of mass, but never gets any less until you pass it.

The curvature increases as you approach the surface from outside. I'm not sure about the behaviour inside the body, and I'm sure not going to calculate the curvature components for the SIS !

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.
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### Re: Spacetime curvature

iamthereforeithink wrote:Interesting stuff here. To make sure I understand the explanations offered, I had a simple follow up question:

Let's say there is a hole running from one end of the earth to the other through the center of the earth. As we know, an object dropped into such a hole (ignoring the effects of rotation), will execute simple harmonic motion with a period of 84 minutes and 23 seconds. However (per my understanding), this period for the SHM is from the reference frame of an observer on the earth's surface. Now, if the object dropped into the hole is me, and I'm wearing a watch, how will my watch record the period of the oscillation?

Presumably you're interested in the answer relative to someone's watch who stays at the surface. The max grav. dilation with respect to him will be when you're at the centre. The overall difference will be a mean over the range r=0 to r= R. You would also theoretically have to include the dilation due to motion, so you'd have the dr metric component which I ignored above. So the answer would involve integrating over the range. It would still be damn small though.
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