Relationship between E=mc^2 and gravity

[pgcurt]

As I understand it, a way to look at E=mc^2 is to think of it as the energy required to accelerate a given amount of mass.

I also understand that mass warps space-time creating an acceleration (gravity) in objects near that mass. Is there a direct correlation between the energy required to accelerate an object and the potential energy (acceleration from gravity) created by the warping of space-time?

I apologize in advance for any bad assumptions I may have made above. I'm mostly an astro-physicist wannabe

 

[pervect]

The way I suggest looking at it is to view energy as causing gravity in GR, rather than mass.

The idea that mass causes gravity is a carryover from Newtonian gravitation.

In GR, it's energy that causes gravity (more precisely, the stress energy tensor).

I also am not fond of relativistic mass, though one sees the explanation you offered in the literature often enough.

You might want to check out does mass change with speed? on this topic.

 

[pgcurt]

That link was helpful. I think I need to get my arms around energy tensors.

On a different note, are you aware of any good sites explaining how to interpret embedding diagrams? I find the bowling ball on a trampoline analogy confusing, as there isn't really any "down". They seem to imply gravity which is what it's supposed to be explaining! What cross-section of space-time is an embedding diagram supposed to represent? What are the axisies? How is acceleration represented? I don't really need all these answered; a good site that explains how to read them would be very helpful. Thanks in advance.

 

[jtbell]

I find the bowling ball on a trampoline analogy confusing, as there isn't really any "down".

The key word is "analogy". This is in fact only a very crude analogy for general relativity, and you shouldn't try to push it very far. All attempts to "explain" modern physics by analogy with parts of classical physics eventually break down, and this analogy breaks down sooner than most.

 

[Santural]

True.
So from your question, pgcurt, I learned something new.
And to the posts below:
Thanks for the embedding diagrams! That answers my deleted question.

 

[smallphi]

That link was helpful. I think I need to get my arms around energy tensors.

On a different note, are you aware of any good sites explaining how to interpret embedding diagrams? I find the bowling ball on a trampoline analogy confusing, as there isn't really any "down". They seem to imply gravity which is what it's supposed to be explaining! What cross-section of space-time is an embedding diagram supposed to represent? What are the axisies? How is acceleration represented? I don't really need all these answered; a good site that explains how to read them would be very helpful. Thanks in advance.

Embedding diagrams visualize the geometry of a two dimensional metric (or 2 dim slice of a metric) as a surface embedded in 3 dimensional space since we have everyday intuition and experience with such surfaces.

I found my current understanding of embedding diagrams in James Hartle's GR book, section 7.7 and I highly recommend reading it.

 

[pervect]

I don't have Hartle's book, though I've heard good things about it.

Most of the sources I've seen use embedding diagrams to describe the geometry of space, but have seen one that uses embedding diagrams to describe the geometry of space-time for the Schwarzschild geometry.

This is http://arxiv.org/PS_cache/gr-qc/pdf/9806/9806123v3.pdf

and came up in another recent thread.

Is this the sort of diagram that Hartle used? By embedding space-time rather than space, it shows how one can get the right results for gravitational time dilation, etc, by drawing space-time diagrams on an ordinary 2-d manifold visualized as the surface of a 3-d object. It describes the space-time geometry around a single massive object, i.e. the Schwarzschild geometry.

I also did a homegrown version that works for only a part of the space-time outside the event horizon of a black hole in

https://www.physicsforums.com/showpost.php?p=1305675&postcount=30

Be warned that this hasn't been checked very thoroughly But it's simpler in some respects than the one in the literature.

Basically time, over a restriced range, maps to the theta coordinate of some surface of revolution, which is a cylinder that tapers down to a point at the event horizon. (Note that the mapping of the event horizon to a point is an unfortunate feature of this simple embedding - the event horizon isn't really a point!).

The radial coordinate r is drawn so that it is always orthogonal to time.

 

[pgcurt]

The PDF is helpful. I've spent a small fortune on GR books for beginners, but find most of them sacrifice accuracy for intuitiveness, which ironically makes them less intuitive for me. The PDF is helpful. I think I'll also invest in Hartle's book.

I had "understand GR" as a new year's resolution for last year. I should've just had "lose 10 pounds" like everyone else!

 

[smallphi]

Spacetime is 4 dimentions total (3 space + 1 time). That is why a general metric depends on 4 coordinates. By setting two of the coordinates to constants (with the black holes they usually fix the two angle coordinates, meaning they restrict attention only to radial motion) you get a metric of only two coordinates. That is called 'slicing the spacetime'.

The metric of two coordinates is embedded in a 3 diminensional space. The 3 dim space is usually Euclidean (3 space dimensions) and Hartle's textbook explains how to embed in it. Everyday people have intuition about Euclidean space that is why it is most often used for embedding. As Hartle points out, not every metric can be smoothly embedded in 3 dim Euclid though.

The article you cited is good but it talks about embedding in a 3 dimensional Minkowski space (2 space + 1 time). The principle of embedding is the same as in 3D Euclidean space only the metric of the embedding space is different. Everyday people don't have intuition about that space unless they had strong course in special relativity and draw and used a lot of t-x spacetime diagrams.

I highly recommend reading Hartle's textbook and doing the exercises. Most science libraries must have it.

 

[George Jones]

On a different note, are you aware of any good sites explaining how to interpret embedding diagrams? I find the bowling ball on a trampoline analogy confusing, as there isn't really any "down". They seem to imply gravity which is what it's supposed to be explaining!

Right.

What cross-section of space-time is an embedding diagram supposed to represent? What are the axisies?

Spacetime is 4-dimensional. Suppressing two dimensions (whichever two you want to suppress) leaves a 2-dimensional surface. Embedding this surface as a (n extrinsically) curved surface in a flat 3-dimensional allows spacetime curvature to be visualized, since the surface started as a 2-dimensional surface in spacetime.

How is acceleration represented?

In relativity, when something is in free fall, like a ball dropped near the surface of the Earth, it doesn't have acceleration. Gravity is represented by curvature, which is a tensor. Non-zero curvature gives rise to tidal forces, about which pervect wrote in another of your threads.

Thus, very roughly, curvature of the surface can represent tidal forces. Because two spacetime dimension are thrown away and two are kept, this procedure can both add and remove spacetime curvature, hence very roughly.

 

[pmb_phy]

As I understand it, a way to look at E=mc^2 is to think of it as the energy required to accelerate a given amount of mass.

E = mc² is the sum of the particle's rest energy and its kinetic energy. It is the kinetic energy which is required to change the speed of a particle.

I also understand that mass warps space-time creating an acceleration (gravity) in objects near that mass.

That depends on the exact configuration of mass but yes, it is mass that can curve spacetime. This is seen through Einstein's equation for gravity where the mass density*c² appears in the T⁰⁰ component of T, the so-called stress-energy-momentum tensor.

Is there a direct correlation between the energy required to accelerate an object and the potential energy (acceleration from gravity) created by the warping of space-time?

The potential energy is not always a well defined (mathematical) quantity in GR but it is always there in some respect. The gravitational potential in GR is taken to be the components of the metric. I guess this is the case when one wishes to describe an analogy with Newtonian mechanics but I've yet to see a book that covers GR not mention this fact. But it is not the curving of spacetime (i.e. the presense of tidal forces) that generates potential energy. It is the gravitational field itself that does that. You have the gravitational field confused with spacetime curvature.

Pete

 

[prasannapakkiam]

I agree with pmb phy... The funny thing happened last year. I played around with Einstein's second postulate that nothing can travel faster then c... I used the conventional field theory of straight lines going into a mass... But say, a stick was spun around; the further up the stick; say a large distance; I derived a formula to show the curvuture of the stick - it cannot be straight as it would exceed c otherwise...

I linked up and came up with a theory - I was excited; only to figure out that 92 years ago, this person called Einstein came up with GR...

Anyway, as you may see; the only important equations that I see are the equations before Relativistic Mass...

 

[haushofer]

E = mc² is the sum of the particle's rest energy and its kinetic energy. It is the kinetic energy which is required to change the speed of a particle.


Pete

That should be E=ymc² for the total energy ;)

 

[pmb_phy]

That should be E=ymc² for the total energy ;)

Yes. Thank you but I'm all too aware of what you speak of. What "m" means depends on how one chooses to define "m". It appears that he used it in the same context as relativistic mass and since I was responding in those terms I didn't bother getting into something the OP probably interested in anyway. It tends to turn into a debate and I'm not interested in debating the topic, ... , again.

Thanks

Pete

 

[George Jones]

The PDF is helpful. I've spent a small fortune on GR books for beginners, but find most of them sacrifice accuracy for intuitiveness, which ironically makes them less intuitive for me. The PDF is helpful. I think I'll also invest in Hartle's book.

I had "understand GR" as a new year's resolution for last year. I should've just had "lose 10 pounds" like everyone else!

I didn't see this post yesterday.

I'm a big fan of Hartle.

Before learning GR, it's useful to see SR from a viewpoint that allows easy (as possible) access to GR. Often, this isn't the point of view from which SR is taught.

Two books that do this are Spacetime Physics by Taylor and Wheeler, and A Traveler's Guide to Spacetime by Moore. The first book is slightly more advanced than the second, and has a wealth of truly outstanding problems. The https://www.amazon.com/dp/071670336X/?tag=pfamazon01-20.

A good plan would be first to go thoroughly through one of the SR books, and then to read Hartle. Some of the people on this forum can provide good help, if needed.