Einstein's equation

[michel123456]

Thanks to IM Egdall who posted in another thread

from this link

 

Einstein's Equation

To state Einstein's equation in simple English, we need to consider a round ball of test particles that are all initially at rest relative to each other. As we have seen, this is a sensible notion only in the limit where the ball is very small. If we start with such a ball of particles, it will, to second order in time, become an ellipsoid as time passes. This should not be too surprising, because any linear transformation applied to a ball gives an ellipsoid, and as the saying goes, ``everything is linear to first order''. Here we get a bit more: the relative velocity of the particles starts out being zero, so to first order in time the ball does not change shape at all: the change is a second-order effect.

emphasis mine

Sorry I don't get it. What does mean "in second order of time"?

[Schrödinger's hat]

Sorry I don't get it. What does mean "in second order of time"?

 

If I had, for example, a ball on the end of a spring.

I pull it back and let it go, then have a look a little while afterwards.

 

The simplest approximation for where it will be in a small amount of time is its current position.

[math] x(t) = x_0 [/math]

This isn't very good, but for a little while, it'll be about where it is. This is a zeroth order approximation.

 

Slightly better is a first order. I look at how fast it's going and add a little bit.

[math]x(t) = x_0 + v_0 t [/math]

This will give us okay answers for slightly longer.

 

But it's also accelerating because of the spring:

[math]x(t) = x_0 + v_0 t + \frac{a_0}{2} t^2[/math]

(second order)

 

and then the change in acceleration and so on.

The nth order comes from the exponent in the time term. So if it's second order in time we go up to [math]t^2[/math]

This works out well for [math]t<1[/math] in most cases because if you raise a small number to a high power you get an even smaller number.

 

Do some reading about taylor series if you want to know a little more.

[michel123456]

O.K. so second order is about acceleration.

 

So IIUC (If I understand Correctly)

At first order of time a round ball of test particles is round. Talking about a ball of particles at rest relative to each other. For an external observer it may be moving at regular velocity, it will still be observed round. And it is round as observed from the inside.

 

At second order of time, accelerating, it will be an ellipsoid. Both for the external observer which observes the ball moving at changing velocity, and for the observer inside the ball.

 

Is that correct?

Edited December 10, 2011 by michel123456

[DrRocket]

Thanks to IM Egdall who posted in another thread

from this link

 

 

emphasis mine

Sorry I don't get it. What does mean "in second order of time"?

 

 

Nothing in that quote makes any sense. If you want to study physics start with a good physics book.

 

As the subject appears to have (indeterminate) connection to special relativity, Wolfgang Rindler's Introduction to Special Relativity might be a good place to start.

[michel123456]

Nothing in that quote makes any sense. If you want to study physics start with a good physics book.

 

Is this comment for me or for J. Baez?

John Baez's Stuff

I'm a mathematical physicist. I have a permanent position in the math department at U. C. Riverside, but I'll be at the Centre for Quantum Technologies in Singapore until Fall 2012. I'm working on information geometry, network theory, and the Azimuth Project, which is a way for scientists, engineers and mathematicians to do something about the global ecological crisis. If you want to help save the planet, please send me an email or say hi on my blog.

[DrRocket]

Is this comment for me or for J. Baez?

 

 

It is for you if you took that quote out of context and for Baez if that is essentially the complete content of what he wrote.

 

You can quote Baez's bio all you want, but it does not really address the issue. I am acquainted with his background and work (and know his former dean personally). He is generally pretty good, but not infallible, and has a reputation more as an expositor than a researcher.

 

Quoting "experts" is not a substitute for a critical reading of what they have written.

Edited December 23, 2011 by DrRocket

[Mark Northrup]

My first foray here. I am working on a blog post regarding a personal annoyance whenever Special Relativity (SR) is discussed. The kerfuffle around the FTL neutrino puzzle announced by the INFN's OPERA team finally pushed me over the edge. To put it bluntly, it seems like anytime SR comes up, science popularizers (including scientists that ought to know better) never actually show the straightforward math behind SR that shows why it is the "cosmic speed limit." As I understand the maths behind SR (I'm American, but "maths" is easier to type than "mathematics")-I have Einstein's 1961 popular exposition "Relativity"-it is basically a divide by zero problem; in Einstein's eq. for kinetic energy, when v^2=c^2 in the denominator, and if the c^2 in the numerator is moved to the left-hand side of the eq., the right-hand side becomes a quantity of mass, divided by a "pure number" (therefore retaining the mass units unsullied) which as the denominator approaches zero, the mass becomes infinite. Graphically, this would be sooooo easy to show in a TV documentary and anyone that has ever encountered a "Div/0" error in a spreadsheet program could grasp the salient point. I just want to make sure my math and general reasoning are sound.

 

Thanks in Advance and Merry Christmas/Happy Holidays/Happy Hanukkah/Sumptuous Solstice or whatever!

 

Mark Northrup

[michel123456]

It is for you if you took that quote out of context and for Baez if that is essentially the complete content of what he wrote.

 

You can quote Baez's bio all you want, but it does not really address the issue. I am acquainted with his background and work (and know his former dean personally). He is generally pretty good, but not infallible, and has a reputation more as an expositor than a researcher.

 

Quoting "experts" is not a substitute for a critical reading of what they have written.

 

The full context is in this link as already posted in the OP.

[DrRocket]

My first foray here. I am working on a blog post regarding a personal annoyance whenever Special Relativity (SR) is discussed. The kerfuffle around the FTL neutrino puzzle announced by the INFN's OPERA team finally pushed me over the edge. To put it bluntly, it seems like anytime SR comes up, science popularizers (including scientists that ought to know better) never actually show the straightforward math behind SR that shows why it is the "cosmic speed limit." As I understand the maths behind SR (I'm American, but "maths" is easier to type than "mathematics")-I have Einstein's 1961 popular exposition "Relativity"-it is basically a divide by zero problem; in Einstein's eq. for kinetic energy, when v^2=c^2 in the denominator, and if the c^2 in the numerator is moved to the left-hand side of the eq., the right-hand side becomes a quantity of mass, divided by a "pure number" (therefore retaining the mass units unsullied) which as the denominator approaches zero, the mass becomes infinite. Graphically, this would be sooooo easy to show in a TV documentary and anyone that has ever encountered a "Div/0" error in a spreadsheet program could grasp the salient point. I just want to make sure my math and general reasoning are sound.

 

Thanks in Advance and Merry Christmas/Happy Holidays/Happy Hanukkah/Sumptuous Solstice or whatever!

 

Mark Northrup

 

In a nutshell the "cosmic speed limit" is the logical consequence of the two fundamental axioms of special relativity: 1) the speed of light is the same in all inertial reference frames and 2) the laws of physics are the same in all inertial reference frames.

 

It turns out that if you assume that anything propagates at some speed, say x, in all inertial reference frames and if you assume axiom 2 then you can deduce the Lorentz transformations with "x" playing the role of "c". "x" is then the "cosmic speed limit". There can be only one such value. One then notes the experimental fact that the speed of light is the same in all inertial reference frames to conclude that x=c. That is the mathematics behind special relativiity. There is no division by 0 involved.

 

If want to go through this in detail a good reference is Introduction to Special Relativity by Wolfgang Rindler.

 

The full context is in this link as already posted in the OP.

 

Then the criticism is for both you and Baez.

 

Your quote is both out-of-context and misleading.

 

What Baez (and Bunn) are attempting to do (I think poorly) is provide a pop-sci explanation of the Einstetin field equations of general relativity. That equation in simplest form is simply

 

[math] E + g \Lambda = \dfrac {8 \pi G}{c^4} T[/math]

 

where [math]E[/math] is the Einstein curvature tensor [math]g[/math] is the metric tensor [math]G[/math] is Newton's universal gravitational constant and [math]T[/math] is the stress-energy tensor.

 

This tensor equation is equavilent to a set of 10 coupled non-linear partial differential equations and to refere to it in the singular as "Einstein's equation" is itself a bit misleading.

 

Baez and Bunn are attempting to explain this very complicated set of equations in extremely simple terms -- to my mind a gross oversimplification, given that the stress energy tensor includes ALL non-gravitational forms of energy. Moreover their statements are, as I said originally, nonsensical hand waving. A spherical ball of test particles, is itself a vague notion, and if initially at rest would stay at rest so the remainder of the paragraph is just more hand waving, as is the notion of changes in shape in terms of "orders in time".

 

Quoting someone who is waving his hands wildly on a subject that you don't understand yourself is not a good strategy.

 

If you want to understand general relativity and the Einstein field equations read a good book. Gravitation by Misner, Thorne and Wheeler is a standard reference. Yvonne Choquet-Bruhat's General Relativity and the Einstein Equations is another excellent book. Those authors are experts in general relativity. Baez is not. The down side is that these books are the real thing and require an investment in time to study them as well as the mathematical background to be able to comprehend their message.

[MigL]

basedI know I'll get repremanded again by the good Doctor, but I've never read Thorne, Misner and Wheeler's Gravitation although it is on my to-do list, nor the other book by Choquet-Bruhat. I did try to work my way through Principles of Physical Cosmology by P.J.E. Peebles and found the going tough because I have no background in differential geometry and some books assume a working knowledge of the subject.

I did download a book based on a series of lecture notes by S. Waner called Introduction to Differential Geometry and General Relativity and it only assumes familiarity with college level algebra and calculus, which makes GR a little easier to understand once you have aquired a basic foundation in differential geometry.

Has anyone else read this, and what did you think ?

Edited December 26, 2011 by MigL

[IM Egdall]

 

Baez and Bunn are attempting to explain this very complicated set of equations in extremely simple terms -- to my mind a gross oversimplification, given that the stress energy tensor includes ALL non-gravitational forms of energy. Moreover their statements are, as I said originally, nonsensical hand waving. A spherical ball of test particles, is itself a vague notion, and if initially at rest would stay at rest so the remainder of the paragraph is just more hand waving, as is the notion of changes in shape in terms of "orders in time".

 

 

I love this forum! I especially learn so much from you.

 

You say the stress-energy tensor in Einstein's field equations of general relativity "include ALL non-gravitiational forms of energy." Are you saying the momenergy contained within the stress-energy tensor accounts for energy of motion as well as gravitational energy? What about other forms of energy, like EM fields and strong force fields etc. Are these also included. Please give us other examples if you have them. And a link if you have it for some references that are not too mathematically dense. Thank you.

 

Oh, and I think Baez's initially spherical ball of test particles is in free-fall, not at rest. So wouldn't they change their relationship with respect each other over time?

 

 

And since Baez is oversimplified, do you know of any other explanation which does not require knowledge of differential geometry which is a better explanation of GR?

[Louis Storn]

I am pretty sure that E=mc2 still works.

[eino]

think this 1+1-1+1=2

 

1+1-1+1 = mc2 if 1 = mc^2 / 2

thebn compare it to einsteins theory

[Klaynos]

!

Moderator Note

eino, please do not post this in other peoples threads. You have your own thread dedicated to this.

Please do not reply to this modnote.

[Temporocitor]

where is the Einstein curvature tensor is the metric tensor is Newton's universal gravitational constant and is the stress-energy tensor.

 

 

What is fascinating is where we allow Lamda's volume definition to use a similar analogy to the elipse mentioned earlier.

 

such that V =(2(PI)^2 R) ((PI) ab) where R<1 and V <= 1

 

As R approaches 0 V must approach 1 for Lamda to <= 1 but <> 0 otherwise if R = 0 and Lamda = -(1/2)V