Questions about MTW Brick: Page 29, 101, 142

[TeTeC]

Hi PF!

I've begun to read the MTW brick. I have a few questions.

Page 29, (1.5)

I don't understand how these equations are calculated. There is a demonstration below, but I don't understand the part where he adds a $\frac{1}{c^{2}}$ factor to the acceleration
$$\frac{Gm_{conv}}{r^{2}}$$.

Page 101, 3 lines before (4.8)

I don't understand how he evaluates $<\textbf{d}y\wedge\textbf{d}z,\textbf{u}>$. There is a definition for evaluating a p-form with a p-vector, but not for p-form with a k-vector.

As a general remark, it's a bit sad that the authors often forget to write the general definitions of the mathematical tools they use.

Page 142, 8th line of paragraph 5.8

[he's talking about the conservation of 4-momentum]

The energy-momentum lost by particles goes into fields; the energy-momentum lost by fields goes into particles.

I don't understand the meaning of this sentence. I understand how particles can lose energy-momentum, but I don't see how "fields" (whatever this is...) can lose or gain energy.

If my questions aren't clear enough, I can type more.

Thanks a lot!

Sébastien.

 

[George Jones]

Hi Sébastien,

I don't have my copy of MTW home with me so I won't be able to look at it until Monday at the earliest, but probably others will give answers before then.

When using latex here, use itex and /itex for math in text, tex /tex (not \tex) for stand-alone math.

Why did you pick MTW for self-study; just curious.

Good luck!

 

[TeTeC]

Thanks for the help provided with Latex ;-) I wondered which magician had edited my post.

I choose this book simply because it seems to be the Bible of this theory. ;-)
Actually, MTW is not my first General Relativity textbook. I've begun with Schutz's (although I haven't read the whole book) because people on forums thought MTW wouldn't be the right book for an introduction to the theory. And I must say that they were right.

 

[HallsofIvy]

For the enlightenment of us benighted persons, what is "MTW" (and how does one read a "brick")?

 

[TeTeC]

It is this book: https://www.amazon.com/dp/0716703440/?tag=pfamazon01-20

(for the brick, well, I guess it's just a french expression I shouldn't have tried to translate.)

 

[shalayka]

It is this book: https://www.amazon.com/dp/0716703440/?tag=pfamazon01-20

(for the brick, well, I guess it's just a french expression I shouldn't have tried to translate.)

Brick is a perfectly suitable analogy. It makes perfect sense in English. No worries.

 

[George Jones]

Hi PF!

I've begun to read the MTW brick. I have a few questions.

In English, MTW is sometimes called "the telephone book".

Page 29, (1.5)

I don't understand how these equations are calculated. There is a demonstration below, but I don't understand the part where he adds a $\frac{1}{c^{2}}$ factor to the acceleration
$$\frac{Gm_{conv}}{r^{2}}$$.

The $c^2$ is there because time is measured in metres, not in second. Let $t'$ be a time in seconds and $t$ be the same time in metres. Then, $t = ct'$. An acceleration in conventional units is then

$$\frac{d^2 x}{dt'^2} = c^2 \frac{d^2 x}{dt'^2}.$$

Then, the $c^2$ is moved to the other side.

Page 101, 3 lines before (4.8)

I don't understand how he evaluates $<\textbf{d}y\wedge\textbf{d}z,\textbf{u}>$. There is a definition for evaluating a p-form with a p-vector, but not for p-form with a k-vector.

As a general remark, it's a bit sad that the authors often forget to write the general definitions of the mathematical tools they use.

This is an example of an interior product (or derivative) of a p-form with a vector. I can't find its definition in MTW; try

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

with p = 2.

Page 142, 8th line of paragraph 5.8

[he's talking about the conservation of 4-momentum]

The energy-momentum lost by particles goes into fields; the energy-momentum lost by fields goes into particles.

I don't understand the meaning of this sentence. I understand how particles can lose energy-momentum, but I don't see how "fields" (whatever this is...) can lose or gain energy.

It is possible to define the energy and momentum of, for example, an electromagnetic field.

Stick a charged particle in an electromagnetic field. Since the charge accelerates, its energy and momentum change. The energy and momentum of the electromagnetic field change in such a way that the energy of momentum of the system of particle and field is conserved. This treated in Jackson, but see also

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

 

[TeTeC]

Thank you! That makes much more sense.

Is it okay if I continue posting some questions here as I progress, or should I make a new thread?

 

[cristo]

Thank you! That makes much more sense.

Is it okay if I continue posting some questions here as I progress, or should I make a new thread?

Sure, you can post them here.

 

[TeTeC]

Hi PF!

Here comes my next set of questions.

I'm trying to understand as deep as possible the equivalence principle. In MTW, there are multiple definitions and consequences of this principle, and I'd like to have a confirmation that my understanding is the right understanding.

1) The definition given in the book is:

In any and every local Lorentz frame, anywhere and anytime in the universe, all the nongravitational) laws of physics must take on their familiar special-relativistic forms.

Equivalently: there is no way, by experiments confined to infinitesimally small regions of spacetime, to distinguish one local Lorentz frame in one region of spacetime from any other local Lorentz frame in the same or any other region. (page 386)

According to paragraph 8.5 page 207, a local Lorentz frame is defined as a coordinate system in which $g_{\mu\nu}(P_{0}) = \eta_{\mu\nu}$. However, I think that this principle doesn't say that in any infinitesimally small region of spacetime, I can find a local Lorentz frame. I think this is made possible by choosing special coordinates: the Riemann normal coordinates. Equation (11.30) and (11.28) (page 287,286) confirm that the coordinate system is indeed lorentzian.

I also think we'll all agree that a local Lorentz frame is an inertial frame (page 217). An observer at rest in a local Lorentz frame is freely falling and moves along a straight line (the definition of an inertial frame).

There is no question here. I'd simply like to have these sentences confirmed. I also wanted to write this in order to make next questions clearer.

2) (page 173) Equation (6.18) gives the line element for the local coordinate system of an accelerated frame:

$$ds^{2} = - (1+g\xi^{1'})^{2}(d\xi^{0'})^{2} + (d\xi^{1'})^{2} + (d\xi^{2'})^{2} + (d\xi^{3'})^{2}$$

After that, the author says that if we pick a small value (or simply 0) for $\xi^{1'}$, we have the line element of a Lorentz coordinate system.

My question is: could we have predicted this result without making any calculation by using the fact that I can find a local Lorentz frame at any point of spacetime?

3) (page 197, middle of the page) The author explains that the Galileo-Einstein principle of equivalence implies that in any given locality one can find a frame of reference in which every neutral test particle, whatever its velocity, is free of acceleration.

It looks like this definition of the equivalence principle includes the result that we can find a local Lorentz frame at any point of spacetime. Therefore, I'm wondering if there is any way to show this fact using the definition of page 386 that I have written up there.

Take into account the fact that I'haven't been further than chapter 12 for now. These chapters mainly introduce the mathematical machinery. Consequently, I still haven't done much physics.

Thanks!

 

[TeTeC]

I don't like to bump a thread, since this is not always very polite... But if you have an answer to any of my questions, don't hesitate to write it.

Thank you.

 

[TeTeC]

Still no answers ?

 

[atyy]

Try Chapter 24 for a discussion of the "local Lorentz frame"

http://www.pma.caltech.edu/Courses/ph136/yr2006/text.html