# MTW Box 21.1 - What can "add and subtract" do for Equation (12)?

• TerryW
TerryW
Gold Member
Homework Statement
Derive Equation (15) from Equation (12)
Relevant Equations
See attachment
I haven't posted for a while and I am still (!) working through some of the things I didn't quite get in MTW Chapter 21.

Here is my latest puzzle.

I want to work out how to get from Equation (12) in the attachment, to Equation (15).

I've tried the "add and subtract" ##\{\frac {(-g)^{\frac12}F^{i0}}{4\pi}A_0\delta t\}_{,i}##

This gives me ##+\{\frac {(-g)^{\frac12}F^{i0}}{4\pi}\delta t\}A_{0,i}## and -##\{\frac {(-g)^{\frac12}F^{i0}}{4\pi}\delta t\}A_{i,0}##

Plus ## \{\frac {(-g)^{\frac12}F^{i0}}{4\pi}\}_{,i}A_0\delta t## and minus ## \{\frac {(-g)^{\frac12}F^{i0}}{4\pi}\}_{,i}A_0\delta t##

All this does is allow me to replace ##\{\frac {(-g)^{\frac12}F^{i0}}{4\pi}A_{i,0}\}## with ##-\{\frac {(-g)^{\frac12}F^{i0}}{4\pi}A_{0,i}\}## which I could have done anyway by index manipulation,

I can then add the two versions of (12) to give a new equation which is $$2\delta S = \int \big[ 2\frac {(-g)^{\frac12}F^{i0}}{4\pi}\delta A_{i}+\{\frac {(-g)^{\frac12}F^{i0}}{4\pi}A_{i,0}-\frac {(-g)^{\frac12}F^{i0}}{4\pi}A_{0,i}\}\delta t -2\mathfrak L\}\big]d^3x$$

What this means is that my result for ##\frac {\delta S}{\delta \Omega}## contains the term ##2F^{i0}(A_{i,0} - A_{0,i})## instead of ##4F^{i0}(A_{i,0} - A_{0,i})##

I then had a look at the Plus and Minus ## \{\frac {(-g)^{\frac12}F^{i0}}{4\pi}\}_{,i}A_0\delta t## terms which I had discarded earlier as they cancel, to see if I could find some extra terms, but I couldn't find anything to fix the problem.

Can anyone point out what I am missing?
RegardsTerryW

#### Attachments

• Box 21.1.png
25.9 KB · Views: 56
Hi Terry, I don't think if you remember my other handle though I hadn't forgotten you queries from MTW I did purchase the paper black version from Amazon.com quite a Heavy lifting.

But I'll get there eventually.
Nowadays I am reading Information Theory by Cover and Thomas for my last degree, hopefully I'll achieve it.
Cheers mate, you'll never be forgotten!

I mean it is paperback.

billtodd said:
Hi Terry, I don't think if you remember my other handle though I hadn't forgotten you queries from MTW I did purchase the paper black version from Amazon.com quite a Heavy lifting.

But I'll get there eventually.
Nowadays I am reading Information Theory by Cover and Thomas for my last degree, hopefully I'll achieve it.
Cheers mate, you'll never be forgotten!
Best of luck with MTW. Should you ever need a steer with any of the problems (I'm currently on Chapter 23), just drop me a message.

CheersTerry W

TerryW said:
Best of luck with MTW. Should you ever need a steer with any of the problems (I'm currently on Chapter 23), just drop me a message.

CheersTerry W
We ain't getting younger, but with no Guts no Glory:

BTW what was your 'other handle'?

TerryW said:
BTW what was your 'other handle'?
Let's just say I am a 21st century polymath...

## What is the context of MTW Box 21.1 in the book?

MTW Box 21.1 is found in the book "Gravitation" by Misner, Thorne, and Wheeler. This section deals with the mathematical techniques used in general relativity, particularly focusing on manipulating equations to derive specific results. Box 21.1 specifically explores the utility of adding and subtracting terms in equations to simplify or transform them.

## What is Equation (12) referred to in MTW Box 21.1?

Equation (12) in MTW Box 21.1 is a specific equation from the book that is being analyzed for its properties and potential simplifications. The exact form of Equation (12) would depend on the context provided in the surrounding text, but it typically involves terms relevant to the field equations of general relativity or related mathematical constructs.

## Why is adding and subtracting terms useful in manipulating equations in general relativity?

Adding and subtracting terms in equations can help isolate specific variables, simplify expressions, and reveal underlying structures that are not immediately apparent. This technique can be particularly useful in general relativity, where complex tensor equations often need to be simplified or transformed to find solutions or make physical interpretations more clear.

## Can you provide an example of how adding and subtracting terms simplifies an equation?

Consider a simple algebraic equation: $$a + b = c$$. By adding and subtracting the same term on both sides, such as $$d$$, we get $$a + b + d - d = c$$. This simplifies to $$a + b = c$$, demonstrating that the equation remains balanced. In more complex equations, especially in general relativity, this technique can help in isolating specific components or simplifying the equation's form, making it easier to solve or interpret.

## How does MTW Box 21.1 help in understanding the manipulation of equations in physics?

MTW Box 21.1 provides a step-by-step guide on using the method of adding and subtracting terms to manipulate equations, offering a practical example within the context of general relativity. This helps readers understand not just the theoretical basis but also the practical application of these mathematical techniques, enhancing their ability to work with complex equations in physics.

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