Schwarzschild Metric Derivation?

[mysearch]

Hi, I was wondering if anybody could help me understand the derivation of the Schwarzschild metric developed by the author of mathpages website. Rather than reproduce all the equations via latex, I have attached a 2-page pdf summary that also points to the mathpages article and explains my present understanding.

While I followed most of the steps, there is one specific step – see equation [13] that was unclear to me. Equally, I could find no obvious way to proceed from equation [12], which I thought would be a valid starting point in order to resolve the last [grr] coefficient.

Would really appreciate any help with the maths and in particular any physical interpretation of the Schwarzschild metric. For example, it would seem that time dilation is a real and measurable effect that can be confirmed on return to flat spacetime, while the idea of length expansion in GR and length contraction in SR appears more ambiguous.Thanks

 

[PeterDonis]

Rather than reproduce all the equations via latex, I have attached a 2-page pdf summary

Instead, why not just link to the mathpages article itself?

 

[m4r35n357]

I think the relevant article is this one. The derivation starts two paragraphs before equation (2).

 

[PeterDonis]

I think the relevant article is this one.

The article referred to in the PDF is this (different) one:

http://mathpages.com/rr/s5-05/5-05.htm

However, I don't see anything corresponding to equations (12) or (13) in the PDF in either of these articles. @mysearch, we need an actual link to the proper article, plus a discussion of what's actually in that article that you don't understand.

 

[PeterDonis]

it would seem that time dilation is a real and measurable effect that can be confirmed on return to flat spacetime

What does "return to flat spacetime" mean?

the idea of length expansion in GR

What are you referring to here?

 

[mysearch]

Instead, why not just link to the mathpages article itself?

I did – see original post indicating that the link was in the pdf. The pdf was simply a summation of my present understanding of the mathpages’ derivation with numbered equations as a point of reference.

I think the relevant article isthis one. The derivation starts two paragraphs before equation (2).

Many thanks. This looks very useful, although I haven’t had time to read into the details as yet.

The article referred to in the PDF is this (different) one:
http://mathpages.com/rr/s5-05/5-05.htm
However, I don't see anything corresponding to equations (12) or (13) in the PDF in either of these articles. @mysearch, we need an actual link to the proper article, plus a discussion of what's actually in that article that you don't understand.

Yes that is the mathpages articles that I linked to in the PDF attachment, although the new article cited may be the explanation I am looking for. As I tried to explain, equation [12] is simply a re-arrangement of the Schwarzschild metric from which I thought I could proceed to resolve the coefficient [grr] but couldn’t. Equation [13] is in the mathpages article after equation (4) in that articles, which I didn’t understand the source – hence post #1.

What does "return to flat spacetime" mean?
What are you referring to here?

If you enter a gravitational field from a position in ‘flat spacetime’, i.e. no gravity – no velocity, the effects of time dilation appear tangible as measured on a clock when returning to your position in ‘flat spacetime’. Threfore, I was questioning whether length expansion in GR, as implied by the Schwarzschild metric, when entering a gravitation field was thought to be real. Along similar lines, SR seems to imply a length contraction associated with velocity. However, it would seem that free-falling with velocity [v] into gravitation field might cancel both these effects on length. While such speculative questions were not really the focus of my original question, I was just interested to hear other people’s thought on this aspect of relativity, i.e. is time dilation ‘real’ but length expansion/contraction more of an observed perception.

Anyway thanks for the link in post #3.

 

[PeterDonis]

If you enter a gravitational field from a position in ‘flat spacetime’, i.e. no gravity – no velocity, the effects of time dilation appear tangible as measured on a clock when returning to your position in ‘flat spacetime’.

Strictly speaking, there is nowhere with "no gravity" in the universe. A better way of putting this would be to say you start from (and return to) a region which is far enough from all gravitating bodies that the effects of gravity (spacetime curvature) are negligible. But "negligible" is not the same as "zero".

length expansion in GR, as implied by the Schwarzschild metric

How does the Schwarzschild metric imply "length expansion"?

SR seems to imply a length contraction associated with velocity.

Yes, for an appropriate definition of "length contraction". But this effect does not change the object itself at all; it's simply an effect of "perspective", of viewing the object from a different frame.

it would seem that free-falling with velocity [v] into gravitation field might cancel both these effects on length.

Have you actually looked at the math instead of just waving your hands?

is time dilation ‘real’ but length expansion/contraction more of an observed perception.

It depends on what you mean by "real" vs. "observed perception". These terms are too vague to lead to a meaningful answer. If you pose specific scenarios, you can certainly use GR to predict what any observer will observe in those scenarios.

 

[m4r35n357]

Anyway thanks for the link in post #3.

For learning I would recommend that over your linked article, which is a bit of an arm-wavey derivation based on Kepler's law. It is interesting mathematically, but not rigorous.

 

[PeterDonis]

Equation [13] is in the mathpages article after equation (4) in that articles, which I didn’t understand the source

Equation (13), the radial geodesic equation, is just a special case of the general geodesic equation, described for example here. The mathpages article doesn't derive this special case, it just gives the result. But basically he is taking the general geodesic equation as given in the Wikipedia article, using $\tau$ instead of $s$ to emphasize that the curve parameter is proper time, and using $r$ for the $\mu$ index and $t$ for the $\alpha$ and $\beta$ indexes, to obtain

$$
\frac{d^2 r}{d \tau^2} = - \Gamma^r{}_{tt} \left( \frac{dt}{d\tau} \right)^2
$$

and then using standard formulas for the Christoffel symbols and the Schwarzschild metric to obtain

$$
\Gamma^r{}_{tt} = - \frac{1}{2 g_{rr}} \frac{\partial g_{tt}}{\partial r}
$$

which, when substituted into the above, gives Equation (13).

 

[mysearch]

For learning I would recommend that over your linked article, which is a bit of an arm-wavey derivation based on Kepler's law. It is interesting mathematically, but not rigorous.

Thank you again for the link in post#3. The author did highlight that it was only meant to be an outline derivation and clearly his other articles cover more detailed mathematical derivations. In part, I was simply looking for some of the physical rationale in support of the mathematics for while I had followed the arguments for time dilation, I was struggling to find any consistent explanation for the gravitational effects on length. However, given the tone of the other ‘responses’ in this thread, I think I will go elsewhere to look for answers.

 

[PeterDonis]

I was struggling to find any consistent explanation for the gravitational effects on length.

Can you be more specific about what you mean by "gravitational effects on length"? For example, can you describe a thought experiment that would show it?

What I mean is, you've described a simple thought experiment to show gravitational time dilation, of a person starting in deep space far away from all gravitating bodies, going down deep into a gravity well, staying there for a while, then coming back out to deep space and comparing his clock with someone who stayed far away in deep space the whole time. The person who went into the gravity well will then show less elapsed time on his clock. Can you describe a thought experiment that will show, in the same sort of fashion, the "gravitational effects on length" that you refer to?