Is Schwarzschild metric more intuitive?

[marlowgs]

The textbooks claim that the weak field (Newtonian) metric is more intuitive than the Schwarzschild metric, but I don’t agree.The time correction factor for the weak field metric is the same as that for the Schwarzschild metric. But for the length correction factor for the weak field metric is smaller than that of the Schwarzschild metric. In special relativity the length correction factor is the reciprocal of the time correction factor and it is also the reciprocal for the Schwarzschild metric. Why should it be more intuitive to use anything but the reciprocal. It seems to me that the weak field metric is less intuitive than the Schwarzschild metric.

 

[WannabeNewton]

"Intuitive" is a very subjective term; it can mean a lot of different things especially in the context of physics. The weak field (Newtonian) metric can be seen as intuitive simply because the connections to the Newtonian potential $\varphi$ and the acceleration $a = -\nabla \varphi$ are immediately apparent and the bridge between the lower limiting case of General Relativity and the (again subjectively more intuitive) Newtonian gravity is evident.

 

[marlowgs]

It makes G.R. a lot easier to understand if you can say that gravitational length contraction is the reciprocal of gravitational time dilation just as it is for S.R., and time dilation just uses the escape velocity (free falling frame velocity) in the Lorentz transformation.

 

[pervect]

It makes G.R. a lot easier to understand if you can say that gravitational length contraction is the reciprocal of gravitational time dilation just as it is for S.R., and time dilation just uses the escape velocity (free falling frame velocity) in the Lorentz transformation.

The fact that $g_{tt}\, g_{rr}$ is not 1 in other coordinates suggests that while this interpretation may be "simple", it's probably not true, at least not in any deep sense.

Isotropic coordinates can be hard to work with (they have more non-zero Christoffel symbols than Schwarzschild coordinates, IIRC), but they have the advantage that the preserve the isotropy of the speed of light.

The isotropy of the speed of light does have advantages when applying one's intuition, which is probably at the root of the textbooks claim.

I think it's ultimately more important to have the mathematical machinery available to check one's intuition - this means being able to move beyond Schwarzschild coordinates if necessary - so one can separate out the "good" intuitions from the "not-so-good" ones.

I'd also like to put a brief plug in for regarding time dilation as an aspect of curvature, rather than to try to interpreting it literally as some "physical" effect.

I'm pretty sure Wald makes some remarks about this, but I don't recall exactly well, and I don't really hae the time to look it up at the moment.

Basically, time dilation is highly coordinate dependent, and focussing too intently on it obscures the broader picture of the close relationship between time an space.

 

[WannabeNewton]

I'm pretty sure Wald makes some remarks about this, but I don't recall exactly well, and I don't really hae the time to look it up at the moment.

He never derives gravitational time dilation but rather only mentions it in passing within a single sentence in chapter 6. However he does derive the gravitational redshift effect using nothing but the geometry of space-time (in particular he really just makes use of the fact that a stationary space-time has a time-like killing vector fields and that the inner product of said killing vector field with the tangent vector to a null geodesic is constant along the null geodesic); this is all on pages 136-137 by the way. Unfortunately, as noted, he doesn't go into any detail at all about gravitational time dilation.

 

[marlowgs]

An extreme mathematical interpretation may be clouding the true underlying physics. If you look at gravity being the result of a field of rest mass energy that has its start at the Schwarzschild radius with a maximum v^2 = c^2, it looks more like other fields. Is there any experimental evidence that can distinguish my simple interpretation (which gives the same length and time correction factors as the Schwarzschild metric) from the more complex coordinates you are referring to?

 

[pervect]

I don't look at gravity as being the result of a field of rest mast energy. I'm not even sure how one would go about defining a "field of rest mass energy". If you happen to mean a scalar field theory, such as http://en.wikipedia.org/w/index.php?title=Scalar_theories_of_gravitation&oldid=544188043, (this represents gravity as being a field that can be described by a single number at every point in space) then the answer is that such theories do make different predictions than GR, which is a tensor theory.

Additionally, all the scalar theories I know about have been falsified (see the wiki article for detail). To simply a bit (perhaps too much), tensor theories represent gravity with a structure that's more complex than a single number at every point in space, making them different than scalar theories.

So from my POV, you are probably already well on your way off to creating a personal theory that's most likely not GR. It is hard to be sure - but that's because it's hard to be sure what your personal theory is. Working a few detailed examples using your theory might clarify it's relation to GR, and to (say) Nordstrom's theories.

 

[marlowgs]

The thing that is making me think fields is that the Newtonian metric is not only the large radius approximation of the G.R. metric but also just a negative radial shift by the Schwarzschild radius. The correction factor squared for length contraction of the Newtonian metric can be rewritten to look like that of the Schwarzschild metric as 1+rs/r = 1/(1-rs/(r+rs)). In this form you can easily see the shift. Could this be saying that the G.R. metric is the same as a Newtonian metric that starts at the Schwarzschild radius?

 

[PAllen]

The thing that is making me think fields is that the Newtonian metric is not only the large radius approximation of the G.R. metric but also just a negative radial shift by the Schwarzschild radius. The correction factor squared for length contraction of the Newtonian metric can be rewritten to look like that of the Schwarzschild metric as 1+rs/r = 1/(1-rs/(r+rs)). In this form you can easily see the shift. Could this be saying that the G.R. metric is the same as a Newtonian metric that starts at the Schwarzschild radius?

This is not fruitful. The universe does not consist of one body with perfect spherical symmetry. Indirect evidence for gravitational waves is very strong, in quantitative agreement with GR. Unless you want to produce a toy theory incompatible with observation at the outset, you must grapple with this.

 

[marlowgs]

You’re making me laugh, but your not answering my question. I would love to know where I’m going wrong. Maybe then I can get a good night’s sleep.

 

[PAllen]

You’re making me laugh, but your not answering my question. I would love to know where I’m going wrong. Maybe then I can get a good night’s sleep.

Equally, you make me laugh. Your approach is analogous to noticing that 3,5,7 are prime and wanting to develop a 'theory' that odd number are prime and deal with the exceptions later.

 

[WannabeNewton]

What in the world is "the GR metric"? Is it an arbitrary solution to the EFEs? In that case this is moot because an arbitrary space-time solution to the EFEs is obviously not asymptotically flat and "large radius" has no meaning in general.

 

[marlowgs]

I was referring to the Schwarzschild metric with the term "G.R. metric". The 1/(1-rs/r) factor approaches 1+rs/r as r approaches infinity.

 

[Nugatory]

I was referring to the Schwarzschild metric with the term "G.R. metric". The 1/(1-rs/r) factor approaches 1+rs/r as r approaches infinity.

Right, but the Schwarzschild metric is just one solution of the Einstein Field Equations, one that only applies to a universe that contains a singe spherically symmetric non-rotating non-charged mass. That makes it a very useful approximation for a large number of real problems, but it's not a good starting point for building a more general theory.