D.W. Sciama's Theory of Inertia: Origin & Discussion

[Charlie G]

I was wondering if anyone is familiar with D.W. Sciama's theory of inertia based on Mach's principle? I came across the theory in a book or selected writings on motion I found at a used book sale and found it very compelling.

Just curious to hear any fallacies in his reasoning or negative experimental results from anyone who is familiar with the work.

Thanks.

 

[Jonathan Scott]

I was wondering if anyone is familiar with D.W. Sciama's theory of inertia based on Mach's principle? I came across the theory in a book or selected writings on motion I found at a used book sale and found it very compelling.

Just curious to hear any fallacies in his reasoning or negative experimental results from anyone who is familiar with the work.

Thanks.

Sciama's explanation of inertia is beautifully neat, but it is based on a simple analogy between gravity and electromagnetism which is of the same order of accuracy as Newtonian theory combined with Special Relativity, so it is more of an illustrative idea than a full-fledged theory.

If the same idea is applied to General Relativity, it suggests that inertia would simply be caused by the linear frame-dragging effect of the entire universe, and this means that the gravitational "constant" G would actually be given by an expression of the form

$$G = \frac{k}{\sum_i m_i/r_i c^2}$$

where $k$ is a simple constant (probably 1/4 for maximum compatibility with GR,
at least according to Nordtvedt) and the sum is for all masses and their distances from the observation point.
There are various alternative gravity theories which are to some extent based on Mach's
Principle (such as Brans-Dicke theory) and these typically require at least some effective variation in G as in the above expression.

If this idea of a varying G were true, it would conflict with GR, which assumes that G is constant. It would mean that G could vary both with location in space and with time, but experiments have placed very severe constraints on any such variation.

Variation with space is not necessarily ruled out by experiment, because in the simplest form of Sciama's idea, the variation in G due to local masses manifests as the varying gravitational potential, and when the potential is converted back to Newton's form, the G which appears describes the effect of all non-local masses, which is effectively constant.

Variation with time is more severely constrained, in that if there is any variation occurring at present, it appears to be on a time scale greater than the age of the universe, which seems to rule out any simple model based on Sciama's idea.

Another closely related aspect of Sciama's idea is that rotation is relative to the rotational frame-dragging effect of all the masses in the universe. This too apparently requires G to be variable in a similar way. This effect is described as the "sum for inertia" and is mentioned for example in MTW "Gravitation".

Personally, I find Sciama's ideas compelling, and I consider it very disappointing that GR appears to be provably incompatible with them, even though there appears to be a surprisingly strong coincidence that both the linear and rotational frame-dragging effects of the whole universe appear to be around the right order of magnitude.

Given that GR is having a lot of problems explaining experimental observations (galaxy rotation curves requiring "dark matter" with increasingly implausible properties, weak lensing results, anomalous redshifts), I personally suspect that GR itself needs some fixing, and I think that Sciama's Machian ideas may provide some useful foundations for a more successful theory.

 

[Charlie G]

Thanks for the response,

I was very surprised when I read the sketch of his idea, I had never heard of any possible explanation of inertia, I had thought it was one of those mysteries where we hadn't formed any notions making it intelligible.

I definitely plan to keep an eye on the idea as it would be great to see ideas build off of it or a variation of his ideas made into the foundation of a theory consistent with our present observations.

Also, besides the one mentioned in your previous post, are there any other possible explanations of inertia that are accessible to a high-school student not yet able to do calculus?

 

[PeterDonis]

Another closely related aspect of Sciama's idea is that rotation is relative to the rotational frame-dragging effect of all the masses in the universe. This too apparently requires G to be variable in a similar way. This effect is described as the "sum for inertia" and is mentioned for example in MTW "Gravitation".

I don't have my copy of MTW available at the moment, but from what I remember, the "sum for inertia" was just a way of describing how it's convenient to separate the frame-dragging effect of a nearby rotating object (e.g., the Earth) from the frame-dragging effect of the rest of the mass-energy in the universe (and indeed, I don't believe it's limited to rotation, it also includes the linear frame-dragging effects that cause objects moving under uniform acceleration, for example, to feel weight). This works just fine in standard GR, with G constant; no variability of G is required. Wheeler co-authored another whole book with Cuifolini, *Gravitation and Inertia*, which expands on this a lot more.

 

[Jonathan Scott]

I don't have my copy of MTW available at the moment, but from what I remember, the "sum for inertia" was just a way of describing how it's convenient to separate the frame-dragging effect of a nearby rotating object (e.g., the Earth) from the frame-dragging effect of the rest of the mass-energy in the universe (and indeed, I don't believe it's limited to rotation, it also includes the linear frame-dragging effects that cause objects moving under uniform acceleration, for example, to feel weight). This works just fine in standard GR, with G constant; no variability of G is required. Wheeler co-authored another whole book with Cuifolini, *Gravitation and Inertia*, which expands on this a lot more.

MTW section 21.12 calls attention to the weird fact that GR requires that the "sum for inertia" is effectively constant, even during the early stages of the universe, when the ratio of the size to the mass would be expected to be completely different, by orders of magnitude. However, it then simply asserts that GR tells us that the "effective" value must be constant.

I have a copy of "Gravitation and Inertia" which I bought specifically to see if it could resolve this paradox. It is certainly all about frame-dragging effects and inertia, and contains some useful explanations of how it all works in terms of the Sciama model, and detailed calculations of local frame-dragging effects together with details of how they could be experimentally observed.

However, it does not answer the point which Einstein himself noted in about 1922 which was essentially that if inertia (and hence G) were determined by the distribution of masses, then this would mean that different distributions in space and time would give rise to variations in G, which would be incompatible with GR. Instead, the book refers to Sciama's approach as a "poor man's model of inertia". It includes some checks that the effect is of the right order of magnitude but then claims that a more accurate model requires knowledge of the "initial value" conditions, which effectively determine the shape and structure of the universe. I found this very unsatisfying.

It is perfectly reasonable to say that as we don't know the exact shape and size of the universe, we can't do the integral to get the exact value for inertia. However, if the inertial mass is affected by the distribution of masses, it should also be affected by local masses, such as those in the solar system and within the galaxy, and even though those masses only contribute a few billionths of the overall potential, their theoretical contribution should imply that the effective value of G would have to be different within the galaxy and outside it, and I don't feel that Wheeler ever managed to come up with a satisfactory answer to this paradox.

 

[PeterDonis]

It is perfectly reasonable to say that as we don't know the exact shape and size of the universe, we can't do the integral to get the exact value for inertia. However, if the inertial mass is affected by the distribution of masses, it should also be affected by local masses, such as those in the solar system and within the galaxy, and even though those masses only contribute a few billionths of the overall potential, their theoretical contribution should imply that the effective value of G would have to be different within the galaxy and outside it, and I don't feel that Wheeler ever managed to come up with a satisfactory answer to this paradox.

I guess I've never seen the paradox here. Instead, I see at least three distinct phenomena which, though they are often conflated in these discussions, could perfectly well have separate explanations (that is, even if there is ultimately one fundamental physical theory that explains them all, say for example superstring theory, it doesn't have to explain them all "by the same route", so to speak):

(1) At a given event in spacetime, which states of motion are inertial and which are not? I think GR takes care of this (at least at the classical level--whether quantum gravity changes anything or not remains to be seen, but I think quantum physics is more important for the other phenomena below), and the way GR takes care of it, I think, incorporates Mach's principle to the extent it can be for this purpose (because the inertial states of motion are determined by the solution of the Einstein field equations, which depend on the distribution of stress-energy in the Universe).

(2) Given a certain quantity of stress-energy, how much spacetime curvature does it produce? (Roughly, this means how much does it cause the answer to #1 above to change from event to event?) Under GR, the constant G (more precisely, $8 \pi G / c^{2}$ if we use units of mass for stress-energy, or $8 \pi G / c^{4}$ if we use units of energy) is what determines this. In one sense, it's just a conversion factor between conventional mass or energy units and geometric units, the same way the speed of light is a conversion factor between time and distance units. But in another sense, G could be a "coupling constant" in the quantum field theory sense, meaning that the value we observe would be the result of underlying quantum physics at a deeper level (and might change, the way other coupling constants in quantum field theory change as the energy of the process used to measure them changes). In neither of these senses, though, would G be affected by the large-scale distribution of stress-energy in the Universe, as far as I can see.

(3) How much force is required to make a given amount of stress-energy follow a given non-geodesic worldline (i.e., have a given proper acceleration)? This is where "inertial mass" comes in, but again, I'm not sure I see how it has to be connected to the distribution of stress-energy in the Universe, since in order to tell whether a worldline is geodesic or not we have to already have answers to #1 and #2 above, so we've already accounted for the effects of that distribution. I'm inclined to view this, like #2, as a question that ultimately needs to be answered by something like a fundamental quantum field theory, for example the way the Standard Model of particle physics has fundamental particles that are all massless (meaning that basically they follow null geodesic worldlines, and can't be "accelerated"), but which acquire mass through the Higgs mechanism (meaning that they sort of "bounce around" along short null geodesic segments in different directions, so at a macro level it looks like they are moving on an "average" worldline that's timelike, and which could be geodesic or not depending on the specific details of the situation).

 

[Jonathan Scott]

If we accept (as Wheeler apparently did) that inertia and rotation are effectively due to the frame-dragging effects of the universe, then even though we don't know the detailed m and r values for the rest of the universe, this implies that the effective sum of Gm/rc² for every mass in the universe is a simple constant.

The paradox then arises from the fact that this cannot hold all the time, everywhere.

For example, if we move some local masses around (or move a test object around them) we get variable local frame-dragging effects, while the more distant effects remain unchanged. If these local effects are included in the effective sum, it cannot remain constant unless something else changes, specifically G, which would however be inconsistent with GR.

 

[PeterDonis]

if we move some local masses around (or move a test object around them) we get variable local frame-dragging effects, while the more distant effects remain unchanged. If these local effects are included in the effective sum, it cannot remain constant unless something else changes, specifically G, which would however be inconsistent with GR.

Well, in GR, the local frame-dragging effects are modeled perfectly well without having to change G. They make a small change in what I labeled #1 above--in which states of motion are inertial and which are not--but no change in G is required to model this; you're just changing the local solution to the EFE that you're using (roughly speaking, from the Schwarzschild to the Kerr solution), without changing the asymptotic boundary conditions (which account for the effects of the rest of the matter in the universe).

In other words, local frame dragging, according to GR, doesn't change the inertial mass of anything--it doesn't have to; it just changes the local spacetime geometry a little bit. Extending this logic to the rest of the universe, according to GR, the frame dragging effects of the rest of the universe determine the asymptotic spacetime geometry against which the local frame dragging effects are seen as a small perturbation. This allows that "inertia and rotation are effectively due to the frame-dragging effects of the universe" without G having to change. According to the numbering of issues in my last post, #1 is determined by the distribution of stress-energy, but that doesn't require that #2 or #3 must also be determined by that distribution.

I do agree that the "asymptotic spacetime geometry" bit doesn't really constitute a final solution to the issue, unless, as Einstein noted, the universe is closed. But I don't see how, for example, a viewpoint like Sciama's would be any better off in that regard.