Inertia, Mach's principle, frame dragging and galaxy rotation
By Wolfgang G. Gasser
1999-11-01 – How was Dark Matter calculated?
Jeffrey H asked:
Does anyone know what formula for gravity they used when they simulated the Milky Way on a supercomputer and discovered a need for additional mass ("dark matter")?
Louis S Pogoda Jr replied:
I think you're assuming the problem is harder to see than it is. Matter was first found to be "missing" back in the 1930's by Dutch astronomer van Oort. He looked at stars in the solar neighborhood whose motions were known. He found that, if you assume these nearby stars are part of the Milky Way and not just "passing through", they were moving too fast to be gravitationally bound by the visible mass of the galaxy. Based on the velocities of nearby stars, there must be about three times more mass in the galaxy than is visible.
Subsequent studies seem to show that the larger the scale at which you consider things, the more mass that seems to be missing.
That's very interesting. Been wondering about that myself. Now I'm wondering why a large black hole at the center of our galaxy couldn't explain this excess speed instead of dark matter. Do you know?
Mark Folsom replied:
I think that the orbital velocity profile, speed versus radius, of stars in the galaxy isn't consistent with a large central mass. Rather, it appears that the excess mass is distributed over a large volume.
Jeffrey H asked:
I wonder if they used Newtonian gravity or Einsteinian?
Nathan Urban answered:
Newtonian. You can show that Einsteinian corrections are negligible.
There is a very simple solution to at least a part of this dark-matter problem: the inersis hypothesis, i.e. the assumption that inertial motions do not follow straight lines but depend on the motions of the surrounding objects. The inersis concept can be considered a quantitative version of Mach's principle, based on 'weighted averages' of the velocities of the surrounding material objects (particles).
Inersis can also be considered a relational ether which is dragged by all particles proportional to their mass and inversely proportional to their distance (i.e. proportional to lost gravitational potential).
According to this concept, the higher velocities of stars in galaxies are caused by the same effect as Mercury's non-classical perihelion shift. Whereas the deviations from classical mechanics are still very small in planetary systems, they become the higher, the larger the scale at which one considers things.
The inersis concept as first developed in 1987 is descibed in: Der Abhängigkeitsraum (in German)
Assume an isolated group of six identical galaxies forming an equilateral hexagon. Further assume a little satellite galaxy s whose distance from galaxy 1 is the same as the distance between two neighboring galaxies.
4 . 1 s
In the short term the shape of the group can remain stable if the six galaxies orbit at a given velocity v their common center.
If we assume that inersis of s is affected only by these six galaxies, we get at the current location of s a tangential inersis velocity of 27% of the rotation speed of the group. This velocity is added to the velocity which is needed for s to compensate gravitational attraction of the group. (See reference above).
In Perihelverschiebung in der Relativitätstheorie (also in German) the analogous explanation of the non-classical perihelion shifts of the planets can be found. Mercury's shift results primarily from the Sun's rotation. The small shift of Mars however results primarily from Jupiter.
It should be very easy to introduce the inersis concept into existing simulations of galaxies. I'm sure that much better agreement with empirical data will be the result. Maybe also the problem of the spiral forms of galaxies can be solved in a simpler and more transparent way.
2007-07-28 – Inertia & Mach's principle & frame dragging
Let us assume that the total gravitational force acting on a freely moving test body in a galaxy is always exactly compensated by an artificial force. In this case the motion of the test body is subject to the principle of inertia.
As a thought experiment let us further assume that except for the test body, the whole galaxy is artificially set into motion in a given direction (i.e. the same velocity vector is added to every part of the galaxy).
The question arises whether this setting-into-motion of the galaxy affects our test body or not.
The answer according to the classical principle of inertia is: the test body is not affected by the galaxy.
On the other hand, the test body is embedded in the gravitational field of the galaxy. Thus, the hypothesis that the test body is somehow affected by the setting-into-motion of the galaxy is also reasonable.
If there is nothing outside the galaxy, then Mach's principle suggests that the test body's velocity with respect to the galaxy does not change while the whole galaxy is set into motion, because inertia depends on the other masses which consist only of the galaxy. This also implies frame dragging because the frame of the test body is dragged by the galaxy.
If inertia somehow depends on the other masses, shouldn't nearby masses have a bigger effect than more distant masses? A simple hypothesis is that the effect of a distant object on inertial motion of a test body is proportional to the mass of the distant object and inversely proportional to the distance, i.e. proportional to gravitational potential lost by the test body due to the distant object (what we can call the gravitational dependence on the distant object). This means that inertial motion of the test body is dragged by all these objects. The resulting drag can be calculated by weighted averages.
Let us introduce a coordinate system where the mass center of a galaxy is at rest and define an inersis vector field in the following way: the inersis velocity vector at any point is the average of the velocity vectors of all parts of the galaxy, weighted according to gravitational dependence.
If the kinematics and the mass distribution of a galaxy are known then it is easy to calculate the inersis vector field. This vector field obviously rotates around the center of the galaxy. E.g. near Earth, the weight of the stars in our neighborhood is stronger than the weight of the stars on the other side of the Milky Way moving in the opposite direction. Therefore, the inersis vectors in our galactic region point roughly to the same direction as the movement of this region.
Let us assume that such a quantitative version of Mach's principle is realized in nature. Then a test body moving at the velocity of the corresponding inersis vector is at rest with respect to the averaged movements of the surrounding masses. If the total gravitational force acting on such a test body were always exactly compensated by an artificial force, then the test body would rotate around the galaxy only by inertial motion (or rather by inertial rest).
The validity of this hypothesis also entails that in order to analyze the dynamics of a galaxy one should at first subtract the corresponding inersis vectors from the velocities of the galactic objects before applying the currently used methods.
Maybe somebody can answer these questions:
o Is it clear what I mean by 'inersis velocity vector field'?
o Is it easy to calculate this velocity vector field starting with an already existing numerical simulation of a galaxy?
o What is the rotation curve of the inersis vector field in comparison with the rotation curve of the rotation speeds? (see: galaxy rotation problem)
o Could the inersis vectors account for the discrepancy between the observed rotation speeds and the predictions of Newtonian dynamics?
2007-07-30 – Inertia & Mach's principle & frame dragging & galaxy rotation
As a google search for inersis vector reveals only two hits, both written by you, and those in terms of some theory based on ether unique to you, I think you can safely assume no one knows what you mean or can give answers to your question.
Please let me know if one of the following points concerning my hypothesis is not clear:
1) Inertial motion of a test body is influenced by the changes in motion of all objects, due to which the test body has lost gravitational potential. That means, velocity changes of all these objects tend to induce an analogous velocity change on the inertial movement of the test body.
2) If there is only one massive object then any change in velocity of this object leads to an identical change in velocity of the test body's inertial movement.
3) If there are n objects, then the n velocity-changes dv1, dv2, ... dvn affect the inertial movement of the test particle. If the gravitational potential losses of the test body due the n objects are respectively Φ1, Φ2, ... Φn, then the resulting velocity change of the test body is the average of the velocity change vectors dvi weighted according to the respective gravitational potential losses Φi: dv = (Φ1 dv1 + ... + Φn dvn) / (Φ1 + ... + Φn)
In many cases, the approach of starting with the velocity vectors vi is more convenient than starting with the velocity changes dvi:
4) We introduce a coordinate system with at its origin the mass center of a galaxy. The galaxy is described at any given time by n objects each having position vector xi, velocity vector vi and mass mi.
5) The total loss in gravitational potential Φ[x] at point x of the coordinate system is the sum of the potential losses Φ1, Φ2, ... Φn due to the n objects: Φ[x] = Φ1[x] + Φ2[x] + ... + Φn[x]
6) At any point x, a velocity vector w[x] can be defined as the average of the velocities vi of the n objects, weighted according to the corresponding potential losses Φi[x].
7) We can call this vector field w[x] inersis, following the stasis vector field of Bruce Harvey: "Taking a broader view we find that stasis is a vector field existing throughout all space and varying from point to point as we move around the solar system, between the stars and from galaxy to galaxy."
8) The inersis vector field rotates around a galaxy in a similar way (but weaker) than the parts of the galaxy do.
2007-07-31 – Inertia & Mach's principle & frame dragging & galaxy rotation
You use some well-known notions like 'inertial motion', but in the common sense?
I start with this definition of 'inertial motion':
A test body's movement is inertial if all gravitational forces acting on the test body are compensated by artificial forces (ignoring other forces such as electromagnetic and contact forces).
Then we can ask for the properties of 'inertial motion'. The answer according to the classical principle of inertia is: rectilinear motion at constant speed.
I deal here with an alternative answer to this question, namely the assumption that inertial motion of the test body is not completely independent of the movements of the other objects (a variant of Mach's principle).
If one defines inertial motion apriori as "rectilinear motion at constant velocity" then my alternative hypothesis obviously seems impossible.
Maybe the following (translated extract from 1988) could make my point of view more understandable:
"The principle of inertia (Newton's first law) states that a body with no forces acting on it moves uniformly on a straight line. Yet this is an empty statement insofar as there are no bodies without forces acting on them. Another version states that the body moves uniformly in the absence of other physical bodies acting on it. However, also this is a meaningless statement, unless 'absolute space' is postulated.
Albert Einstein spoke of the movement of a body which is far enough away from all other bodies. But also in this version coming closest to reality, the real problem is not solved but only deferred. Under what conditions is a body far enough away from all other so that one can speak of rectilinear movement at the same speed (with respect to what?).
We should search for a suitable (realistic) form of the principle of inertia. A test body moving e.g. in our Milky Way is gravitationally attracted from all sides by other bodies. From the mass distribution of the Milky Way (and the rest of the Universe) we can in principle calculate the combined gravitational acceleration. If the test body is accelerated artificially by the opposite of this acceleration vector, then the question of this "force compensated" movement gets a concrete physical meaning.
This question may be answered in two principally different ways:
o By analogy with the so-called Galilean principle of relativity, this force-compensated movement of the test body remains independent from the movement of all other celestial bodies, i.e. rectilinear and uniform with respect to geometric (absolute) space.
o The test body's movement still remains bound to the movements of the other celestial bodies.
Are you assuming background, absolute space and time?
At least insofar as for dealing with the dark-matter problem classical approximations are enough, I assume absolute space and time.
Are you tempting at a foundation of classical mechanics or gravitation theory?
Here I only want to find out whether this rather simple inersis hypothesis (originally introduced for other reasons) can resolve the dark matter problem.
Inertial motion of a test body is influenced by the changes in motion of all objects, due to which the test body has lost gravitational potential.
What is "loss of gravitation potential"?
We can also call it gravitational dependence. Gravitational potential loss is simply ½ vescape2 where vescape is escape velocity. Our gravitational dependence on Earth is ΦEarth = ½ (11.2 km/s)2 = 63 km2/s2. Our loss due to the Sun ΦSun is around 900 km2/s2.
If there is only one massive object then any change in velocity of this object leads to an identical change in velocity of the test body's inertial movement.
If there is only one massive object, what changes its velocity?
This is only a thought experiment for didactical purposes.
We introduce a coordinate system with at its origin the mass center of a galaxy. The galaxy is described at any given time by n objects each having position vector xi, velocity vector vi and mass mi.
What for a coordinate system, how its axes are defined?
A simple Cartesian coordinate system with the galactic plane preferably in the x-y-plane.
We can call this vector field w[x] inersis, following the stasis vector field of Bruce Harvey: "Taking a broader view we find that stasis is a vector field existing throughout all space and varying from point to point as we move around the solar system, between the stars and from galaxy to galaxy."
Ok, you can do that. What is the meaning of it?
The Sun has lost potential energy due to all objects of the Milky Way (and also due to other galaxies). Ignoring the other galaxies we can calculate for the Sun the average velocity of all these objects weighted according to the corresponding gravitational dependence (i.e. proportional to the object's mass and inversely proportional to its distance from the Sun). We get a resulting velocity vector roughly parallel to the Sun's movement in the galaxy.
Let us assume that this inersis velocity is around 30 km/s for the Sun and that the Sun's orbital speed is 220 km/s. In this case we would have to subtract this 'inertial drag' of 30 km/s from the 220 km/s and use the remaining 190 km/s to test whether the Sun has a stable orbit at a rather constant distance from the galactic center. (The 190 km/s and not the 220 km/s would also be relevant to kinetic energy with respect to the galactic center).