Discuss Relativity

[Jadzia]

Referring back to Shaw's post:

Minkowski Space-Time include three spatial dimensions and one time dimension,

which can be reparametrized in polar coordinates as

In General Relativity the metric is not static throughout space-time, it is effected by mass-energy. So if we put a mass M at the origin of the above representation what we get is

I'd like to know the following:

Suppose a cubic volume of space was measured out, side length x metres.
It has volume x^3 cubic metres. It is centred at distance D from the origin.

Then we suddenly introduce the mass M at the origin, and it warps space around it. What happens to the volume of the cube? Presumably the mass stretches the space and the cubic volume is distorted.

If we let x tend to zero, what is the ratio H of the volumes (new/old), as a function of M and D?

This kind of measures the local intensity of the warping.

If we assume that the spatial warping is the result of the flux of gravitons from M, we expect the attenuation of H (with increasing distance D) to be the same order as the attenuation of the graviton density, which... is it inverse square??

 

[Ronald Held]

How did the central mass get there and how long did it take?

 

[Jadzia]

I don't know. I didn't think that was important. But I've never studied relativity.

(1) It moved there from the opposite side, taking an infinitely long time
(2) It moved there from the opposite side, moving into position at light speed.
(3) It grew as a result of minute particles falling into the origin from all directions, taking an infinitely long time.
(4) It grew as a result of minute particles falling into the origin from all directions, at light speed.
(5) The whole mass just appeared there with no apparent journey. Can we compare that with spontaneous meson production?

 

[Jadzia]

I'm noticing that the tensor shown is only non-static in the radial direction, so it appears that the relativistic space warping is basically one dimensional. So is it a simple as pulling the coefficient it out of the metric?

ie, H = ds/dr = (1 - 2M/r) ^ (-1/2)

I notice that for r>2M (outside event horizon, or normal non-singular gravity), ds/dr > 1, suggesting that the spatial warping is a stretching/expansion of space towards the mass, much like those rubber sheet things.

ie, in the vicinity of mass, there is more space than there would otherwise be.

When we introduce an event horizon, this stretching becomes asymptotic, creating an infinite 'tube' of space.

Comments please. I would like to understand this forbidding subject.

 

[Shaw]

Referring back to Shaw's post... I'd like to know the following:

Suppose a cubic volume of space was measured out, side length x metres.
It has volume x^3 cubic metres. It is centred at distance D from the origin.

Then we suddenly introduce the mass M at the origin, and it warps space around it. What happens to the volume of the cube? Presumably the mass stretches the space and the cubic volume is distorted.

If we let x tend to zero, what is the ratio H of the volumes (new/old), as a function of M and D?

This kind of measures the local intensity of the warping.

If we assume that the spatial warping is the result of the flux of gravitons from M, we expect the attenuation of H (with increasing distance D) to be the same order as the attenuation of the graviton density, which... is it inverse square??

I'd be happy to speak to Classical Physics stuff (which includes Relativity Theories), but graviton are a construct of attempts at quantum gravity (which I have no expertise in).

As for the General Relativity stuff, these are great questions which are very helpful in gaining an understanding of what is happening.

Much like the techniques of Lagrangian Mechanics, it is best to pick a coordinate system that works with what we are dealing with. And as we are talking about change of volume based on some mass M, not only are spherical coordinates nice, but having the volume being effected being a spherical region rather than a cube is also very helpful too.

I'm going to outline this quickly as I am in the middle of a ton of work I need to finish (which is why I haven't been posting much over the last week).

(1) It moved there from the opposite side, taking an infinitely long time
(2) It moved there from the opposite side, moving into position at light speed.
(3) It grew as a result of minute particles falling into the origin from all directions, taking an infinitely long time.
(4) It grew as a result of minute particles falling into the origin from all directions, at light speed.
(5) The whole mass just appeared there with no apparent journey. Can we compare that with spontaneous meson production?

Here is the thing to note, we don't need all of the dimensions for getting a handle on this. So switching from Minkowski 3+1 space to Minkowski 2+1 space (or even Minkowski 1+1 space) might be helpful in simplifying the math and making the geometrical effects more noticeable. So in the Minkowski 2+1 case we would look at the effect on the area of a disk rather than on the volume of a sphere (and in the Minkowski 1+1 case we would be looking at the change in length of a region centered at the picked origin). In each case we shrink the equations by removing excess dimensions (some of which aren't in direct play in these questions). The Minkowski 2+1 case should look like this...

Without doing anything at all, the last two sets of equations display the relative differences of a before and after state of a spherical region before and after the introduction of a large mass M at the origin (the second to the last set can be thought of as the final set with M=0). Same for a mass M growing. If we assume that it is growing from a uniform mass dispersion about the origin (which once started to fall towards the center did so in uniform spherical densities), then their effect of the region could be ignored inside the changing spherical densities, and as adding to the total mass of the M outside of them until they are literally part of the center mass. In this case one could look at the growing of M over time and the effect of differing values of M as a function of the radius. You could even reduce this further and look at the changes as a spherical surface density of mass shrunk through the region toward the center mass... or even a spherical mass density at a fixed distance from the center mass (which would provide an understanding of how a Dyson sphere would effect such a region).

When considering all this, one need only remember that apparent (instantaneous) volume (or area) is primarily a function of the radius component of our spherical coordinate system. Looking at the effects over time needs to be done by keeping in mind that the effects of any change in central mass move outwards from the center at the speed of light. So if the volume you have picked is large enough for this to be effected (say like a mass about that of the Sun and a volume of a radius of about the size of the Earth's orbit where the radius is about 9 light minutes), then you can watch over time as that instantaneous introduction of a mass would effect that region (over a period of about 9 minutes).

Which, of course, starts to bring us into a place where you might want to do some applications of this. In which case, much of what I've done to simplify the equations has to be undone. Generally I like to have mass, distance and time to all be in the same units, and have constants like the speed of light and the gravitational constant (G) equal to "1". I also did this to make the geometry of the equation show through more as it was the related geometry I was attempting to show.

And again, I'm sorry for not being versed in the nature of particle physics... the closest I got to that type of stuff was a study of Gauge Theories in which the other forces of nature (electromagnetic, weak and strong) are treated as connections over fiber bundles (additional dimensions) which display group like structures (specifically spherical groups which become non-Abelian in the cases of the weak and strong forces)...in other words, it is all geometry rather than particles.

So I'd be very happy to discuss Gauge Theories in addition to Relativity for those who are interested, specially as it makes use of a number of vary interesting areas of mathematics such as fiber bundles and Lie Groups (again, as my time permits).

 

[Shaw]

... I would like to understand this forbidding subject.

Actually, it looks like you are starting to get a handle on it.

As with many subjects, a mystique is built up around this that is often propagated by those who enjoy the prestige of it. It is not easy, but the idea that only a few people in the world can understand Relativity is a myth I'd like to see undone.

Granted, the major steps you made came from knowing enough to ask the right questions while looking at this (and that insight shouldn't be discounted in any way), but I see no reason why someone with your foundations should ever be told that this subject is in any way outside of your grasp (as you've already proven so nicely ).

When we introduce an event horizon, this stretching becomes asymptotic, creating an infinite 'tube' of space.

And this is why a number of attempts have been made to create a new geometric understanding of that region and (in effect) glue that geometry to the geometry outside at the event horizon.

Unfortunately, this really starts to step over the line between physics and mathematics as we have no evidence of what is happening inside of the event horizon to compare any of our models with.

 

[Jadzia]

Thanks Shaw this is brilliant. Very grateful. I'm going to study it now.

 

[Zachary Smith]

Seems to me that if the "mass" introduced into the volume of space is unique in origin (i.e. NOT drawing any of its source "essence" from the space itself), then the "space would: 1. need to expand to accomodate the new presence then 2.contract as a function of the mass warping the space around it in the form of gravity. By what degree, I haven't the faintest idea.

If the source "essence" of the "mass" IS drawn from the space, you're not really adding or removing anything; just moving the constituent components around.

Then again, I may out of my depth here.

 

[Shaw]

Seems to me that if the "mass" introduced into the volume of space is unique in origin (i.e. NOT drawing any of its source "essence" from the space itself), then the "space would: 1. need to expand to accomodate the new presence then 2.contract as a function of the mass warping the space around it in the form of gravity. By what degree, I haven't the faintest idea.

If the source "essence" of the "mass" IS drawn from the space, you're not really adding or removing anything; just moving the constituent components around.

Then again, I may out of my depth here.

Well, a lot of this can be made to work out nicely (or at least nicer) if we look for invariant situations. Looking directly at Jadzia's questions (which really are pretty much the core questions that were put forward when General Relativity was introduced... only they needed to wait for the mathematics to catch up back then), we can arrange the experiments so that the mathematics doesn't require a lot of reworking (which is always a pain to be avoided if possible).

(1) It moved there from the opposite side, taking an infinitely long time.​

In this case, what we could do is have are test region be relatively large, but have a much smaller sphere of a fixed radius move in from one side until it reaches the origin of our system and examine the effects on the sphere as it moves in.

And in this case as long as the speed of the smaller sphere is not anywhere close to relativistic velocities, we should be able to generalize all of our results.

(2) It moved there from the opposite side, moving into position at light speed.​

Same set up as above, but in this case the relative speeds are going to mean that not only are we going to see the effect of gravitation (General Relativity) on the geometry of the smaller sphere, but also Special Relativity (which effects time, length and apparent mass in the direction of travel).

(3) It grew as a result of minute particles falling into the origin from all directions, taking an infinitely long time.​

This is the situation where you start with a symmetric mass distribution and at the origin of our test region introduce a brake in the symmetry that starts the near by particles to start falling in. This is a little messy in the math area as what you would be concerned with is collapsing spherical shells of mass density that (once outside of any given shell) would effect the rest of the region as if it was already concentrated at the origin.

That is why I suggested looking at a single mass density shell first, get a feel for how it effects the region, then you can integrate over an infinite number of infinitely thin shells to see the large scale effect.

(4) It grew as a result of minute particles falling into the origin from all directions, at light speed.​

Same setup as above, but rather than letting the braking of the symmetry create the falling towards the center, we would look at mass density shells with relativistic initial kinetic energy rushing towards the center.

(5) The whole mass just appeared there with no apparent journey.​

This one is actually very important as it was this type of thought experiment that gave rise to the gravitational wave theory. Because the introduction (and/or removal) of a large mass doesn't effect the rest of space-time instantaneously, there would be a gravitational wave that would propagate out from the center.

As I recall from a talk by Kip Thorne back when I was in school, such a wave might proceed any light based information from such an event. So having a gravitational wave detector in place would give astronomers a sort of early warning system of where to look for interesting phenomena like supernova.


But beyond these thought experiments, Jadzia nailed the basic geometry of what is happening near large masses. And when taken in conjunction with what happens to time in those same regions, one can actually gain a pretty solid understanding of how gravitation works locally.

Now when you start looking at the large scale effects of General Relativity on the universe, a lot more issues come into play. Like the topology of the universe, the expansion rate (which can introduce special relativistic effects between otherwise seemingly stationary objects) and the initial mass/energy density of the universe (assuming conservation of energy within the universe as a whole).

So yeah, when you move beyond the local effects to these larger scale effects (which was what you seemed to be saying when asking about the "essence" of the "mass"), that is where this stuff can really start to become really interesting.


One of the last things I was working on in this area before I started my (overly long) break from school was a look at a "universe" with the spatial topology of a 3-sphere where the radial component wasn't within the universe itself but was actually a function of the passage of time, r(t). It was a re-deriving of the Schwarzschild equations with a closed small universe with a fixed amount of mass-energy M. Even though this dealt with aspects of General Relativity and Singularity Theory, it was mainly a mathematics paper as I was ignoring all other aspects of nature within the model to just look at just the geometry of that (overly simplistic) scenario.

Which, in turn, is one of the nice things about being part of the mathematics community rather than the physics community. In mathematics you can study anything you want where as in physics you often get pushed into following whatever the conventional wisdom is at the moment. So for me pure mathematics offered a higher degree of freedom than theoretical physics, specially at my old school (where the best theoretical physicists were in the mathematics department)... but that is starting to veer off topic (though I thought I should explain part of the reason why I went towards mathematics rather than physics).

 

[Jadzia]

Looking at the simplest 1+1 Minkowski space, now that we can calculate the local degree of stretching in the radial direction (r), we can introduce an hypothetical spatial dimension (I'm calling it y) and represent this stretching as extrusion of the "flat" surface into that dimension. To aid visualisation of warped space, this allows us to move away from the Minkowsi metric, back to a Euclidean metric, and to represent warped space through the intrinsic geometry of this surface manifold.

ie, given the metric

ds^2 = (1 - 2M/r) ^ (-1) dr^2


we represent the manifold with ds^2 = dr^2 + dy^2, where y is the new hypothetical dimension. Equating these gives:

(1 - 2M/r) ^ (-1) dr^2 = dr^2 + dy^2


dy^2 = [r/(r-2M)-1] dr^2


dy/dr = [2M/(r-2M)] ^ (1/2)

Integrating this we get the equation of the manifold:

y = 2.[2M.(r-2M)] ^ (1/2)

which is basically an inverse parabola, with it's turning point shifted horizontally to r=2M.

Since there is this radial symmetry to the spatial warping, we can conveniently imagine a "surface of revolution" of this curve into the other two spatial dimensions. This gives us a 3 dimensional manifold embedded in a 4 dimensional space, that we can just about visualise.

Contrary to what I was expecting, this doesn't create an infinite tube of space at the event horizon, which makes me think I've gone wrong. Curiously the space ends abruptly at the event horizon, unless we also consider the other half of the paraboloid reflected on the -y axis. However, any geodesic crossing that sphere of turning points doesn't come out again. I'm also surprised that this manifold is a paraboloid, and not a hyperboloid, having no horizontal asymptote as r tends to infinity. Something is amiss.

 

[Shaw]

I need to rush off to some appointments today, so I'm not sure if I'm just not seeing this in a rush, but where is your time component? In both special and general relativity all paths need to be studied as "time-like" paths between events, and without the time component what you end up with is "space-like" paths between points (paths on the surface of space stuck in a frame of time).

So while the space is approaching a tube like shape, time is slowing to a stop. That is why there is talk about unzipping the geometry at this point and looking at something else.

Further, and this is where the name comes from, what you are noticing is that the surface is no longer differentiable... and a non-differentiable point on a manifold is called a singularity (hence the use of the term in the study of black holes).

We are cut off (both physically and mathematically) from what happens on the other side of the event horizon... which gives rise to the no naked singularity rule (meaning nothing can have a causal path from within the event horizon), which in turn was broken if you believe that the big bang was a singularity and everything here has a causal path back to it. So what we have are spatial singularities (black holes) and time singularities (big bang and maybe big crunch).

Does anything exist on the other side of an event horizon? The mathematics used for General Relativity falls short in attempting to answer this question. And before we discovered their existence, the name black hole was a derogatory term given to research in this area as it just seemed like a mathematical curiosity when first put forward by people like Oppenheimer (who didn't live long enough to see his ideas vindicated).

Ooops... now I'm late.


Edit: I don't want to leave you hanging, but what you want to be looking at is called de Sitter space. I haven't taken the time to read the Wikipedia article, so only take it as a starting point (I generally don't trust Wikipedia articles as a rule).

 

[Jadzia]

Hi Shaw.

I checked through my calculus in my afternoon break today and I think its right.

I trimmed the time term from the metric as it didn't seem relevant to the spatial geometry. Back in my original question, I was looking to understand the shape of the warped space through measurements of lengths/areas/volumes and how they are affected by a nearby mass. I feel that the paraboloid I've got here represents the geometry of the space as far as laying out an array of tape measures goes. Euclidean measurements of length/area/volume on that paraboloid match the corresponding Minkowski measurements of the warped space.

But the event horizon it isn't as I expected, because the integral of this "asymptotic stretching" does actually converge to finite limit. The shape of the integrand is O[r^(-1/2)]. In calculus, a shape of O[1/r] represents the threshold of convergence, as it integrates to ln(r), the function with the weakest asymptote.

What this means is that we don't need an infinite number of metre rulers to measure the Minkowski distance from an orbiting planet to the event horizon.

But correct me if I'm wrong

 

[Shaw]

Just a heads up to everyone...

Jadzia's math was correct, I was talking with her via PM about the time aspects (to make sure I didn't make a mistake in how I characterized it). But yeah, in the absence of the time components what she came up with is not only correct, but as she found it matches the Flamm's paraboloid (which I hadn't heard of before).

Just an update so that it doesn't seem like this topic was left hanging.

 

[Jadzia]

Given that mass stretches the space, and increases the amount of space in the vicinity of the mass, as shown by the paraboloid. And given that the cumulative spatial stretching is finite when measured from a distant point to the event horizon, it prompts the question: Just how much more space do we have?

This stretching increases the amount of space, while the event horizon removes a spherical region too. So does it balance out? Is there more or less space? The answer wil obviously be different depending on the number of spatial dimensions we consider this warping to occur in.

We can calculate the arc length from the event horizon r=2M to a point at radius r=L from the centre of mass, we effectively find the arc length of a parabola, as this Flamm's paraboloid has a parabolic profile.

It's pretty basic calculus but it fills a couple of pages that I'm not typing out here! Just briefly.

The event horizon r=2M I'll now denote by its radius r=r0.

Arc length S = INTEGRAL from r=r0 to r=L of {r/(r-r0)}^(1/2) dr

change of coordinates y = 2(r0(r-r0))^(1/2)

S = (1/(2.r0)) INTEGRAL from r=0 to r=2(r0(L-r0))^(1/2) of {(2.r0)^2 + y^2}^(1/2) dy

which if my rusty calculus is correct gives:

S = (L(L-r0))^(1/2) + r0.LOG{ 2[r0(L-r0)]^(1/2) + [4.r0.L]^(1/2) } - r0.LOG[2.r0]




So how does this differ from flat space where M=0?

We look at the difference in the arc lengths = S-L

In the case of the VERY LARGE, it is the limit as L tends to infinity. So assume that L is much larger than r0.


S-L --> (L.L)^(1/2) + r0.LOG{ 2[r0.L]^(1/2) + [4.r0.L]^(1/2) } - r0.LOG[2.r0] - L

--> r0.LOG{ 2.[r0.L]^(1/2) / r0 }

--> O[LOG{L}]


This can be depicted through a conversion table

Distance.....Distance
S, if M=0....S, for some mass M>0

10.............11
100...........102
1000..........1003
10000........10004
10^n.........10^n + n

The logarithm function doesn't have any horizontal asymptotes, so this is an INFINITE amount of additional space that mass M adds to the universe, although it is spread thinly over great distances. Also as we know, the gravitational effects propogate at light speed so it would never reach infinite additional space in finite time, but it would be a continuous unbounded expansion.

 

[Zachary Smith]

^^maybe I wasn't so far off. *heh*

 

[Jadzia]

The next thing to look at is time dilation in the vicinity of mass.

This term can be pulled from the metric too.

dt'/dt = (1-2M/r)^(1/2)

where,
t is the passage of time for a relatively stationary distant observer
t' is the passage of time in the vicinity of the mass (at distance r from it)

Outside of the event horizon, or in the normal space around a star, dt'/dt < 1, suggesting that proximity to the mass causes time to slow.

At r=2M, at the event horizon, dt'/dt = 0, suggesting that time slows to a standstill as we move closer to the event horizon.

At infinite distance, dt'/dt = 1, suggesting no time dilation effect at infinite distance.



How do we interpret this?

Suppose we look at two planets in a black hole star system BX are populated.

BX I is deeper into the gravity well such that the time there is observed to flow at 50% the rate from the perspective of the more distant planet, BX II.


When BX I residents look up to the skies, they see BX II moving twice as quick through its orbit than classical inverse square law suggests. And television broadcasts from BXII show images at double speed, while in conversations over the telephone they hear their neighbours speak twice as fast.

There is also potential for computer processing power multiplication. Beings on BX I can send information to BX II for computers to process, and the computer sends the solution back. From their perspective, the BX II computers are twice as powerful.


But perhaps the BX I beings have drawn the short sword? By the way the galaxy is arranged, this slowed time is restricted to these gravitational islands around stars, making their inhabitants almost like prisoners of time. Most places in the galaxy are aging faster than these islands, and in some sense there is a 'normal rate' where star masses are relatively small, locations at sensible distances from stars, and moving relatively slowly.

Should you be living in a gravity well with slowed time, for the threat of aliens wanting to invade, you'd be putting yourself at a distinct disadvantage to have your time slowed relative to them. Their ships would appear to approach twice as quickly, giving essentially half the time to activate the planetary defence systems.

Your computers would appear to run 50% slower than identical computers on the alien ships, meaning that in theory it could always strategically outmaneouvre you. They could also out think you as their minds operate twice as fast as yours.

The rest of the galaxy's inhabitants advances technologically at twice the rate you do, meaning that you'll have inferior defences which are mismatched to your attacker's weaponry.

But on the plus side, you experience more of the age of the universe.



I wonder if there is an opposite effect? To create an island of space where the flow of time is faster inside than it is relative to distant interstellar space?

A region you can enter into briefly as a kind of temporal stasis. A island region where you can spend a year of your life, and when you exit only 6 months (or less) has passed for everyone else.



Well yes there is, look at different kind of gravity wells: Those described by the Reissner-Nordström metric.

This is an artificially upgraded black hole, by adding an electrical charge.

Independent of the mass, the intense electronic charge changes the mechanics of the space and time warping.

This is the metric:

Here we see that the time dilation is different. A high electronic charge would normally mean repelling forces that would blow a planet apart, but a black hole is held together by irresistible gravity, so it can hold together.

dt'/dt = (1 - 2M/r +Q^2/r^2) ^ (1/2)

Clearly now, as charge Q increases, there comes a point where the time dilation experienced at BX I can increase above 1, meaning that in the gravitational island around the black hole, time can be compressed instead of dilated. Computers (and people) placed in the gravity well could process information more quickly than those at distance.

 

[Jadzia]

This evening's study is returning back to the Schwarzchild solution and thinking about what spatial warping will occur with a whole galaxy full of stars.

The Einstein equations are not linear, so we can't just superimpose solutions and expect it all to work out nicely. I don't know if a transform has ever been worked out to do this; a machine which inputs two relativistic gravitational fields, and outputs their sum.

What I notice about the Schwarzchild star is that the stretching is one dimensional towards the mass, and any significant space-time curvature is highly localised to a few million kms around the star. Beyond that distance, the curvature is very low, so solutions to the Einstein equations may be treated as approximately linear in this region. Meaning that we may superimpose space-warping solutions without incurring much error for modelling the large scale mechanics of stellar clusters, and principally where I'm heading here is the relativistic dynamics of our own galaxy.

The second feature of the Schwarzchild solution is that the time warping apparently mirrors the space warping. I'll maintain this relationship. This figure is a scalar quantity, ie is independent of the direction of the spatial warping. So having superimposed our space warping solutions, we can calculate the total spatial divergence at a point, and invert this figure for the degree of time warping for that region.

At this stage I'm unsure how to combine these three scalar components of space warping: as a root-sum-square, or as a new/old volume ratio. These are both coordinate system independent combinations. I'll think about that later.


My Method:
I'll treat the galaxy as a 2-dimensional disc of mass M, radius R which is composed of a uniform spread of an infinite number of infinitessimal masses dM, each creating a weak relativistic distortion of space and time. For any point along the radius of that disc (distance r from the core), we integrate the sum of these effects to find the total spatial divergence. This will be done numerically for various radii r from 0 upto 3R, and profile graphs created.


I'll move to AI units now.

ds/dr= (1-2Gm/rc^2)^(-1/2)

Let A = 2Gm/c^2. Although compiled with universal constants, this represents the mass of the galaxy.

ds/dr = (1-A/r)^(-1/2)

Far from the star, where A/r is small, a binomial expansion of the derivative will be approximate.

ds/dr ~ 1 + (-1/2).(-A/r) + (-1/2)(-3/2).(-A/r)^2 / 2! + ...

ds/dr ~ 1 + A/2r + O[1/r^2]

This error term is insignificant when r is large, so may be omitted.

So we take ds/dr = 1 + A/2r

The expansion is simply (ds-dr)/dr = A/2r

This is linear in m, so the integration of over the disc area is natural as dM tends to zero.

How on Qo'Nos do I do this integral? Hmmm..

Right, what is the polar form of an off centre circle? Cosine rule...

H = r.cos(x) + [ R^2 - ( r.sin(x) )^2 ]^(1/2)

INTEGRAL (x=0 to 2.PI) INTEGRAL (h=0 to H) [ A/2h .h] dh.dx
NB: The second h in the integrand represents the arc expansion of polar coordinates

= INTEGRAL (x=0 to 2.PI) [ AH/2 ] dx

For the two component directions, we now introduce component terms into the integrand. These are absolute values, since the stretching is the same whether the star is infront or behind us. I'm assuming that adjacent gravitational fields amplify rather than cancel-out their space warping effects in the null gravity point inbetween them.

Galactic Radial component = A/2 . INTEGRAL (x=0 to 2.PI) [ H.|cos(x)| ] .dx
Galactic Angular component = A/2 . INTEGRAL (x=0 to 2.PI) [ H.|sin(x)| ] .dx

These will obviously reduce to two half integrals.

Let sin(x)=y.R/r

Galactic Radial component = A/2 . INTEGRAL (x=0 to 2.PI) [ H.|cos(x)| ] .dx

= A/2 . {0 + 2.R^2/r .LOOP_INTEGRAL (x=-PI/2 to PI/2) (1-y^2)^(1/2) .dy

= A.R^2/r . [r/R .sin(x). (1-r^2/R^2. sin(x)^2 )^(1/2) + (1/2).INV_SIN(r.sin(x)/R) ] (with x=PI/2)

Letting a=r/R, the proportion of distance to the rim of the galactic disc.

So, the radial component of space-warping at distance r=aR from galactic core is...

= A.R.[ (1-a^2)^(1/2) + INV_SIN(a) / (2a) ]
----------------------------------------------

How nice. I don't think that R should be there as a product though maybe A/R but not AR.

Although to be fair, I got through about 4 sheets of paper before finding the right substitution. It's been a while. This is probably all nonsense anyway.


Now for the angular component:

This time let cos(x)=y.(R^2-r^2)^(1/2)/r

Galactic Angular component = A/2 . INTEGRAL (x=0 to 2.PI) [ H.|sin(x)| ] .dx

= A/2 . {0 + 2.(R^2-r^2)/r .LOOP_INTEGRAL (x=0 to PI) (1+y^2)^(1/2) .dy

= A.(R^2-r^2)/r . [(y/2).(1+y^2)^(1/2) + (1/2).LOG[y + (1+y^2)^(1/2)]] (limits x=0 , x= PI/2)

= A.(R^2-r^2)/r . { a/(1-a^2) + (1/2).LOG [(1+a)/(1-a)] }

So, the angular component of space-warping at distance r=aR from galactic core is...

= A/R . { 1 + (1-a^2)/(2a).LOG [(1+a)/(1-a)] }
----------------------------------------------

That one looks better.

Pictures to follow.

This calculus has been quite fun actually. I'm surprised I still remember it all, so it must be like riding a bike.

 

[Jadzia]

Oh I know what I did wrong that made that funny looking R appear. That m should be the area-density of the galactic disc, not the mass as such.

So A is revised: m becomes m/(PI.R^2). Or rather, keep m and A as they are, and I'll carry this correction directly into the final formulae:

Radial dilation = A/(PI.R).[ (1-a^2)^(1/2) + INV_SIN(a) / (2a) ]

Angular dilation = A/(PI.R) . { 1 + (1-a^2)/(2a).LOG [(1+a)/(1-a)] }


Sorted. You don't know how much that has been bugging me the past couple of hours

Bedtime Jadzia... BEDTIME !! Now you know the universe works... you can dream of shoes and ships and sealing wax and cabbages and kings, and why the sea is boiling hot and whether pigs have wings


UPDATE

Despite getting the cane across my hands for separating space and time, which is apparently forbidden in relativity, I've decided to upload the graphic for these warping functions.