Recently I've been spending some time trying to get a handle on General Relativity. I've never been satisfied with objections to my preference that black holes don't really exist so I thought it would be a good idea to see if I could determine for myself one way or another whether the idea has merit.

I've been looking at the Schwarzschild metric and trying to get a feel for it, or even what "metric" means. Back in the early days when Einstein was working out GR (General Relativity) this guy Schwarzschild worked out a closed-form exact solution for a spherical non-rotating, uniform density, non-charged mass, at least the part outside the sphere. First we start with a term r sub s which represents the radius in meters of a black hole of mass M:

where M is the mass in kilograms, c is the speed of light in meters per second, and G is the gravitational constant:

When all is done everything cancels out except for meters. The r sub s term grows linearly with mass and is equivalent to the radius required for the mass to have an escape velocity of the speed of light. Working the math out for the mass of the earth we get:

or about 9 millimeters, which is pretty small. Here is the Schwarzschild metric:

This is polar coordinates, and

Represents a single angle movement along the surface of the unit sphere. The theta angle is equivalent to the angle from the north pole of the earth (latitude), and the phi angle is equivalent to longitude. When theta is small phi sweeps a small circle. When theta is 90 degrees phi will sweep the equator. Both circles are centered on the axis of rotation of the earth.

I'm only interested in the radius dimension and a single angle dimension, which is the d omega term. Imagine a slice of the earth where a plane intersects the earth, going through the north and south pole. We end up with a solid flat disc. The radius dimension represents the distance from the center, and the omega angle is the angle away from the north pole (top of the disc, or at 12 o'clock). When we multiply a radius times a small angle in radians we get a distance along a circle, and that's what the r times d omega term is, we've just squared it, like the other terms.

First of all note this expression is repeated twice:

The r term is the radius we're interested in. We only care about the region

because I don't believe there is anything inside the "black hole", nor can we even get down all the way to the Schwarzschild radius. Note that in the expression evaluates to zero if r is equal to r sub s. This is important. When r is really big compared to r sub s the expression evaluates to something closer and closer to 1. And as r gets down closer and closer to r sub s the expression gets closer and closer to 0.

Ok now I'm going to simplify things a little.

So now

The d tau term represents the "proper" time of something that moved along some path in spacetime. Suppose you carry a clock from one corner of a block to another. The proper time of the clock is how much time it measured over the course of the path. Since you were in the gravity of the earth and since you were moving, less time will have elapsed compared to an identical stationary clock many light years away in empty space. The key thing about general relativity is the natural progression of clocks changes depending on local conditions. On the surface of the earth at sea level a clock advances slower than a clock on top of a mountain. This is experimentally known to be true to very high accuracy, and measured results agree very well with general relativity's predictions.

Consider the case where we have a clock at radius r from the center of the mass and we're standing still. The dr term is zero since we're not moving up or down, and the d omega term is zero since we're not moving sideways. The metric simplifies to

The dt represents a small increment in time as experienced by a distant clock in empty space. Since k is between 0 and 1, the square root of k also will be, so the d tau term will end up being smaller than the dt term. This is an example of time dilation. Suppose we set r to 4/3 r sub s:

So that tells us at that distance out from the center of the "black hole" our progression of time is 50% the speed of the distant reference clock. Note that we wouldn't perceive the slowdown, since our own internal clock is slowed down also, as is all our mental processes, all the biology of our bodies, all the physical processes nearby... everything around us is slowed down the same amount so we don't notice anything. Only when we compare how much time has passed for the distant reference clock do we know less time has passed for us.

We've just worked out the time dilation for a clock remaining motionless in the vicinity of a mass. You can substitute in whatever values you want for r and see what the dilation is. When r is really really big, there's practically no time dilation at all. If we consider the case where r is equal to r sub s, k goes to zero! This suggests that no matter how much time passes on the distant reference clock, no time at all passes for us. It's like time has stopped completely! I want to avoid this problem for now, that's why I want to consider values of r that are arbitrarily close to r sub s but not exactly equal to it.

In this paper I'm only going to consider two possible speeds. We've already examined the first one, which is where we have a speed of 0 and aren't moving at all. The other speed I want to consider is if we're moving at the speed of light. In this case we are just considering the case of a photon of light. Since we have mass like any clock we're likely to use, we can't actually go along with this experiment. Real matter can never be accelerated all the way up to the speed of light. I want to avoid any in-between speeds, because I don't fully understand general relativity and all I've considered are the two extremes: not moving at all, and moving at the speed of light.

To move along a path at some speed in between would involve the evaluation of complex path integrals, and likely more intricate details of GR, such as tensor math. I'd be out of my depth. But if we consider the instantaneous direction of a beam of light we have all the GR equations we need already.

The d tau term in the case of a non-moving clock shows the effect of time dilation. When we consider the case of a photon of light, the d tau term must evaluate to zero. For a photon of light, moving at the highest speed possible, no proper time passes at all! Our equation gets simpler immediately:

Let's consider the case where we're motionless at some distance r from the center of the mass and we shine a flashlight sideways, making sure it isn't pointed at all down towards the center of mass, or up away from it. In this case the dr term is zero and our equation is:

Isn't that interesting! Remember we've normalized the speed of light to be 1, so this equation is telling us that the speed of light moving sideways is slowed down by exactly the same amount as the time dilation! In retrospect I suppose we could have guessed that. After all the thing that is special about the speed of light is that it is always the same, for all observers, in all situations. Countless experiments have been conducted trying to disprove this, but as far as I know none have succeeded.

If there were no time dilation the sqrt(k) term would be one, and so for every unit of time dt we'd move 1 unit of distance, since the speed of light is 1. But since there is some time dilation, light is only able to move a shorter distance.

So this equation has demonstrated that light slows down by the same amount as our clock, when it is moving sideways. We're considering just the instant when it happens to be moving exactly perpendicular to the line from our position to the center of the mass. Of course, an instant later the beam will no longer be moving exactly perpendicular and will likely be moving outward, at increasing distance from the center of mass. No worries, I only wanted to consider the instantaneous situation where the light is moving exactly sideways, and we're done with that case.

Now let's consider the case where a beam of light is moving only in the radial direction, and not at all sideways. In this case let's have the person off at a large distance in empty space shine a beam of light at us as we remain motionless at a fixed distance r from the center of mass. Let's consider the instant when the light passes our position. Since the light isn't moving sideways we can leave out the d omega term:

Now that is a very interesting result! We no longer have the square root of k term, we just have the k term. When the light was moving sideways, it was slowed down by the square root of k. When the light is moving straight up or down, it is slowed down by k! That means it is slowed down even more. When we were at 4/3 r sub s the time dilation ended up being 50%. But in the case of the light beam moving precisely in the radial direction, the slowdown is the square of that amount, so light is moving only at 25% speed! It turns out this rule applies regardless of the dilation factor. If our clock is moving at 1/10th the speed of the distant clock, light moving up or down will move at 1/100th of the normal speed "outside", and it will move 1/10th the outside speed if it is moving sideways.

That means that when light is moving radially (up or down), the time dilation is applied twice. I believe this is significant. It doesn't matter whether the light is moving up or down, the slowdown is the same. Consider the implications. What if the light continues downwards, towards the center of mass? The r value will be decreasing, so the k value will be getting closer and closer to 0. This means the time dilation factor is getting even smaller, and so the speed of light will slow down even more. The end result is inescapable. The closer and closer the beam of light gets to the r sub s radius, the slower and slower it progresses... compared to the speed of light "outside". At the same time the speed of light for sideways movement is slowing down, just by not as much. The slowdown sideways goes down as the square root of the slowdown in the vertical direction.

The net result is things really slowdown when we try to fall down the gravity well and get to the event horizon, but they don't slowdown anywhere near as much if we move in the sideways direction. Try to visualize the situation. What's happening is movement is getting more and more constrained in the up + down direction, compared to movement in the sideways direction. We're becoming flatland! Unless we aimed the light beam just right there will be more and more opportunity for it to get deflected sideways.

To me it's intuitively obvious that the light can never complete its trip down to the event horizon of the so-called black hole. It either veers sideways or comes to a complete standstill. Since nothing can move faster than light, nor can matter even move as fast as light, and since light can't even make it down to the event horizon, it's intuitively obvious to me that matter can't complete the journey either. The mass of the collapsing star or whatever can't compact enough to ever reach a point where the time dilation in the vertical direction gets to 0. The mass just keeps slowing down more and more. The event horizon never forms, and there is no black hole, and no continued contraction down into a point of singularity.

All this comes from the reasonable and intuitive interpretation of the Schwarzschild metric, which he introduced around 1915.

No one has ever actually touched a black hole, or been near one. A dense object taking forever to collapse is pretty much indestinguishable from a so-called black hole. We have never observed evidence of the existence of the event horizon anywhere. All of this is just theory! One thing I find ridiculous about wikipedia's discussion of the Schwarzschild metric is how they dismiss the interpretation that there is a sort of singularity at r = r sub s. They admit it (here) , because it's clear it would take an infinite amount of time for the matter to collapse down to the Schwarzschild radius and so form the event horizon. Their solution is to introduce different representations of the geometry! Because the Schwarzchild metric calls for time to stop at r sub s, it must be wrong... It's as if the belief in black holes is a religion, they must exist, and they must continue collapsing down in a finite time to a singularity of zero volume and infinite density.

It isn't at all clear to me why they want black holes to be real. I very much prefer if they aren't real, and that the matter falling into them isn't trapped forever. See my other three documents about black holes, btw. It is clear to me they're just inventing math willy-nilly to enable their beliefs. We have a lot of experimental and observational evidence that general relativity is accurate in the realms we have access to. The situation around massively dense collapsing objects is entirely theoretical. When we're dealing with competing theories in the absence of empirical evidence, it becomes more a matter of personal bias which one we prefer to adopt.

If massively dense objects cannot collapse into black holes, I'd think a lot of things would get simpler. I've already explored the idea of their role as matter recyclers. The nonexistence of black holes would fit quite nicely with an eternal, ever changing universe that didn't begin with a big bang, and won't end in a dark heat death, and isn't expanding or contracting. That sort of universe is much more believable and appealing to me.

Anyway just a bit more math. I wanted to work out the time it takes for light to traverse a radial path from r sub a to r sub b. We start with the metric with the dr case:

We want to solve this as a differential equation. We get all the r terms on one side and all the t terms on the other:

Now we want to integrate both sides:

WolframAlpha is a good math resource. I entered "integrate dr/(1-k/r)", using k since I couldn't figure out how to enter r sub s. It came back with:

We're left with an equation for t in terms of r and r sub s. The t value represents the time that would be measured by a reference clock in empty space. Converting this to a definite integral from r sub a to r sub b and making sure the sign is positive:

Again note the speed of light c is 1. The r sub a minus r sub b term on the right is just the ordinary time it takes light to cross the distance in flat space. The term on the left was unexpected. The log function uses base e (it's the natural log). It's around 2.783... The log part is telling us that every time our distance from the Schwarzschild radius of the mass changes by a factor of e, up or down, we want to add a time delay period of how long it takes light to traverse the Schwarzschild radius in flat space, which is a constant.

This equation is troublesome. The tiniest distance realistically considered relevant to the physics of our universe is on the order of the Planck length, which is

Going from one meter down to that distance and taking the log base e yields 80.11. Suppose we're dealing with a Schwarzschild radius for the mass of the sun, which is

or just under 3 kilometers. Light takes 9.93 microseconds to go that far in flat space. Multiply that by 80.11 and we get

That's how much real time passes for light to drop from 1 meter above the sun's Schwarzschild radius to one Planck length above it. What would the time dilation be down there then? It's the square root of the Planck length divided by the sun's Schwarzschild radius, which is 7.3766*10^-20. That means for one second to elapse "inside", 429.6 billion years must pass by "outside".

That equation doesn't fit with my earlier discussion of time dilation. Something seems off... But actually it's correct. Light does travel quite fast, after all, and it does only need to travel a shorter and shorter distance so the effect of the time dilation is cancelled out. When we get into the realm of the really small, though, we can't say what's going on. There's the problem of the Uncertainty Principle to deal with. It may be meaningless to presume that the light ray can keep pointing exactly radial to the mass as we get closer and closer to the Schwarzchild radius. Once it veers sideways even the tiniest bit it's game over.

Rearranging terms in our dr / dt equation makes it more intuitive:

Consider the case where r is just slightly larger than r sub s. We have r which would be a fairly large value in comparison, and we end up with:

As epsilon goes to zero the term before dr goes to infinity. This is again telling us that every tiny change in dr takes an unlimited amount of time outside. It turns out we don't really need wolfram to integrate this:

The integral of dx/x is log_e(x)+c, which ought to be common knowledge but is readily discoverable on the internet. So we're left with:

Which is what wolfram gave us, except they combined the "-r sub s" term in with the constant. We can't actually evaluate this at r = r sub s since log of 0 is negative infinity. I'm comfortable with this result.

The nature of how the Schwarzschild metric implies 3D space flattening down to 2D as we get down to the Schwarzchild radius is very interesting. Space is occupied by matter, and the matter interacts with other nearby matter. In our 3D realm we're used to there being no direction bias in the interaction, from our point of view the interactions are the same whether we go sideways or up and down. But squashed down into the 2D realm near a massive object things are different. There is an enormous bias for interaction "sideways" as compared to up and down. At a time dilation of .001, 1000x as much "sideways" activity is going on compared to up and down activity. I think that is deeply significant. The implication is that maybe our familiarity with 3D is really a case where we're living in a 4D space, only there is a huge time "bias" for matter-matter interactions in the 3 dimensions of space we're familiar with.

Having messed with the Schwarzchild metric and its implications, it seems to me there is no need to even consider space as actually being warped at all. Rather we can have perfectly flat euclidian space, and all that changes is how readily nearby points can interact. Movement is a form of interaction. The metric implies the relative speed of movement can vary based on direction. It seems to me a lot simpler if we can think of the universe as flat euclidian space, but we can add more flat dimensions whenever convenient, as opposed to thinking in terms of curved/warped space.