Ultimate space simulation software

PlutonianEmpire
Pioneer
Posts: 441
Joined: 02 Nov 2016
Location: MinneSNOWta
Contact:

Science and Astronomy Questions

I read this article today (http://www.iflscience.com/plants-and-an ... explosion/), and I began wondering, would it be possible in any way, shape, or form, to take advantage of tumor biology and use it to seed alien terras, or during the seeding phase of terraforming desert planets such as Mars?
Specs: Dell Inspiron 5547 (Laptop); 8 gigabytes of RAM; Processor: Intel® Core™ i5-4210U CPU @ 1.70GHz (4 CPUs), ~2.4GHz; Operating System: Windows 7 Home Premium 64-bit; Graphics: Intel® HD Graphics 4400 (That's all there is )

Terran
Space Tourist
Posts: 39
Joined: 24 Nov 2016

Science and Astronomy Questions

Watsisname wrote:
Source of the post Example where this doesn't work in general relativity:
A classic test of general relativity is the gravitational lensing of light.  The revelation is actually not that light rays bend, but that they bend twice as much!  The Newtonian calculation predicts half of the correct lensing angle.  Using general relativity and treating the effect as arising from time dilation also predicts half of the correct angle.  Einstein himself initially fell for this trap when predicting the bending of light around the Sun (and was lucky to fix this before the observation was made!)  The correct calculation requires the complete Schwarzschild metric where both time and space are distorted, and demonstrates that the spatial geometry is not Euclidean.

Woah! Ok... That is very interesting... I tried to find Einstein's original equation to compare to the Schwarzchild metric but unfortunately the internet isn't very helpful. Do you by any chance know the equations for that model that predicted 1/2 what it was observed to be? I want to check for something...
Watsisname wrote:
Source of the post So I don't think it's good to be so married to a model as to avoid the potential for better ones.  A better model describes a wider range of phenomena more accurately.

I don't doubt they will get the right answers, I just feel that they might be interpreting them incorrectly, like that story with the blind men and the elephant, but in reverse. Like creating the math equations for inertia but saying thats because mass gets tired, you get the right predictions but you answer with incorrect conclusions. Something about quantum mechanics really frustrates me, like they are looking at the problems backwards. It feels like they are answering their questions with the same question. Quantum scientists say that a particle is a wave of probability, and that it can be at any place at any time (probabilistically), to me however, a particle can only be at one place at ONE time, but can be in many places at many times, that what if we are not seeing waves of probability, what if we are seeing cross-sections of a field or plane in a higher dimensional timeline. Thats what I have been looking for... I am starting to suspect that we have been neglecting the time part of our universe, arguably the weirdest part. I want to see where this perspective leads me...
To reality, through nothing, from un-everything.

Watsisname
Science Officer
Posts: 1532
Joined: 06 Sep 2016
Location: Bellingham, WA

Science and Astronomy Questions

Terran wrote:
Source of the post I tried to find Einstein's original equation to compare to the Schwarzchild metric but unfortunately the internet isn't very helpful. Do you by any chance know the equations for that model that predicted 1/2 what it was observed to be?

Yes, this page outlines the basic idea behind how Einstein did it.  In particular, Einstein's original (1911) prediction of half the correct deflection angle arises from the use of equation 1, which is a flat (Minkowskian) space-time metric except with a distortion to the time component due to the presence of the mass.

The corrected prediction uses the Schwarzschild metric, where both the time and spatial coordinates are distorted.  Einstein refers to this correction in his 1915 paper describing the anomalous perihelion precession of Mercury's orbit.

Terran wrote:
Source of the post I don't doubt they will get the right answers, I just feel that they might be interpreting them incorrectly, like that story with the blind men and the elephant, but in reverse. Like creating the math equations for inertia but saying thats because mass gets tired, you get the right predictions but you answer with incorrect conclusions.

Ah, interpretations.   If "mass gets tired" can be expressed in the form of a mathematical model which correctly makes all the predictions of classical mechanics, then it is a valid interpretation.  Otherwise, it's wrong.

Another example:  We could re-interpret general relativity as an effect on measuring rods and clocks without invoking space-time curvature at all, and this would be a different interpretation that gives the same experimental results.  We are free to use whichever interpretation we like, provided that it works in the sense of making experimentally testable and potentially falsifiable predictions.

Quantum mechanics has a lot of interpretations:

Many people are fond of choosing, studying, and debating these interpretations.  But those who are interested in doing calculations are not highly concerned about them.  When I solve a problem I do not care about the nature of wave function collapse, or whether reality is deterministic or not.  It doesn't matter.  What I care about is that the Schrodinger equation can be applied to a quantum mechanical system and produce the correct predictions.

The Schrodinger equation is an equation that acts on a wave function, and the wave function gives the probability of finding the particle in some region of space.  If you set up many boxes, each with one particle inside, and then simultaneously measure the locations of every particle in every box, then this correctly predicts the distribution of results you will get.

If we focus instead on a single measurement of a particle in a single box, where will it be?  There is no known theory of physics that can give the answer!  Only the probabilistic answer.  Once the measurement is made, the location is set, and a (sufficiently quickly made) repeated measurement will produce the same result.

This really makes it seem that the particle "really was there" at that location all along, before the measurement was made.  Afterall, the repeated measurements show the same thing, right?  And it is further tempting to believe because the notion that the location of a particle is probabilistic before we measure it is very contrary to how we are used to thinking about how the world works.  Classically, nothing behaves like that.  But there are a few problems with making that conclusion:

1)  At first glance it seems to be unfalsifiable.  (How do you prove where the particle is before measuring where it is?)

2)  Reality is not obligated to behave the way we think it should.  Our intuitions are ill-suited for understanding how things behave quantum mechanically, or at speeds close to the speed of light.

3)  Repeated measurements (made sufficiently quickly) coming up with the same result do not imply anything about where the particle was before the first measurement.  The Schrodinger equation says that by making the first measurement, the wave function of the particle collapses to a narrow spike around that measured value, and then proceeds to spread out again.

4)  It gets better:  Bell's Theorem showed that no successful theory of quantum mechanics can be made where the measured location of the particle was determined by some function of variables that we don't have access to.  If the particle really was there all along, or if our current probabilistic predictions are just a consequence of "missing something more fundamental" (the hidden variables), then those hidden variables must operate nonlocally (faster than light).

Following the impact of Bell's Theorem, the philosophical can of worms opened up to unleash a multitude of ways of interpreting why quantum mechanics works this way.  But again, interpretation is secondary to a mathematical framework that allows us to do calculations to predict the results of experiment.

Terran
Space Tourist
Posts: 39
Joined: 24 Nov 2016

Science and Astronomy Questions

Watsisname, huh... I see how the spatial distortion is required... But it still isn't sitting with me the fact that you can observe said spatial distortion. A spatial distortion would be a topological distortion and would only be observable by an outside observer. Perhaps if the intrinsic nature of the curvature created a compression and expansion but not a deflection... To make my life easier I reduced major equations to simple geometric ones. "Theta" = sin^-1(v/c), Time dilation = Normal Time * cos(theta)^-1 (or no inverse depending if you are an outside observer or not), Mass Dilation = Mass * cos(theta)^-1 (mass can be swapped out with time for some reason, works with other equations :/). velocity = c * sin(theta), length dilation = length * sin(theta), And so on... I just noticed how length and velocity share a part of the equation and was wondering... Could velocity dilation account for it if you equalized the speed of light and length? Or am I connecting dots in the wrong places? And with quantum mechanics, it looks like what you would expect with multiple time components (yes, I could just be seeing what I want to see, but then again, if everyone saw the same thing no one would question if it was wrong or right, so I am going to try to prove them wrong, if they are wrong then I am right, if I am wrong they are right, I guess we'll see which is it in the future XD) I feel like I am keeping this discussion alive for too long, perhaps I should create a dedicated post? Or am I getting ahead of myself?
To reality, through nothing, from un-everything.

Watsisname
Science Officer
Posts: 1532
Joined: 06 Sep 2016
Location: Bellingham, WA

Science and Astronomy Questions

Terran wrote:
Source of the post But it still isn't sitting with me the fact that you can observe said spatial distortion. A spatial distortion would be a topological distortion and would only be observable by an outside observer.

All distortion to space and time described in general relativity is intrinsic curvature (a property of the surface or manifold itself), and can be measured by those within it.  If the distortion was instead extrinsic (a property of how the surface or manifold is embedded in a higher dimension), then it would indeed be a problem.  Not only would that curvature be unmeasurable, it would also not explain gravitation!  Gravitation causes a deviation of geodesics in space-time, and that can only occur with intrinsic space-time curvature.

A simple example of extrinsic curvature:
Curl up a flat sheet of paper to produce a cylinder.  The geometry everywhere on the cylinder's surface is spatially flat.  Straight parallel lines remain parallel, and all triangles have interior angles which sum to 180°.  By rolling up the paper to a cylinder, we only change how the surface was embedded in 3 dimensions, not the geometry of the paper itself.

A simple example of intrinsic curvature:
The surface of a sphere.  There is no way to unroll a sphere to produce a flat sheet (thus the distortions present in any and all 2D maps of the Earth.)  The surface itself is curved.  And we can measure this curvature without seeing it from outside, too.  Ants (sufficiently smart ones) on a sphere could determine that they live on a sphere without ever leaving the surface.  Indeed, without even needing to travel all the way around it.

All they need to do is construct triangles of sufficient size and measure the sum of interior angles with sufficient precision -- a difference from 180° indicates the surface is curved.  Or they could trace parallel lines that are everywhere straight along the surface, and observe that they don't remain parallel.  Just like if a bunch of people start marching due north from different points along the equator, they discover the distance between them decreases, even though they all started out moving exactly parallel and they all walk straight.  If they could march all the way to the North pole, they discover they all meet up.  Straight parallel lines converge on a positively curved surface.

Perhaps if the intrinsic nature of the curvature created a compression and expansion but not a deflection... To make my life easier I reduced major equations to simple geometric ones.

It's probably good to do a sanity check.  To make sure your formulas work correctly, use them (show me) to calculate the deflection angle of a light ray just grazing the Sun's surface.  Then calculate the deflection angle for when the light ray exactly skims the radial coordinate r=3GM/c^2.

Also, a suggestion:  In relativity we do not like to use "mass inflation".  You can, but it throws away one of the rare things which turns out to be invariant.  In relativity, quantities which can be made invariant are like diamonds.  Do not throw away diamonds!   Say instead that mass is invariant and it is rather the energy and momenta that change, according to m^2 = E^2 - p^2 (in units where c=1).

Terran wrote:
Source of the post And with quantum mechanics, it looks like what you would expect with multiple time components

Does it?  Calculate the wave function for an electron in a 1-dimensional conductor of length L, using a theory with multiple time components.  Normalize it and plot your result.  What is the probability of a measurement finding the electron between L/3 and 2L/3?

Terran wrote:
Source of the post yes, I could just be seeing what I want to see, but then again, if everyone saw the same thing no one would question if it was wrong or right, so I am going to try to prove them wrong, if they are wrong then I am right, if I am wrong they are right, I guess we'll see which is it in the future XD

Theories in physics don't establish themselves because nobody questions them.  They arise because a lot of people did question them by testing them -- making sure that their predictions were consistent with observations.  The predictions of general relativity are still tested to this day.

I have had the pleasure of verifying for myself numerous results of modern physics, including some of the more famous and technically difficult tests like measuring the quantization of charge by levitating oil droplets.   Not because I thought it was wrong, but because the experience of carrying out those experiments, understanding the theory behind them, and performing the data analysis, are useful for a physicist's development.

Questioning something like the interpretation of relativity or quantum mechanics, or trying to replace it with your own interpretation, is all fine and well.  But it's wise to make sure you correctly understand them and can go through the rigor of applying them before you do.

As for continuing the discussion here or splitting to a new thread, I think it's fine either way.  If you'd like to split I'll be happy to do it for you.

A-L-E-X
Star Engineer
Posts: 1301
Joined: 06 Mar 2017

Science and Astronomy Questions

Watsisname wrote:
A-L-E-X wrote:
Source of the post Yes, I love math too!  Which is why I was so puzzled by it being called "gross."  I mean raw sewage is gross (or sewage of any kind)- but math?!  No!

A-L-E-X wrote:
Source of the post How long will it be before we understand the details of the collapse and rotating black holes do you think? Do we need a workable theory of quantum gravity first (perhaps that will also settle the question of black hole cosmology too.)

I think theorists are very slowly getting there, but it's an exceptionally difficult mathematics problem made more difficult by the weird property of rotating black holes which causes their interior geometry to depend on what happens in the future as well as in the past.

You can get a sense for why this is the case just by looking at the Kerr metric as is -- things that fall into it never hit the singularity, but rather cross relativistically with more matter falling in -- including itself.  You end up with three singularites -- the initial one at the center, plus a "past" singularity which fell in after the hole formed but before you, and a "future" singularity which is coming in after you.  Thus the interior of the black hole depends on what happens in the future!

That property still becomes relevant when studying what should actually happen in a real rotating black hole, not just the Kerr vacuum solution.

Rotating black holes are weird!

A further difficulty is, as you said, what actually happens to the matter distribution as it approaches singularity conditions, or when the space-time curvature becomes very large.  A huge hurdle here is that different researchers have all sorts of ideas on how to approach the problem (how to bridge toward a theory of quantum gravitation), but they are enormously difficult to test experimentally.

So for the time being, when asked what really happens in an astrophysical, rotating black hole?

A-L-E-X wrote:
Source of the post I saw you were using transcendental math up there, what's your opinion of Stephen Hawking using Imaginary Time to eliminate the Big Bang singularity problem?  This would also be workable for black holes.

I'm not too knowledgeable on this, and I'm definitely not as smart as Hawking, so I wouldn't know what to conclude about it.  Sometimes very complex problems may have elegant solutions, but I have a difficult time knowing how that approach would be helpful for this one.

Wat, that looks like something Einstein might have scribbled   Now I see why what you and Doc said and what I believe is the most elegant solution to all this, a universe in which past, present and future coexists.  So you actually only have one singularity, but because of the way we perceive time, it seems like three.  The three parts would be linked together through time but would really be one.

A-L-E-X
Star Engineer
Posts: 1301
Joined: 06 Mar 2017

Science and Astronomy Questions

Watsisname wrote:
If the gravitons can't escape, then they don't generate a black hole, and so they escape.  If they escape, then they generate a black hole, and so they can't escape.

Paradox?  It sure seems that way!  Asking how gravitons escape a black hole is a bit like asking how gravity escapes itself -- a quick road into an endless loop.  How do we break the loop?  Abandon the notion that gravitons must escape from anything.  Say instead that they are what make the black hole!  Gravitons, or the fields they generate, are what give the marching orders to other masses in the vicinity.

Thinking in the context of general relativity (without gravitons) may help.  How does an object near a black hole "know" that there is mass hidden away inside the horizon?  It doesn't!  It only knows the simplest rule: "always move straight".  It moves on a locally straight line according to the space-time geometry of its immediate vicinity, and that geometry is warped.  Why is the geometry warped?

The field equations of general relativity say that the singularity of the black hole distorts the geometry immediately around it, just like poking at a rubber sheet.  But that distortion can't be localized.  The singularity can't just cause a tiny dimple that suddenly cuts into flat geometry.  Just as how your poking into a rubber sheet distorts the whole sheet, the distortion around the singularity must further distort the geometry around that, reaching out until the whole geometry is warped in a smooth manner given by the Schwarzschild metric.  This is how the singularity extends its influence beyond the event horizon.  It doesn't violate the one-way rule of the horizon, it makes the one-way rule of the horizon, by generating the geometry.

A quick detour:
We now know that there are gravitational waves, and that those also move at the speed of light.  But gravitational waves definitely do not escape from inside a horizon.  This is because gravitational waves are self-propagating changes in the space-time, rather than the static geometry attributed to the source mass.  So they must follow the rules of the geometry that already exists around their source.  They can't climb out of a black hole if they were generated inside one -- they are trapped.

In fact this must happen anytime two black holes merge together.  Some of the gravitational waves will spiral around inside the resulting black hole and make a new, particularly nasty type of singularity -- one which not only has infinite curvature, but infinitely rapidly oscillating changes in curvature, like the function y=sin(1/x)/x.  Those who read Kip Thorne's Science of Interstellar book may have seen it as the "BKL singularity".

Kip Thorne is awesome He also wrote about Traversible Wormholes and was a consultant for Interstellar.
Wat, what do you think of the theories that say that gravity leaks in from outside our universe (it is used to explain why it's so weak compared to the other forces.)

Watsisname
Science Officer
Posts: 1532
Joined: 06 Sep 2016
Location: Bellingham, WA

Science and Astronomy Questions

A-L-E-X wrote:
Source of the post Wat, that looks like something Einstein might have scribbled

It is not the longest calculation I've ever done, but it is definitely one of the most "oh god, the symbols..." reaction inducing ones.    The calculation is basically just verifying that the expectation value of the energy of an electron whose wave function covers two states is just the average of the energies of those two states.  Not particularly surprising or interesting, aside from the realization that a wave function for an electron being in two states is a valid solution in quantum mechanics.  (In fact this is what happens during electron transitions -- the probability of being in one state or the other shifts smoothly over some transition time.)

A-L-E-X wrote:
Source of the post Now I see why what you and Doc said and what I believe is the most elegant solution to all this, a universe in which past, present and future coexists.  So you actually only have one singularity, but because of the way we perceive time, it seems like three.  The three parts would be linked together through time but would really be one.

I don't think this implies that past, present, and future coexist.  It's analogous to saying that if someone dropped a coin in a wishing well yesterday, and you toss one in today, and another person will toss a third in tomorrow, then if those coins overlap it means yesterday today and tomorrow all exist simultaneously.  I don't think that logically follows.  They are different coins, and each being thrown into the well is a different event.  The future event didn't influence the past or present events, it influenced the world lines that followed from them, because their paths are stuck in the well.

A-L-E-X wrote:
Source of the post
Wat, what do you think of the theories that say that gravity leaks in from outside our universe (it is used to explain why it's so weak compared to the other forces.)

It sounds good in a handwaving sense, not in a rigorous or predictively useful sense... at least as far as I'm aware.  I don't spend much time thinking about theories if they are not testable.

A-L-E-X
Star Engineer
Posts: 1301
Joined: 06 Mar 2017

Science and Astronomy Questions

Watsisname wrote:
A-L-E-X wrote:
Source of the post Wat, that looks like something Einstein might have scribbled

It is not the longest calculation I've ever done, but it is definitely one of the most "oh god, the symbols..." reaction inducing ones.    The calculation is basically just verifying that the expectation value of the energy of an electron whose wave function covers two states is just the average of the energies of those two states.  Not particularly surprising or interesting, aside from the realization that a wave function for an electron being in two states is a valid solution in quantum mechanics.  (In fact this is what happens during electron transitions -- the probability of being in one state or the other shifts smoothly over some transition time.)

A-L-E-X wrote:
Source of the post Now I see why what you and Doc said and what I believe is the most elegant solution to all this, a universe in which past, present and future coexists.  So you actually only have one singularity, but because of the way we perceive time, it seems like three.  The three parts would be linked together through time but would really be one.

I don't think this implies that past, present, and future coexist.  It's analogous to saying that if someone dropped a coin in a wishing well yesterday, and you toss one in today, and another person will toss a third in tomorrow, then if those coins overlap it means yesterday today and tomorrow all exist simultaneously.  I don't think that logically follows.  They are different coins, and each being thrown into the well is a different event.  The future event didn't influence the past or present events, it influenced the world lines that followed from them, because their paths are stuck in the well.

A-L-E-X wrote:
Source of the post
Wat, what do you think of the theories that say that gravity leaks in from outside our universe (it is used to explain why it's so weak compared to the other forces.)

It sounds good in a handwaving sense, not in a rigorous or predictively useful sense... at least as far as I'm aware.  I don't spend much time thinking about theories if they are not testable.

Right, but this makes me wonder if the past present and future are predetermined (if time is a landscape), a la superdeterminism, in other words we get rid of causality, if everything is predetermined then "cause" doesn't matter, the outcome is fixed.

Watsisname
Science Officer
Posts: 1532
Joined: 06 Sep 2016
Location: Bellingham, WA

Science and Astronomy Questions

They could be, but I'm not aware of anything that convincingly pushes favor towards or against it besides personal philosophies and wish-casting.  Like (or even highly related to) the various interpretations of quantum mechanics.

A-L-E-X
Star Engineer
Posts: 1301
Joined: 06 Mar 2017

Science and Astronomy Questions

Watsisname wrote:
They could be, but I'm not aware of anything that convincingly pushes favor towards or against it besides personal philosophies and wish-casting.  Like (or even highly related to) the various interpretations of quantum mechanics.

Yes, there's so many of them it's bewildering.  I know Doc favors pilot wave theory

midtskogen
World Builder
Posts: 755
Joined: 11 Dec 2016
Location: Oslo, Norway
Contact:

Science and Astronomy Questions

In a multiverse one could perhaps argue that the past, present and future are fixed and predetermined, but our path through the multiple universes of the predetermined multiverse is not, perhaps fundamentally indeterminable.
NIL DIFFICILE VOLENTI

A-L-E-X
Star Engineer
Posts: 1301
Joined: 06 Mar 2017

Science and Astronomy Questions

midtskogen wrote:
In a multiverse one could perhaps argue that the past, present and future are fixed and predetermined, but our path through the multiple universes of the predetermined multiverse is not, perhaps fundamentally indeterminable.

It does make me wonder, glad to see you thinking along the same lines When I saw the shape of the hopf fibration which has been outputted by supercomputer simulations of two supermassive black holes colliding, it made me think of a recurring big bounce with all pasts, presents and futures coexisting, with timelines emerging at each bounce and separating via inflation and converging again just before the next one (deflation).  In other words, our decisions do not create the timelines, they were always there to begin with (since the beginning of that cycle anyway.)

Terran
Space Tourist
Posts: 39
Joined: 24 Nov 2016

Science and Astronomy Questions

Watsisname wrote:
Source of the post Then calculate the deflection angle for when the light ray exactly skims the radial coordinate r=3GM/c^2.

I'm gonna try to do the sun one later. I have never gotten this far so, I'm gonna try! So far I have gotten a time dilation of t*cos(sin^-1(root(2*((3*G*M)/(c^2))*gravity)/c))^(1 or -1), and a spatial length contraction of length*(root(2*((3*G*M)/c^2)*gravity)/c)... My hand held white board may not be enough! I shall use my bigger white board! k... I already know I am wrong, not my simplifications I did but rather trying to implement it with my limited knowledge... I got something like cos^(-1)((6.72852×10^34 3^(-v/2) t)/M), where v is the perspective (1 or -1), t is time, M is mass... Heres most of my math:
► Show Spoiler
Maybe if I knew more calculus, and took more time on the problem, I might get something close to the actual solution. Don't have time and the patience to figure out how this math changes as the light changes position, I do realize thats one issue... But doing the math wrong at least gives me some insights into the problem, and its complexity. The geometric simplifications still work though: Because root(1-x^2) is always equal to cos(sin^-1(x)), so I just switched things around to make my life easier.
Watsisname wrote:
Source of the post Does it?  Calculate the wave function for an electron in a 1-dimensional conductor of length L, using a theory with multiple time components.  Normalize it and plot your result.  What is the probability of a measurement finding the electron between L/3 and 2L/3?

Ah! You got me! I have made little effort into the math that would come from such a conclusion, so far what I have said was rather an observational "connection". Let me elaborate on my observation:

► Show Spoiler

So in conclusion: No, I am not qualified to answer these kinds of questions. Probably not even ready to ask them yet. But I will try, and I see I have a way to go. XD
To reality, through nothing, from un-everything.

Watsisname
Science Officer
Posts: 1532
Joined: 06 Sep 2016
Location: Bellingham, WA

Science and Astronomy Questions

Terran wrote:
Source of the post Heres most of my math:

Problem:  You cannot use g=GM/rfor this case.  The Newtonian expression for gravitational acceleration fails badly near the Schwarzschild radius.  (I would say it's usefully accurate only as close as a few hundred M).  So your calculations are definitely going to be incorrect for r=3M.

Terran wrote:
Source of the post Maybe if I knew more calculus, and took more time on the problem, I might get something close to the actual solution. Don't have time and the patience to figure out how this math changes as the light changes position, I do realize thats one issue... But doing the math wrong at least gives me some insights into the problem, and its complexity.

Yes, welcome to general relativity.   This is not a trivial problem to solve.

I know of three ways to solve it correctly:

1)  Begin with the metric.  Then write the equation of geodesic deviation for a photon.  Calculate the connection coefficients, and then integrate the resulting equation.  (This is probably the most tedious method).

2)  Derive the equations of motion for a mass in a gravitational field from the effective potential, using the energy and angular momentum as the constants of motion.  These are differential equations in the form (dr/dτ, dφ/dτ and dτ/dt).  Remove the proper time τ from the equations by using the chain rule.  Then take the limit that the mass m goes to zero (to describe a light pulse), to get the equations of motion of a photon in terms of the Schwarzschild coordinate time t.   Then to find the trajectory, integrate.

3)  Take the previous equations of motion for the photon, and re-express in terms of dφ/dr to get the incremental change in angle dφ for each incremental change in radius dr.  Finally, again, integrate.

Terran wrote:
Source of the post Ah! You got me! I have made little effort into the math that would come from such a conclusion, so far what I have said was rather an observational "connection". Let me elaborate on my observation:

Well, that is why I forced the question.   How are you making an observational connection if does not come from performing an observation?  How can one be justified in saying "X looks like what would be expected from Y", if the expectation from Y did not arise from experiment or by math?  What you're doing is presenting wishful thinking and handwaving.  An alternative model of quantum mechanics can't be taken as correct if it is not quantitatively rigorous and generates the right answers.

The right answer by the way is 0.609, if the electron is in the ground state.  (Notice this is much greater than the 1/3 expected if the electron had equal probability of being anywhere along the length L.)  If the electron were in the 1st excited state then the answer becomes 0.19955.  (Less than the 1/3 that would be obtained if the particle had equal probability of being everywhere along the length).  These results come from obtaining the wave function through the Schrodinger equation, and they agree with experiment.  (In fact, what may seem like an over-simplified 1-dimensional case is actually testable by using carbon nanotubes).

Who is online

Users browsing this forum: Google [Bot] and 1 guest