Schwarzschild Metric Geodesic vs. Inverse Square Law of Gravity (How CLOSE They Are Visually)

I posted this video on youtube on January 28, 2020 showing how the inverse square law really is an extremely good approximation of a geodesic on a Schwarzschild Metric:

https://www.youtube.com/watch?v=KUKcp_2rd-Y


The graphs I use in the video are: (dy/dx)^2 = (2*3.985*10^14)/y for the inverse square law and (dy/dx)^2= (1-0.0089/y)^2*(299,000,000^2 – 299,000,000^2*(1-0.0089/y)) for the Schwarzschild metric geodesic


For the zoom of the graph, I first use the ranges:

X range is -6K to 6K

Y range is 0 to 1.4 x 10^7

Because the distance, Y is measured from the center of the earth instead of the earth’s surface, and earth’s radius is pretty big, 6.371 million meters.

I also zoom WAY in close later in the video, 

X range is -5 to 8

Y range is 0.0079 to 0.0099

To show the graphs aren't EXACTLY the same.

My general strategy was to take the inverse square law d/dx of dy/dx = k/y^2 where x is time and y is distance from the center of gravity, and turn it into a graph.  Then I took the relevant part of the Schwarszchild metric, ds^2=(1-rs/r)c^2*dt^2 – dr^2/(1-rs/r), and tried to figure out how to do a geodesic on it.  

Going to ideas from Sam Lilley’s book, it gave me a kind of formula for what a geodesic on a metric is.  If you draw a graph with a r axis and a t axis, you can think of dr and dt as infinitesimal distances on this graph.  On a graph where r and t would be flat spacetime, dr and dt would be equal everywhere.  But warped space has varying dr and dt at different points.  

We can think of ds as the hypotenuse of a triangle completing the distance moved dt in the t direction and then moved dr in the r direction.  This isn’t entirely accurate with the Schwarzschild metric because I think ds is actually smaller than either dt or dx here, but it’ll at least give us a rough idea of what’s going on.  If we take this picture here, and imagine that ds is ALWAYS proportional to dt, no matter how long or short dt is, we’ll get a geodesic – a line that doesn’t bend to the left or right as you move across it.  So we just need ds to be equal to some constant times dt and plug that into our metric to get a geodesic on that metric.  

So the next part of the strategy was to set ds equal to a constant B times (1-rs/r)dt.  Then plug it into the metric.  

After messy algebra, the equations for the graphs used in the video popped out.

No One Respects a Man Who.....

No one respects a man who has neither the patience to wait nor the guts to act.

Why is not being an atheist my fault?

I work hard to get more educated in math and science and fail time and time again, while I see other people going to college at age 14 on full scholarships, seemingly with ease. My brain is just not as large as theirs, period. Nature favors them. So if being smarter than a normal person is a requirement for being an atheist why is it my fault that I still believe in God? Will not believing in God suddenly increase my mathematical ability? I doubt it. And would the atheist community respect me more if I expanded my mind by studying mathematics more without dropping my faith, or if I just said there was no God and gave up studying math altogether? I'll admit that the stereotypical person that follows religion has a lazy attitude towards increasing their scientific knowledge, but I haven't been one of them. For the past three years I've struggled to improve, and I think I have, but I've got a long way to go, and I do not think I will stop any time soon because I've found a lot of beauty in the truths of mathematics, just as atheists have.

Justification

Scientists should have this problem more often than they actually have it.  A person with little scientific background due to lack of opportunity says, "Hey, how does gravity work?"  The scientist says, "The curvature of space-time causes gravity to work."  The person replies, "Wow, that's neat.  Can you give me more detail?"

The scientist gives more detail, in layman's terms.  The person seems skeptical and wants more justification.  The scientist gives a full justification with math that's well beyond the layperson's capabilities.  The person is faced with the decision on whether or not to trust the scientist's work without actually being able to understand it.  Most will.  But what if instead the person said, "I'm sorry, I just can't see it.  Can you make this justification any easier?"

What does the scientist say then? "Just trust the experts"? Trusting authority figures isn't scientific proof of anything.  But clearly the layperson doesn't have the luxury of going to school to understand the curvature of space-time due to other preferences and responsibilities.  The layperson could still say, "I'm sorry, that all looks like jibber jabber to me.  As far as I can tell, space-time curvature and gravity may have no relation whatsoever because that justification means nothing to my limited mind."

Is this person entitled to such an opinion?  I would say so, even though it's most likely a foolish opinion to have.  The layperson's only other option is to research the topic of space-time curvature in their own spare time, if such time is even available.

It's just not good for scientists to get into the habit of saying, "Trust us, we're the experts.  We have data to back it up too, even though it may look like gibberish to you."  Politicians do the same thing. 

We're Not Trying To Set a Survival Record Here.....

Why do we evolve?  Do we evolve merely to have a better chance for our species to survive as long as it can?  Yeah, probably, but I think there's more to it than that.  Much more.  We evolve to see and experience beauty.  Our desire to experience and understand beauty is more than a survival mechanism.  Much more.  Evolving is about more than survival.  It's about an attempt to become closer and closer to our imaginative ideal.  We want to be like God.  And God gives us a heck of an opportunity to explore his creation so that we can strive for that ideal.

Why Is The Goal Important?

A lot of times I see people working frantically to promote or denounce ideas like they're the most important thing in the world.

We need faster computers!  We need longer lifespans!  We need less pain!  We need more people to know the truth!  We need less people to disagree with us!  We need less manual labor!  We need our species to continue indefinitely!

All of these ideas are well and good.  But WHY are they good?  Ecclesiastes teaches us that despite all of our hard work, the universe is going to carry on regardless.  Did the universe NEED life to survive during its first however many billion years?  Nah.  So why does it need life now?

The list of needs I described above are actually really nothing more than preferences.  I know I certainly want a longer life for me and my friends where we aren't suffering crippling diseases.  And I would like to think that my life will have some small positive influence on humanity for generations to come.  And most sensible people would feel the same way.  But the cold hard fact is the universe doesn't need it.  It doesn't need us to have utilitarian laws that will be better for the prolonging of our species.  It doesn't need intelligent life or protozoa even.  I imagine it's fine just being floating rocks and gases.  Or even just a plain old big crunch singularity.

Once we recognize that all of these things we are being pressured to achieve are really just preferences more than necessities, the urgency behind them is no longer quite so urgent.  They're not really urgent at all on a universal scale.  Only a preference scale.  This revelation has some liberating properties but it is also accompanied by a lot of emptiness.  It's easy to feel worthless when you realize you aren't needed.

Some people accept that nothing we ever do has any inherent worth, but I think most do not.  I think most people either turn to religion or try to plan out how our species will continue to evolve into more complex lifeforms that will survive forever, perhaps even preventing a big crunch.  The latter seems sillier than the former to me, but that's just a personal preference.

I've often questioned the goal of evolution.  Why the drive to survive?  Why the drive to procreate?  Why adapt to anything when the universe doesn't need you to do it?  I believe these important questions are usually left unanswered.  They are certainly outside the realm of science because all science can do is draw conclusions from fossil records.  But why get so concerned about finding the truth if all it does is lead you to an unsolvable axiom?  Are you seriously not going to ponder that axiom at all?  You REALLY want to discover the exact details about how life formed over the centuries, but never get the least bit curious about why the universe set this evolution of more and more complex lifeforms in motion to begin with?

I study physics all the time and I still ponder over and over again why something can move indefinitely without energy unless a force stops it.  Granted it doesn't make me much money to do so.  Still, I can't help being curious..

Accused of Being Scientifically Lazy?

I have a feeling my belief in the supernatural will one day get someone to accuse me of being scientifically lazy.  It's true that I'm not near as intelligent as many physicists, but for a person who does not have a job in the scientific field I've put a lot of hours into scientific study.  I actually have a STEM degree.  If I were more gifted intellectually I might actually have a career in one of the STEM fields, but alas I'm not that gifted intellectually.  God did not bless me with the intelligence that he blessed many physicists with.  Even if I put in the same exact amount of hours of effort they did they're just flat out more gifted and I still couldn't match their success.

I looked up the CEO of Yahoo and it's not like that person devoted her whole life to computer science.  She had hobbies too.  Part of the difference between her success and mine is due to intellectual effort, but much of it is not.  Most of it is most likely due to natural God-given ability. 

So the next time I'm accused of being a scientific idiot, I will make the claim that I actually have put in a lot of effort into my studies long, long after that effort wore out its economic usefulness.  I did it for the love.  And the reason I lack understanding should be clear.  It is because God chose not to give it to me.  Is it MY fault that my intellect is not above a certain level despite all of my study?  Is it MY fault I can barely understand special relativity after years of pondering and consulting texts while others zoom through general relativity in their first semester?   Call me inferior to Stephen Hawking.  That's fine.  And of course it's true.  But then explain WHY.  Explain why.  Try explaining how something not outside of manmade effort blessed that man's mind.