"Infinitesimal" would imply that it has non-zero size.
I'm not normally pedantic but since you are started it...
"Infinitesimal" means "greater than zero, but smaller than any real number greater than zero". A point of zero volume would have zero mass even it had infinite density. An point of infinitesimal volume and infinite density would have a non-zero finite mass.
It's somewhat of a misconception that the singularity is this infinitesimal point with infinite density. In mathematics, the term "singularity" is really synonymous with "undefined" - ie we don't know what happens at the singularity.
Alright, strike the "infinitely-dense object" from my initial post and replace it with "singularity". Both my questions still stand. GR has been shown to be inaccurate/incomplete at small scales anyway, so if the formulas can't make predictions, why is that even a big deal? Possibly even a bigger deal than the places where it does make predictions, and those predictions are known to be wrong? Why is there so much research into singularities, their influence on the universe, their nakedness etc when we shouldn't even trust the premise?
I've stayed away from responding to this question because I'm woefully underqualified to answer any questions about it. Fortunately, my research advisor-- while not especially renowned in this field-- is qualified to at least examine some open questions and we happened to talk about it around a year ago.
Tub, from many physicists' perspectives, you are exactly right in your skepticism. In the words of my advisor, "Infinities always cause problems in physics," and the more you think about it, the more you realize it's true. We often use integration over the entire domain but if you take a step back, you realize that it has to break down eventually and the best you can hope for is that there is no "heavy tail" and the correction terms are very small. One example of this is in quantum field theory when we examine the perturbation of particle scattering experiments. Our perturbation technically includes instances where an electron flies out to Jupiter, orbits it three times, and then returns to the laboratory before detection and we assume that the inclusion of this and other "absurd" terms won't affect our prediction (and so far, theory and prediction match up quite nicely).
By my understanding, researchers in general relativity took the infinities (or singularities) as gospel when they were developing theories of black holes. I personally had an astronomy professor as an undergrad exasperatedly insist to me that black holes collapse to a single point infinitely fast, even though I was trying to simply argue, effectively, "No one has seen the interior of a black hole, so the mass being concentrated at a single point is indistinguishable from the mass being concentrated just within (or even just outside) the theoretical event horizon." Today, more general relativists and cosmologists are moving away from language that asserts or implies that the infinities/singularities are physical in nature.
Take all that with a grain of salt. You'll want to talk to someone who has the mathematical background to say what's going on with both general relativity and quantum mechanics.
An point of infinitesimal volume and infinite density would have a non-zero finite mass.
"Infinitesimal" means explicitly "non-zero". If the object has non-zero volume and a finite mass, its density cannot be infinite. If it had infinite density, then it would have infinite mass.
An point of infinitesimal volume and infinite density would have a non-zero finite mass.
"Infinitesimal" means explicitly "non-zero". If the object has non-zero volume and a finite mass, its density cannot be infinite. If it had infinite density, then it would have infinite mass.
No, thatguy is right. Study limits. Or study non-standard analysis.
An point of infinitesimal volume and infinite density would have a non-zero finite mass.
"Infinitesimal" means explicitly "non-zero". If the object has non-zero volume and a finite mass, its density cannot be infinite. If it had infinite density, then it would have infinite mass.
No, thatguy is right. Study limits. Or study non-standard analysis.
Ok, explain to me how an object with non-zero volume and finite mass has infinite density.
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thatguy wrote:
"Infinitesimal" means "greater than zero, but smaller than any real number greater than zero"
Warp, let n be a positive real number, and consider n/''infinitesimal''. As infinitesimal is smaller then any real number greater than zero, n/infinitesimal is bigger then any real positive number. If you say n/infinitesimal is m, being m any big real number, there will be a number k so that n/k = m. And as k is greater than zero, infinitesimal <k> m, for every real m. Thus, n/infinitesimal is infinity.
Firstly, there is no such a thing as "the smallest real number larger than 0." Such a real number doesn't exist. Likewise there is no such a thing as "a real number that's larger than all other real numbers".
Secondly, you are arguing for n/k to be infinity without n being infinity nor k being 0. That is a contradiction.
Firstly, there is no such a thing as "the smallest real number larger than 0." Such a real number doesn't exist. Likewise there is no such a thing as "a real number that's larger than all other real numbers".
Secondly, you are arguing for n/k to be infinity without n being infinity nor k being 0. That is a contradiction.
This is why I suggested you study limits. You are not speaking from mathematical experience, only your intuition. When we take the limit as n approaches 0 from the right, a positive number divided by n approaches infinity (it grows arbitrarily large).
The language comes even closer to the mathematical definitions and theorems when we use non-standard analysis. In that formulation, there is an infinitesimal (hyper)real number greater than 0 but less than all other real numbers and its reciprocal is infinite. All of calculus can then be re-derived from this formulation.
This is why I suggested you study limits. You are not speaking from mathematical experience, only your intuition. When we take the limit as n approaches 0 from the right, a positive number divided by n approaches infinity (it grows arbitrarily large).
You are still arguing that the result of the limit becomes infinity without the divisor becoming 0.
At the limit the divisor is 0, and the result is infinity. You can't have the divisor as non-zero and the result infinity.
Firstly, there is no such a thing as "the smallest real number larger than 0." Such a real number doesn't exist. Likewise there is no such a thing as "a real number that's larger than all other real numbers".
Secondly, you are arguing for n/k to be infinity without n being infinity nor k being 0. That is a contradiction.
I think BrunoVisnadi wasn't talking about the reals, but about the hyperreals: https://en.wikipedia.org/wiki/Hyperreal_number (infinitesimals and infinities don't exist in the real numbers, because the Archimedean property holds; instead they're expressed in first-order formulae of a certain form, whose proofs make use of the AP).
All syllogisms have three parts, therefore this is not a syllogism.
This is why I suggested you study limits. You are not speaking from mathematical experience, only your intuition. When we take the limit as n approaches 0 from the right, a positive number divided by n approaches infinity (it grows arbitrarily large).
You are still arguing that the result of the limit becomes infinity without the divisor becoming 0.
At the limit the divisor is 0, and the result is infinity. You can't have the divisor as non-zero and the result infinity.
Okay, guys! Pack it up! A forum member on a video game website with no formal education in higher mathematics just found a flaw in over 300 years of mathematics, completely dismantling calculus and all of the subjects that follow from it. Please collect your Fields Medal, Warp.
In general relativity, for non-spinning and non-charged black holes, all the mass that falls eventually reaches the center. The geometric center of the BH is point such that: for all r > 0, the sphere of radius r around this point has a mass m0.
Therefore, the density of such a sphere tends to infinity, as r tends to zero. The same thing would occur at the moment 0 of the Big Bang.
Many physicists expect that general relativity and quantum mechanics break down at Planck length scales.
Okay, guys! Pack it up! A forum member on a video game website with no formal education in higher mathematics just found a flaw in over 300 years of mathematics, completely dismantling calculus and all of the subjects that follow from it. Please collect your Fields Medal, Warp.
Is sarcasm a new form of mathematical proof that I haven't heard of?
A volume of non-zero size has finite mass but infinite density. Perhaps it's you who should be collecting those medals. You are inventing new physics.
Okay, guys! Pack it up! A forum member on a video game website with no formal education in higher mathematics just found a flaw in over 300 years of mathematics, completely dismantling calculus and all of the subjects that follow from it. Please collect your Fields Medal, Warp.
Is sarcasm a new form of mathematical proof that I haven't heard of?
A volume of non-zero size has finite mass but infinite density. Perhaps it's you who should be collecting those medals. You are inventing new physics.
You're a smart guy, Warp, and we're not your math instructors. Go study limits and/or non-standard analysis. They aren't that hard.
You're a smart guy, Warp, and we're not your math instructors. Go study limits and/or non-standard analysis. They aren't that hard.
So you are saying that with n/k, when k approaches zero, the result approaches infinity, and therefore the result is infinity without k being zero.
So "approaches zero" means "is not zero", and "approaches infinity" means "is infinity". Because reasons.
Yes, makes a lot of sense now. How didn't I see that before?
You're a smart guy, Warp, and we're not your math instructors. Go study limits and/or non-standard analysis. They aren't that hard.
So you are saying that with n/k, when k approaches zero, the result approaches infinity, and therefore the result is infinity without k being zero.
So "approaches zero" means "is not zero", and "approaches infinity" means "is infinity". Because reasons.
Yes, makes a lot of sense now. How didn't I see that before?
The reciprocal of an infinitesimal number is infinite in non-standard analysis.
Also, it's a dirty little secret that people are often imprecise with their language when it is generally understood what everyone is talking about. This happens in math, physics, computer science, and elsewhere. We don't really need to draw a distinction between "approaches infinity" and "is infinite" because we've all gone through those rigorous calculus classes and we know what is meant, even if it is not technically precise. Well, we didn't need to draw that distinction until you chimed in, at least.
So "approaches zero" means "is not zero", and "approaches infinity" means "is infinity". Because reasons.
Because mathematical rigor.
The defining traits of a positive "infinitesimal" are (1) it is greater than zero and (2) is smaller than any positive real number you come up with.
Likewise, the defining traits of a positive "infinity" are (A) it is greater than zero, and (B) it is greater than any positive real number you come up with.
The definitions for negative infinities and negative infinitesimals should be obvious. Also, note that the similarities in these traits are rather striking: in fact, you might say (and in non-standard analysis, it is said) that infinities are reciprocals of infinitesimals (and the converse).
In standard analysis, on the other hand, this means that neither infinities nor infinitesimals are numbers, but concepts: they represent limiting processes. So "approaches zero" means "infinitesimal" means "nonzero" by definition; and "approaches infinity" means "infinity" because infinity is that limiting process. When the limiting process is well-defined (I am not going into that), "approaches zero" can indeed be zero; as a rule of thumb, any case when you have a singularity is not well-defined (but you can't assume this!).
My assumption, after having only briefly glanced over the full publication, is that it's because of an effect similar to Shell Theorem. There are graphs in the publication that support this assumption. Do note that the publication is free if you click the PDF link, so you could read it for yourself if you're really interested =P[/url]
I understand that future time travel is theoretically possible, because when you start to approach the speed of light. Time speeds up. However, I'm curious if there is anything in the laws of physics which allow time travel into the past.
I understand that future time travel is theoretically possible, because when you start to approach the speed of light. Time speeds up. However, I'm curious if there is anything in the laws of physics which allow time travel into the past.
Both of those answers depend on the nature of the end of the universe.
If the universe ends in a Big Rip, the answer very strongly depends on the nature of dark energy, and whether or not the ratio of its pressure and its energy density is less than, greater than or equal to -1. The ship would not survive the Big Rip.
If the universe ends in Heat Death, this could take as much as 10^2100 years to finalize, but the universe will be exceedingly uninteresting for us after about 10 to 100 quadrillion years. The ship would survive this. It would take between 71 and 76 years to complete the journey to the uninteresting bit.
Universe ends via vacuum decay. No telling how long this might take. So no good estimate.
Big Crunch will almost certainly not happen thanks to dark energy.
But overall it just might be easier to link you this: http://nathangeffen.webfactional.com/spacetravel/spacetravel.php
Build a man a fire, warm him for a day,
Set a man on fire, warm him for the rest of his life.
