1

Just Chat! / Re: where has "eternal student" gone.

« on: Yesterday at 23:45:57 »

Hi.

      Thanks for your kind words.    I am well recovered from my bout of illness, thank you, although it did take out several weeks of productivity.   Please take good care of yourself and make sure that you are fully recovered from your own illness.

      I have just about caught up on everything I was supposed to do but have decided that I really can't spend the same amount of time on this forum in the future.  I'm just a house keeper and carer now rather than any sort of professional mathematician but I've been letting my main duties slide a bit.  So the house is falling apart and the garden is going to start annoying the neighbours etc. 

     This is the "just chat" section but that's already far too much about me and my situation.    I see that you (paul cotter) are still contributing well,  that is very much appreciated (by a few of us at least).

    Turning this discussion to the website or forum in general:    One of the things it can do well is provide a place for people like myself to read something or discuss something scientific.   There are always going to be a few people who can't or just don't work in a professional capacity as any sort of scientist anymore.   In principle, this forum is then a very good thing and there are always moderators and other experts who will try and provide some sensible discusson.   It's also extremely rewarding when you are actually able to help someone else understand something or just consider something new.    However, everything has a downside.  Scientists can be a bit single minded and get obsessed about something.  Using a forum is no exception, it can be addictive and it probably has been for me.   Overall you need to try and be objective about what you are doing here.   For example, over half of the new posts are really just trying to advertise something.   It's important to step away from the computer and remind yourself that, realistically, you aren't going to achieve very much here.   On the average day you'll be lucky to discuss something sensibly with one person,  most of the time the other person(s) are just advertising their own products and ideas and nothing you do will make a jot of difference.   In fact the more you interact with them, the higher the thread will go in the search tables like Google and that was just exactly what they wanted anyway.  So you get token responses from them and that's all.   Don't get me wrong, it is not and it should not all be about bringing them around to your opinion.  Quite the contrary, you should spend the time to check that what their saying is or is not sensible and consider shifting your own position.   However, you can then lose hours reading references and making your own Google searches before deciding that your time is being wasted.

   Anyway, that's it.   I'm too easily obsessed and lead myself to believe that what is done here could be as real or important as discussing something face-to-face  or  I.R.L.     Meanwhile many of the other people making posts just see the thing as more of a game... they can and will waste your time and not think twice about it.   Their goal (I would think) is just to further their own ends, such as advertising their own products or even just something utterly pointless, like making themselves seem important for a short while by presenting some information of dubious quality as if they are actually in some position of authority.   

    I'll make a special mention to all the moderators.  They really do give up their time and bring some expertise to the site.    There are also a few regular users that are great but I'm not going to mention them here because I'll end up offending those I miss out.   Anyway, anyone who gives up their time and tries to read something that other people have written, that's impressive stuff in my opinion.   Just stay healthy and get off the computer every now and again.

Best Wishes.

2

New Theories / Re: Do a Finite Amount of Infinitesimals Sum to a Finite Amount?

« on: Yesterday at 16:28:43 »

Thanks Paul.
  I've been doing stuff and things elsewhere.  I hope you and everyone else are well.

3

New Theories / Re: Do a Finite Amount of Infinitesimals Sum to a Finite Amount?

« on: Yesterday at 01:58:41 »

Hi all,

There are  (I think)  2 answers that are reasonable mainstream mathematics.

1.   What do you mean by an "infinitesimal"?    The usual definition is that this is a number with a modulus (size) which is less than any strictly positive number.    This may need some clarification:
   An infinitesimal is a number    x    such that   |x|   <     1/n      for every  natural number n.
     <=>    x is an infinitesimal      if and only if        |x| is less than     1 ,  1/2 ,   1/3,   1/4,  1/5,    1/6,   .... etc....       
   
    If you examine this definition carefully then the only Real number that is an infinitesimal is 0.
   

If you add together a finite number of  zero's   then you do always end up with  0.
As previously mentioned,  there is some ambiguity in your (talanum1) use of the phrase   "a finite amount".   I assume this means a finite number of infinitesimals are added together.
   Anyway, keeping it simple....   In the Real numbers.... there is only one infinitesimal and it's 0.  Add together any finite number of them and you still get 0.    An infinite sum is not defined in the basic axioms of the field of Real numbers.   So the overall answer is quite dull....   there is only one infinitesimal in the ordinary Real numbers and it's 0.    You can't get anything bigger than this by just adding a finite number of them together.


2.      Obviously then 0 is not a very interesting or useful infinitesimal.   Leaving the ordinary Real numbers behind we can construct a new field.   The most common one which turns out to be useful is the field of Hyperreal numbers.   In that you can have infinitesimals that are not just 0.    More information can be found in various places.   Here's Wikipedia's entry:
   https://en.wikipedia.org/wiki/Hyperreal_number

   Using Hyperreal numbers and the infinitesimals they provide will gives us "infinitesimal calculus" exactly as it is commonly used to explain or think about calculus  (integration and differentiation).  It's these sort of ideas they (teachers) would have used to motivate and explain calculus at school.  Non-mathematicians don't need to worry too much about the formal details of the Hyperreal numbers, it just does turn out to let you develop and explain calculus in a very intuitive way.  Developing calculus with the Hyperreal numbers is called "Non-standard Analysis"  even though it is actually the more intuitive way of developing calculus and historically this was how calculus was first thought about.

    As an alternative you can stick solidly and rigorously with the Real numbers and use standard Real Analysis to develop the idea of limits - from which calculus will follow.   This "Standard Analysis" was developed precisely because Mathematicians were never happy about using or assuming the existence of infinitesimals.   So it was developed almost immediately with the public acceptance of calculus.   Non-standard Analysis was pushed to the back and frowned upon as if it was a dumbed-down and good enough for school only explanation of what is happening.  Non-standard Analysis regained some recognition much later, especially due to work of people like Robinson who laid down most of the formal mathematics for the Hyperreal numbers.   At the current time, you have a choice about how you explain calculus, both standard and non-standard analysis are equally valid.

   Anyway.... your (talanum1) question only gets interesting when you use something more than just the ordinary Real numbers.  If you keep everything Real then the only number which is "infinitely small" is 0 and precisely just 0.

Best Wishes to everyone.

4

Physics, Astronomy & Cosmology / Re: Why can't i understand the andromeda paradox?

« on: 08/07/2022 15:11:53 »

Hi.

I'm glad you're getting better @paul cotter .

A higher speed of one participant would give a greater slope to your green line.

   Yes, that's absolutely correct.   However, that's not where the emphasis is usually placed.
The emphasis is usually put on the massive distance between Earth and the Andromeda galaxy.   As such even a tiny slope on the green line results in a huge difference in deciding which event is happening now in Andromeda.
 
  So, the common development of the Andromeda paradox is that even if you did something like just turn or shake your head in different directions,  that slow movement is important when magnified over huge distances.   It's an effect of special relativity that doesn't require space rockets and travel at speeds close to that of light etc.

Best Wishes.

5

Just Chat! / Re: The coronation of "eternal student", why was i not invited?

« on: 06/07/2022 12:10:49 »

Hi.

   I'm not annoyed.    The last post was meant to be humorous it just evidently didn't work well, sorry.
I should probably be catching up on all the housework I haven't done while I was ill,  instead of posting too much for the next few days.   Meanwhile, I'm glad you're still OK but I should imagine you will need to keep taking it easy for a few more days.

Best Wishes.

6

Physics, Astronomy & Cosmology / Re: Why can't i understand the andromeda paradox?

« on: 05/07/2022 16:31:28 »

Hi.
    No worries @paul cotter ,  rest up and get well soon.
Best Wishes.

7

Physics, Astronomy & Cosmology / Re: Why can't i understand the andromeda paradox?

« on: 05/07/2022 15:00:42 »

Hi again.

    Well look, here's the thing from first principles...

    Start with a person called Paul on planet earth.   For ease of notation and calculation we will set up an inertial reference frame S  with co-ordinates (space, time) = (x,t) .  We'll only need one spatial dimension (just choose a Cartesian frame so that planet earth and the Andromeda galaxy lie along the x-axis).  We'll have Paul stationary in the frame S and, as usual, we'll put him at the origin.
    To keep it all simple, we'll have the Andromeda galaxy at a distance of  3 million light years from earth and we'll have it remain stationary in the frame S.     (In reality it's about 2.5 million light years away and you probably know it's on a collision course - none of that's necessary for the Andromeda Paradox, we just need it at a large distance).

    Let's draw a spacetime diagram now and put some important worldlines on it.


Andromeda1.png (12.8 kB . 1304x767 - viewed 2280 times)

   Since Paul and the Andromeda Galaxy are stationary in this frame, their worldlines run parallel to the t axis.   Paul's is precisely along the t-axis.

    Now the general idea is that there are some Aliens living in the Andromeda galaxy and they will decide to invade earth.    Let's mark some important events on the spacetime diagram.     Event (x_A, t_m )   =   where and when the Andromedans have a meeting to consider invading Earth.       Event  (x_A , t_L ) =  where and when the Andromedans launch the invasion fleet.
   Paul considers events lying along the x-axis to be events occurring at time t = 0,  or  "right now".   So, for Paul,  the Andromedans are having a meeting to decide if they will invade.


Andromeda2.png (12.89 kB . 1304x767 - viewed 2285 times)

   Now Paul has a friend,  let's call them Bicycle Brenda,  who is riding a bicycle past Paul at a low speed and just happens to be travelling toward the Andromeda Galaxy.   So Let's consider how Brenda sees the universe next...
   This is just standard stuff in Special Relativity,  Brenda sees Paul looking slightly thinner than if he was at rest,  this is length contraction  etc.   However, since the speed of Brenda relative to Paul is so small, it's not like this really matters or is significant.   Anyway, the easiest way to determine how Brenda sees the world is to construct another inertial frame around Brenda where she is the one at rest at the origin.   We'll set up a second frame S'  in which Brenda is at rest and at the origin.   We'll synchronise the times t and t' = 0 when Paul and Brenda are precisely at the same place (I'll say just passing each other but for the Mathematics it will fine if Brenda has ridden right through Paul at the event t=t'=0 and x=x'=0).   Anyway,  the frames   S with co-ordinates (x,t)   and    S'  with co-ordinates (x',t') are just Lorentz boosts of each other as usual,   with a tiny off-set velocity v = bicycle speed = 2 mph.

    We need the Lorentz transformations to see how the events in S are mapped to events in S',  we'll just focus on the time coordinate t'

t'  =   
   You can check this equation yourself, see https://en.wikipedia.org/wiki/Lorentz_transformation , for example.  Spend a moment and remind yourself how this equation can be used.    Given an event with values (x,t) specified in the frame S,   this equation determines the t' value that would be assigned to the corresponding event in frame S'.    We could determine the corresponding x' value with another equation - but we won't need that for this example. 

    Brenda considers all events in the universe to be happening "right now"       if and only     they have t' coordinate = 0 in the frame S'.    They can have any  x' co-ordinate (that just tells her how far away the events are in space),  just so long as  t' =0  (that means they are happening at t'=0 or right now, where "now" means when Brenda passed Paul and said "Hi!"  or  "Sorry I just ran you over" etc).
    We want to mark all the events on our spacetime diagram that are happening right now for Brenda.  However, our spacetime diagram only has an axis marked up for the frame S.
    Set t'=0  in the above Lorentz transformation equation and determine the set of all events (described with co-ordinates  x,t in S) that are happening "right now" for Brenda,  that's what we will mark on our diagram.

   From the Lorentz transformation equation we obtain:
0 =  t'   =      
   ≠ 0    ,  so divide ,   re-arrange and obtain 

  t' = 0        <=>        t = 

On the R.H.S.   we have the relationship  between  t and x  (the co-ordinates in S  that describe events happening now throughout the universe for Brenda.
    v =  small positive value =  bicycle speed in S and it's in the +ve x direction.
    c²  =  a massive number.

v/c²   is therefore a tiny positive number.      t =    (v/c²) . x   describes a straight line on the diagram that is almost parallel to the x-axis.   However the key is that it is NOT quite parallel to the x-axis, it is angled upward a tiny bit.   Moreover the Andromeda galaxy is a long way along the x-axis, so let's  put this line of constant  t'  on our spacetime diagram and see what happens.....


Andromeda3.png (16.34 kB . 1304x767 - viewed 2286 times)

   The Green line shows all the events throughout the universe that are happening now (t'=0) for Brenda.
Hopefully you can see that, as far as Brenda is concerned, when she passed Paul and said "Hi!"  the invasion from Andromeda has been launched.   Meanwhile, for Paul the meeting to decide whether to invade Earth or not hasn't finished yet.

Anyway, I hope that will help to see the gist of the argument for the Andromeda paradox.
As @Halc mentioned earlier,   you need to be carefull not to say,  Brenda "sees" the invasion has been launched.    There's no way light has reached Brenda's eyes yet, she isn't going to "see" the ships coming.   It's only that the launch is actually happening now for Brenda even though there is no practical way she could know.  So, for example, she couldn't tell Paul that it was coming.... unless she has some  extra-sensory perception and says "you know, it's funny but every time I ride my bike in that direction I just feel like some invasion has been launched".  In which case Paul would reply, "have you tried changing to a lower gear?  You might be getting yourself a little out of breath and dizzy".

Best Wishes.

8

Physics, Astronomy & Cosmology / Re: Why can't i understand the andromeda paradox?

« on: 05/07/2022 12:52:43 »

Hi.

    Hope you are well.   This is is obviously one of the most interesting posts available today.  I have a strong suspicion that @Halc is about to write a post since the forum is reporting him as viewing this post at the moment.    He obviously has a head start over me, so I'm just going to wait and see what gets done.
      For the short term, I'm just going to glance through some YT videos that might exist and see if I can recommend one.

Best Wishes.

9

New Theories / Re: How Many Numbers Exist?

« on: 03/07/2022 03:26:19 »

Hi.

Is there a set of numbers which contains more than algebraic numbers but less than real numbers?

   If you just want a SET of numbers,  yes,  many of them.
There's an infinite set of distinct transcendental numbers,   T.   These are real numbers that are not algebraic.   Actually MOST of the real numbers are these,   the algebraic numbers are countable so they are a tiny drop in the ocean compared to the transcendental numbers.
   Anyway, you could just keep adding the transcendental numbers one at a time until you got bored.

   You can cut down some of those intermediate sets of numbers if you put more of a restriction on your set of numbers.   For example, instead of just asking for a set of numbers, insist that the new set of numbers is always a proper Field in its own right.   These would be called Field Extensions.   (Example:   If you try to add π,  then the field operations automatically generate  π², π³, ....,  1/π, ...., -π, .... (705 + 3/π), ....   so  to ensure the set really is closed under field operations you'd need all of these numbers added in one go).
  None the less, I still think there's an infinite set of simple field extensions you could find.  (I haven't spent too long thinking about it but I reckon you could try a simple field extension by adding π and show that √π is still missing,  so extend again with √π and show  the √(√π) is still missing..... etc.....    the root of what you've just extended with should always be missing).

Best Wishes.

10

Physics, Astronomy & Cosmology / Re: What is a black hole made of?

« on: 27/06/2022 19:01:32 »

Hi.

   Is David joining us as a member on the forum?   You could use a false name etc.   Anyway if you aren't, then your answer will be be available in a podcast shortly.   If you are joining us then please say something here and we'll have a discussion.

Best Wishes.

11

Physics, Astronomy & Cosmology / Re: Why is inflation needed in the bbt?

« on: 25/06/2022 15:32:44 »

Hi.

from an engineering perspective(which is what I am limited to) ∇.b=0 precludes the possibility of magnetic monopoles.

   Yes, more or less.

   We haven't observed any magnetic poles,  that is written in mathematical notation as  ∇.B  = 0.
That's all that is,  just a way of writing that phrase in mathematical notation.   It's not a proof that magnetic monopoles don't exist.    If they were found, then that equation in the set of Maxwell's equations would need to be changed. However, it would only need changing for those situations where there is a magnetic monopole in the experiment or situation being modelled.    Where there isn't any magnetic monopole then the equation obviously still holds.
   The existing theory of classical electromagnetism can be considered as some "proof" or evidence that magnetic monopoles are very rare and certainly don't contribute to many situations.   Maxwell's equations including  ∇.B  = 0   have proved very accurate and reliable in predicting and modeling electromagnetic phenomena.   If  ∇.B  is significantly not 0 in some situations we should have noticed it by now because Maxwell's equations would have given us predictions that didn't match the observations.
     However, that's as much as we can say - if they exist, then they will be rare and not important for the situations we have studied or modelled so far.   ∇.B(x, t)  ≈ 0   at all positions x in space and times t   should remain a very useful approximation.

I would love to hear some elaboration on this, in particular why the bbt requires these monopoles.

    It's not exactly the Big Bang Theory that demands monopoles.   The existence of magnetic monopoles is suggested by several more specific theories in physics.  For example, standard particle theories and superstring theory.   Of which the standard model of particle physics is probably the most important and most widely accepted, superstring theory is an optional addition. 
    So it's possible to imagine that the bare-bones of the ideas for the BBT could persist while some of the fine details for things like particle synthesis just after the Big bang are adapted.   Physicist's would like to find monopoles because that would support the overall conglomeration of ideas that make up our best theories of cosmology including things like particle synthesis.   These early moments of the Universe including processes like nucleosynthesis are all included under the umbrella term "the Big Bang Theory".

Best Wishes.

12

Physics, Astronomy & Cosmology / Re: Is the inverse square law only approximately correct in general relativity?

« on: 25/06/2022 14:33:45 »

Hi.

     I'm always very grateful for anyones time spent in conversation.  So thank you very much @evan_au.
However, I can't always agree with everything and that is the nature of discussion after all.   What you've said isn't wrong, it's just misleading.   Although, equally, what I've said before could be considered misleading.
    We need to establish a few things and then hopefully we will be seeing and understanding the same things.

I was not thinking about a horizon at a fixed distance.

   OK  -  BUT you should be.    The inverse square law is about the intensity received at a distance, r, from the source.   That is a physical distance, so it is determined by the metric.   It is not determined by reference to a difference in the values assigned to locations in the co-ordinate system we commonly use to describe an expanding universe.
   The usual co-ordinates used in an expanding universe are the called the co-moving co-ordinates.  Galaxy 1 can have fixed co-moving co-ordinates and it's tempting to say it has a fixed position.   Galaxy 2 can also have fixed co-ordinates and we can be tempted to say it has a fixed position.    The co-ordinate differences between the galaxies never changes, so that seems fine.   However, the physical distance between the galaxies is increasing with time,  in that respect it doesn't look like the galaxies are fixed in position in any ordinary sense of the word.   To avoid the confusion, it's best if you just don't say or imagine that either of the galaxies have a fixed position.   Instead, just say that the galaxies are "co-moving with the universe",  or  that they are "co-moving" for short.

But space can expand faster than c, so (in principle) there are distant galaxies that people on Earth could see today, but
 which will not be visible in 10 billion years, because the expansion of space has carried them outside our visible universe.

      This is correct.   However, the light emitted by the distant galaxy will travel an infinite physical distance.   It's just that how far it can go in 1 unit of time will not be sufficient to match the expansion of space that is occurring between the distant galaxy and earth.
      We would place our sphere, where the intensity is being determined, at a fixed physical distance from the distant galaxy and not at a fixed co-moving co-ordinate separation from the distant galaxy.   The light from that distant galaxy will reach the surface of that sphere.    Just to get the image and understanding right,  the surface of that sphere might have started at time t=0 precisely where planet earth was.   However, the surface of that sphere will be a long way from earth at a later time t=1 unit,  when the photons from the distant galaxy finally cross over that sphere.
     We can place a sphere with a fixed physical radius, r, around the distant galaxy (it doesn't matter how large r is)  the photons from that galaxy will (eventually) reach the surface of that sphere.   In no way does this contradict the idea that the photon won't reach earth - the surface of the sphere is nowhere near planet earth when the photons cross over the sphere.

If you posit some particle that travelled at c/10 (and didn't slow down), there would be regions of our visible universe that could never detect these particles, because the space in between is/will be expanding faster than c/10.

   Yes, total agreement.   However, that has nothing to do with the particles crossing over a sphere at some fixed and pre-determined physical distance from the source.

Best Wishes.

13

Physics, Astronomy & Cosmology / Re: How does special relativity explain dimensional components ...

« on: 25/06/2022 13:02:01 »

Hi.

Does time and space have separate components like that?

   Yes and No.

The yes bit:
    4-vectors are what are important in spacetime.    These have 4 components,   3 of them are called spatial components and the other component is called the time component.  You can write the spatial components first and the time component last but it's more common to write the time component first.   It's also fairly common to start numbering the components from 0 and not from 1.   The final slightly confusing thing you might see is that if you had a 4-vector  X   then you may see the components written as  X⁰,  X¹,  X²  and  X³.    Superscripts instead of subscripts like  X₀,  X₁  can be used.
    For example    r =    ( ct,  x,  y,  z)  is the usual way of writing  the position 4-vector of an object.    It has a time component  ct  =    c (the speed of light)  multiplied by the position of the object on the time axis, t.    It has spatial components   x, y, z   which are the position of the object along the x, y and z axis respectively.    Now you could use t as the time component instead of ct but the algebra turns out to be much easier if you use  c (the speed of light) multiplied by t   as the time component.

The "no" bit
1.   There's an unusual way of determining the magnitude of a 4-vector.  You might see it called the "norm" or "Minkowski norm" of a 4-vector.    For simple vectors used in Euclidean space or Newtonian mechanics,  whenever you increase the size of one component the overall magnitude of the vector would increase.   For 4-vectors that's not always the situation,  you can increase the size of one component and end up reducing the overall magnitude of the 4-vector.    The Minkowski metric is described in various places  [For example,   https://phys.libretexts.org/Bookshelves/Modern_Physics/Book%3A_Spiral_Modern_Physics_(D%27Alessandris)/3%3A_Spacetime_and_General_Relativity/3.1%3A_Minkowski_Metric  ].

2.   You are mainly discussing velocities and motion in your posts, rather than just positions.   In ordinary Newtonian mechanics, velocities are just a rate of change of position with respect to the time co-ordinate.   For 4-velocities we can't determine rates of change with respect to a fixed time co-ordinate,  instead we must determine rates of change with respect to what is called the proper time for that object undergoing the motion.
   What this boils down to is that spatial components of the 4-velocity are NOT exactly the spatial components of the ordinary Newtonian or 3-velocity that the object might have.   Instead the spatial components are a multiple of the spatial velocity you would have assigned the object in Newtonian mechanics.  Also it's not a fixed multiple,  the multiple changes according to the Newtonian 3-velocity of the object.    Specifically, the  spatial components of the 4-velocity are given by   γ  (the gamma factor) multiplied by the spatial components of the 3-velocity.

- - - - - - - -
     That might be more detail than you were after.   Overall there is a lot of similarity between  4-velocities used in Special relativity and more conventional velocity vectors you might have seen in Newtonian mechanics.    I've mentioned the differences because, in my limited experience, if we don't then it's human nature to run away with the idea that it's all exactly like Newtonian mechanics and ordinary 3-velocity vectors.   You'll soon hit problems if you do that.
     For example,  it can be useful to consider the magnitude of a 4-velocity vector.  An ordinary object with some positive rest mass always has a 4-velocity vector of magnitude c (the speed of light).   That magnitude can be shared out between the time component and the spatial component of the objects 4-velocity.   An object at rest (in a given frame which we will use to assign the velocity vector) has all of its velocity in the time component while the spatial components would have the value 0.     Meanwhile, an object in motion (in the given frame) has a non-zero value in the spatial components of the 4-velocity and a correspondingly lower value* in the time component.

Best Wishes.

* LATE EDITING:  I don't like this on a second reading.   It's precisely one of those examples where you could have a larger numerical value for the time component but that is actually reducing the overall magnitude of the 4-velocity and not increasing it (because the Minkowski metric subtracts the time component instead of adding it).  It's fair to say the object has less velocity through time and many Pop Sci articles will do this - but it's not correct to imply that the numerical value you find written in the time component of the 4-velocity has to be smaller.

14

Physics, Astronomy & Cosmology / Re: Why is inflation needed in the bbt?

« on: 24/06/2022 15:19:48 »

Hi.

The short article provided by @Origin  is extremely good.   It sets out the three main problems and explains how inflation solves these:

1.  That space does seem to be almost flat.

2.  That distant parts of space seem to be at similar temperatures, suggesting they were in thermal contact at some time.  Ordinary big bang expansion would have kept them apart and kept driving them apart faster than light could have travelled from one region to the other.   No exchange of heat between them should have occurred.  Without inflation, the only possibility is to assume that it was coincidence that all these regions had the same temperature after the big bang.  (That isn't an impossible coincidence, if the Big bang was very symmetric you might get something like that - but it's an assumption that you just don't need to make if the regions were in thermal contact with each other).

3.    Lack of our ability to find magnetic monopoles.

We can probably add two extra problems / observations:
4.    That space is surprisingly homogeneous and isotropic -  it's much the same density everywhere and in every direction.   With inflation you don't need to assume it was always this way just after the big bang.   A massive expansion of a less homogeneous space produces a large region where everything is much more homogeneous.
5.    Conversely, it isn't completely smooth.   There are small irregularities in density and this is what ultimately gives rise to the clumping of matter  -  the formation of galaxies in some places and voids in other places.   

   How 4. and 5.   work together is quite interesting.   They aren't really opposites of each other.    Assume the situation described by 4. happens: 
(i) The lengths over which differences are observed increases - differences in densities observed over a length of say 1 light-second are now only noticeable over a distance of 100 light seconds.
(ii)  The actual size of the density differences is also reduced:  Since the volumes of space were stretched the actual densities are brought closer to 0, so the difference in densities is reduced.
    This sort of change keeps happening.  Meanwhile there will always be small quantum fluctuations happening - things that might produce a difference in density that are observable at lengths of a (tiny) fraction of a light-second.   However, these small changes that were due to quantum fluctuations get stretched and eventually become density differences that are noticeable over lengths of about 1 light-second.  This whole thing keeps going and repeating (it's not a discrete, first this, then that,  it's a much more continuous thing but seeing it as series of small discrete steps is just easier to visualize).    Overall,  the coarse and large scale differences in density keep getting stretched and smoothed away, while the quantum fluctuations keep getting magnified to produce small differences in density that are noticeable over lengths of about 1 light-second.    We end up with a situation where the small differences in density that we want are assured while larger or more coarse differences are equally assured to have been removed.  "We want" these small differences in the sense that these do work well in the models and simulations for galaxy evolution.

    That's about it.   That's the limit of my understanding about inflation.   Sources of information were:
a)  "The History of the Universe", especially Chapter 13,  David H Lyth,  Publisher: Springer,  2016.
b)   NASA website provided by Origin earlier.
b)   Wikipedia:  https://en.wikipedia.org/wiki/Inflation_(cosmology)

A final note worth mentioning is that Inflation remains an optional addition to the Big Bang Theory:
The basic inflationary paradigm is accepted by most physicists, ..... however, a substantial minority of scientists dissent from this position       [Quote from Wikipedia]

Best Wishes.

15

New Theories / Re: Did we really never observed white holes ?

« on: 23/06/2022 23:36:28 »

Hi.

   I'm sorry but I am struggling to understand what has just been said.   Some bits of it do look like existing ideas for white holes.

Best Wishes.

16

New Theories / Re: Can conscious thought act on matter?

« on: 23/06/2022 23:26:31 »

Hi.

I read most of what was written just above.   It says quite a lot of stuff   BUT  none of it seems to be useful.  It doesn't make any predictions or suggest something that can or should be tested.   Science is less concerned about why or what some ultimate truth might be and more concerned with having an explanation and a model that is useful for making predictions.   

Best Wishes.

17

Physics, Astronomy & Cosmology / Re: Is the inverse square law only approximately correct in general relativity?

« on: 21/06/2022 18:46:50 »

Hi.

I see a dimensional conflict...

   Yes.   I almost mentioned this in the post but it was already too long - because all good scientists are trained to check the units and dimensions.

The metric has a form where φ actually has dimesnions of length:

ds²  =     (stuff with an overal dimension of length²)  +  Sin²(θ) dφ². 
We'll leave the θ as a dimensionless angle,   then the  dφ² must have dimensions of length².
 
   Anyway,   In the final expression for the surface area  4πr,     that is   really   2 . (2π) . r    where the  (2π) has dimensions of length.

Best Wishes.

LATE EDITING:  Having a dimension assigned to what looks like an angle can seem very hard to swallow.   If it helps, the space can be considered and conceptualised as something with flat (2-D)  polar co-ordinates  (r, θ)  and a third dimension φ which is the depth into the page.   The depth into the page just happens to permit a range between 0 and 2π.   That helps to see φ as something that can have dimensions of length.   As I mentioned, there's no good way to visualise this very non-Euclidean space and re-drawing everything with that sort of representation just strains understanding elsewhere.

18

Physics, Astronomy & Cosmology / Re: Does The Gravity Of A Black Hole Travel Faster Than The Speed Of Light ?

« on: 15/06/2022 02:02:07 »

Hi.
   Thanks for the extra information, @evan_au .  There's stuff I didn't know there and I may look into Supernova again to see what the latest is.   
   The rest of this post may look like a criticism, sorry.  It isn't meant to be.

Time dilation is so extreme within millimeters of the event horizon that the image of an infalling rock would be very quickly red-shifted into oblivion.

    I think we need to establish where the observer is and assume they are using a frame of reference where they are at rest.   The distant observer = An observer at a fixed radial co-ordinate, r >> Schwarzschild radius.      The rock = the rock or anyone close to the rock and falling in with it.

      We all agree that the rock doesn't spend long outside the event horizon, it just falls in within a finite amount of time as far as its concerned.   However, for the distant observer the rock takes an infinite amount of time to reach the event horizon   (well, certainly if the rock is treated as such a small mass that it doesn't affect the spacetime geometry - a previous post discussed that and mentioned my uncertainty about it).
     As far as the distant observer is concerned, light from the rock is progressively red-shifted and total luminosity from it reduces.   This isn't a quick process.

So: Very few photons, severely red-shifted: The rock would not "float" near the event horizon, it would just disappear.

   Bits of this are OK.   However, it doesn't "just disappear", it fades away.  It's like having a studio engineer with the slowest hands in the world, turning the fader knob so slowly that the universe will end before the stage actually goes dark. 

...forming a black hole, with almost the same mass as the star before it imploded...

    Is that right?  I know stellar collapse varies quite a lot and I'm not aware of the latest ideas for the typical behaviour.   Old texts used to suggest that typically there is a supernova explosion where the outer layers of the star making up about 20% of the original mass of the star is blown away.    Some sources put the amount of matter ejected far higher than that...
About 75% of the mass of the star is ejected into space in the supernova.   

https://imagine.gsfc.nasa.gov/science/objects/stars1.html

Best Wishes.

19

Physics, Astronomy & Cosmology / Re: Does The Gravity Of A Black Hole Travel Faster Than The Speed Of Light ?

« on: 14/06/2022 18:26:02 »

Hi.

Something must be holding light back faster than light itself travels.

    Not really.    The easier way to imagine what is happening is to assume space itself is being pulled in toward the black hole singularity.   So light is travelling as fast as it can through space but it's not good enough, space itself is being pulled into the singularity faster than that.
    This is only an image or conceptual representation but some people have presented the idea as water flowing and people trying to travel through the water.   There's a reasonable animation on YouTube that I'll try and find and add to the post later.

LATE EDITING:   I can't find it in isolation.   You can see it in this video by Brian Greene,   "your daily equation #31", available on YouTube.      Aim for the time   between   4:20   and  6:00.

so it's the propagation of internal gravity waves that stops light ?

   No.

does light even exist inside a black hole ?

   According to theory,  yes it can   (for a while before it hits the singularity).   No one has actually been in there to see it.

Best Wishes.

20

Physics, Astronomy & Cosmology / Re: What are the properties of space?

« on: 13/06/2022 16:28:52 »

Hi.

    I fully understand your comments, @paul cotter  .    Replies are always optional.   The forum is meant to be for interest and when it starts to feel like an obligation or burden then it's time to re-examine what you're doing and why you think you should.
      The comment you made about the metre is a perfectly good one.   However the idea of defining a metre by as how far light travels in 1 / 300 000 000   th  of a second* isn't completely silly or arbitrary and it does open the door to looking at things in different ways.   That means a few more minutes of hopefully interesting discussion, if nothing else.
*LATE EDITING:  Yes, it's actually 1/ 299 792 458  - it's just that the exact numbers aren't important and get in the way of the general idea.

      For example, how would you realistically measure big distances?    Presumably with a radar or laser measuring device,  you'd just time how long it takes to get a signal back.    How would you know if the distance between two objects has increased after 100 years, or if the speed of light has actually decreased as time evolved?    More significantly, would there be any difference to how physics behaves?  These ideas are always worth a moments thought and well worth revisiting every now and again.   Usually you can discern a difference between distance increasing and light slowing down but it get's murky again if you consider that maybe your measurement of  time is also changing.  Maybe there are some important dynamic fields in the universe and the frequency of radiation emitted by Caesium-133 atoms is changing as a result of that, so that the atomic clocks used to define the second are not actually keeping the correct time.
     For simplicity, we can consider trying to view the universe from some other place or point of view.   One easy option is to adopt the language used by those who suggest our Universe might be a giant simulation (much as if it's running on someone's laptop).   If their laptop starts to run more slowly (perhaps because the processor is overheating and it gets throttled-back) then they notice that our ideas of time have changed, it takes longer to get their action hero across a room.   However, the simulations don't notice any difference, everything in the simulation moves slower in the correct proportion.    Similar things happen if distances suddenly doubled (in all directions) but everything can move twice as fast as before.   The computer simulation action hero can still fire an arrow across the room in the same time.   
    Overall then, you have to decide that something(s) is (are) "locked down" or held constant.   Typically you'd start with time and assume the simulations have some awareness of a flow of time and it doesn't matter if it changes relative to the gamer's time outside the laptop, it's constant and fixed for the simulations.  Then build up from there systematically, considering if it's possible for something else (like the speed of light inside the simulation) to change or if it would be indistinguishable from some other combinations of things changing.

Best Wishes.