Graphing tachyons, energy-momentum graph, imaginary mass

*Continuation of the previous two posts introducing tachyons and deriving the energy-momentum relationship

Again, sorry for the pictures but putting in equations on blogger is a futile task

Going to continue to try to figure out the spacelike timelike stuff eventually.....

Thanks again for reading

Derivation of Einstein's Energy, Mass and Momentum Relationship

Sorry for the long delay. I wanted to make my math not be unreadable and hence made the entire post on a google doc and took screenshots. The post, therefore, looks kinda unaligned. 

Do make sure to read the Lorenz factor derivation done in my previous post as this post starts off assuming that you know what the Lorenz factor, γ,  means and that you know what relativistic time/mass mean.

Hope this helps, I'll finally be getting into the astrophysics side of this (tachyons/ energy-momentum/ space-time graphs/ spacelike timelike as soon as I manage to figure that stuff out).

Thanks for reading!

Mehul

Relativity - Deriving the Lorentz Factor

γ or the Lorentz Factor is defined as “the factor by which time, length, and relativistic mass change for an object while that object is moving.” Hence in order to find this constant, let’s define a relationship between time and the relativistic time:

T'=t*(constant)

Where T’ is the relativistic time and t is the time taken. Therefore there must be some constant which is multiplied
to t to get T’. For practicalities sake, let’s define this random constant as γ. Hence:

T'=tγ

Now let’s do some basic geometry:

In the above figure, we can assume for à source emitting a photon at speed c, the distance travelled is simply
c * the time the photon travels for. This makes sense as the distance à photon would travel over five seconds
would simply be (Sorry for the terrible way the equations are written):

5*c=5*3*10^8=1.5*10^9m

At the same time, we can assume that the source s is travelling horizontally at à velocity v which is à fraction
of the velocity c. We however also know that for an observer O, the time will be relative compared to t (T’ = tγ).
Hence we can deduce that the horizontal distance travelled by the source s is vT’.

As the source is moving however, the observer O observes the path of the photon to be:

The problem however now is that the observed distance travelled by the photon is greater than the actual
distance. Hence, as c is à constant, the only way the photon could have travelled à greater distance is if it
has travelled over à greater period of time and hence we multiply the speed of the photon c to the dilated
time T’ (as dilated time for all bradyons is greater to actual time) to get the observed distance travelled by
the photon.

Using more basic geometry, we know using pythagoras’ theorem that:

(cT')^2 = (ct)^2 + (vT')^2

c^2*t^2 = v^2*T'^2 - c^2*T'^2

c^2*t^2 = T'^2 (v^2 - c^2)

On dividing both sides by c2T'2,we get:

t^2 / T'^2 = (c^2 - v^2) / c^2

On square rooting both sides we get:

                            t / T' = sqrroot (1 - v^2 / c^2)                    Eq - 4

As we have previously defined - T'=tγ

Therefore T'/t=γ and hence we have to reciprocate Eq - 4 to get:

    T' / t = sqrroot (1 / 1-v^2 / c^2) = γ                

Hence above is a very simple method of derivation. Again, quick apologies for the messy difficult to read math.

Nuclear Fusion Inside Stars / Fission (Nuclear bombs)

Stars are dense, highly pressurized and extremely hot objects. In these extreme environments, plasma is formed (a state of matter after 'gas') where all the particles get 'unbound' in a way and merge into a singular chargeless fluid.

In this chargeless fluid, as all the particles are unbound and zip around at extremely high speeds, hence many nuclei collide and combine to make a new heavier element. This process releases an extenuatingly large amount of energy. The process was used to make the H-Bomb (many times more powerful than the A-bomb released on Hiroshima and Nagasaki). Essentially nuclei in a plasma state combine and create new nuclei and hence new elements and this process releases energy.

Hence in stars, due to the large amounts of pressure and heat at the cores, nuclear fusion takes place which creates new elements. Our sun in fact constantly fuses hydrogen into helium. However what's really fascinating is the fact that fusion stops at iron. There is a 'binding energy' that is needed to fuse nuclei and iron actually has the greatest binding energy despite not having the greatest molar mass. Hence fusing beyond iron is impossible.

Now moving onto nuclear fission. As some of you may know, isotopes decay through alpha, beta, and gamma radiation and do this based on their half-life. Technitium-99 a common medical isotope decays and has a half-life of 6 hours (extremely short). Alpha and beta radiation both cause a change in the element while gamma does not affect the original element. (Beta decay releases an electron and alpha decay releases a helium nucleus). Hence some scientists started thinking about atoms splitting into constituent elements aka nuclear fission. 

What basically happens is that uranium 235 or plutonium 239 are hit with a neutron beam or with a neutron. The new isotope now undergoes the process of fission and splits into two separate atoms. At this point, three more neutrons are released. If the uranium was in a pile then the new neutrons would irrigate the other uranium and a chain reaction would begin. Each 'fission' of uranium would release 2.5MJ of energy which is well.... a lot. A ton of this uranium would release that much more energy. Hence this chain reaction is what is used in the A-bomb as the large amount of energy/radiation causes the explosion and after-effects.

The missing antimatter problem

Some of you may have heard of the "missing antimatter problem" and in this post, I want to go into detail on what it is and what it means.

Starting off, currently there is way more matter than antimatter. Most things ranging from our books to large stellar bodies all consist of matter while there is very less antimatter. However, the big bang should have released an equal amount of matter and antimatter. Hence scientists are trying to find out the cause of this disparity.



To give a brief overview of what antimatter is, antimatter particles share the same mass as matter particles however they have opposite charges ie. a positive positron (antielectron) is simply the opposite of a negatively charged electron. When matter and antimatter come into contact, they both immediately annihilate and release energy usually in the form of gamma rays



Scientists recently were able to create and analyze an antihydrogen to a significant degree of precision and what's truly fascinating is the lack of difference between the two ie. both the hydrogen atom and the antihydrogen atom had no significant differences.

Currently, there is no explanation for this disparity however one theory has gained some traction. Due to the near infinite volume of the universe, we barely know anything about it. Hence it's not a stretch to say just like matter has formed galaxies, there also might be antimatter galaxies somewhere far in the universe. This is only speculation though as there is no real proof or alternatives.



I'll make sure to keep updating if something interesting is found.

Until then,
Adieu

How we discovered that the universe is expanding

To understand the acceleration of the expansion of the universe, we first have to understand what redshifting is and to understand redshifting we have to understand some basic information about the wave-particle duality of light.

As most physics students know, light has wave and particle-like characteristics and hence has a wavelength and frequency. In simple terms, scientists measure the light from a 1a supernova. However, due to the universe expanding, the light redshifts and scientists can calculate the redshift to figure out the expansion of the universe.

The idea of redshift is based on the fact that if light is attempting to reach somewhere, but the place is moving away at an extremely fast pace. Then light's wavelength increases and hence the colour turns red (visible light spectrum)




Hence while observing light from distant stars, scientists observed that the light was undergoing 'red shifting' and found out that the red shifting was increasing at an accelerated rate and hence the only way for this to happen is if space and hence the universe were to be expanding.

Now that we've gone through the how, let's go through the why. Currently this expansion of the universe is considered to be there due to 'dark energy' which is well...... we don't know. What we do know however is that something is causing this expansion and we have hence coined this something as dark energy. The existence of dark energy and expansion of the universe have led to some scenarios of the future of the universe but I'll go through that in another blog post.

Signing out,
Mehul.

White Holes / Kerr Black Holes / Multiverse Penrose theory Presentation

Hi guys,

Below is a link to my presentation on white holes/Kerr black holes/multiverse Penrose theory.

You can use this for whatever.

https://drive.google.com/file/d/1T7-OoMVPr3QkDWAFh5P0Ql9vcoRhzCgz/view?usp=sharing

Thanks guys,
Mehul