Relativity, an Introduction

[Erasmus00]

Relativity is often discussed in this and the strange claims forum. However, it seems to me many of the people who proffer opinions on the topic don't seem to understand many of the basics. Therefore, I've decided to take it upon myself to present a brief overview of some key concepts, starting with special relativity. I will try to keep the math level to algebra, although at times I have a feeling the only way to get from point A to point B will be calculus.

 

This first post will develop the idea of Lorentz transformations between reference frames. I'll try to add to it when I have time. My introduction to relativistic physics came largely from Feynman's lectures on physics, Gravity by Hartle, and An Introduction to Mechanics by Kleppner and Kalenkov, so I'm sure this will draw heavily on their presentation/pedagogy.

 

As I'm not writing history, I will take as a starting point that light moves at a speed c regardless of the reference frame. This was supported in Einstein's time by both theory and experiment (Maxwell's EM theory, and the Michelson interferometer experiments). Up untill Einstein, the equations to move between two observers reference frames were galilean. That is, to move from observer A (who uses x, y,z,t for his reference frame) to observer B (who uses x',y',z',t' and is traveling at velocity V relative to observer A) the following equations are used. (Note, for simplicity, I'm assuming at time 0 the two frames coincide, and I've used my freedom to orient the axis such that all motion is in the x direction).

 

x' = x-Vt (1)

y' = y (2)

z' = z (3)

t' = t (4)

 

Taking the derivative of equation 1 with respect to time, however, we get

 

v'= v- V (5)

This is the addition of velocities that we are used to, however, it is fairly obvious this equation can yield results above c, which we do not want. So what do we do? The only way to keep the velocity of light a constant in all reference frames is to allow time to change as we move from observer to observer.

 

So, lets postulate the most general transformation between two inertial frames.

 

x' = Ax+Bt (6)

y' = y (7)

z'= z (8)

t' = Cx+Dt (9)

 

The transformation is assumed linear, otherwise we couldn't have a simple 1 to 1 correspondance between events in each system. A nonlinear transformation could have acceleration in one system, even if we don't have acceleration in another. Clearly unacceptable.

 

Now, we do four thought experiments in the hopes of arriving at A, B, C, and D.

 

Experiment 1: Observer in the (x,y) system sees the origin of (x',y') move along x axis with velocity v.

 

Coordinates in (x,y) are x = vt (10)

Coordinates in (x',y') are x' = 0. (11)

Using this equation 6 becomes 0 =Avt+Bt, so B= -Av, and equation 6 now reads

x' = A(x-vt).

 

Experiment 2: Observer in the (x',y') sees the origin of (x,y) move along x' axis with velocity -v.

Coordiantes in (x,y) are x = 0. (12)

Coordinate in (x',y') are x'=-vt' (13)

Working with 6 and 9, we get

A(0-vt) = -v(0+Dt), or A=D.

Now we have x'=A(x-vt) and t'=Cx+At.

 

Now, experiment 3, A light pulse sent out from the origin along the x axis at time t= 0 . Its location is given by:

x= ct (14)

x'=ct' (15)

Using these in our transformation equations, we get C = -Av/c^2

 

Now, we just need a value for A. To get this value, we do one last experiment.

Experiment 4: A light pulse is emitted straight up the y axis at t=0.

In the (x,y) system, its position is given by

x = 0 (16)

y = ct (17)

In the (x',y') we get

x'^2 + y'^2 = (ct')^2 (to the moving observer, the light travels at an angle)

 

Using these and our transform equations, we arrive at A = 1/sqrt(1-v^2/c^2). We use the positive root so that when v=0, x' = x.

 

So, our general transformation is:

 

x' = 1/sqrt(1-v^2/c^2) [x-vt] (18)

y'=y

z'=z

t'= 1/sqrt(1-v^2/c^2)[t-vx/c^2] (19)

 

Notice, that when v/c <<1, then 1-v^2/c^2 is approximately 1, so 1/sqrt(1-v^2/c^2) is approxiamtely 1, and our transformations become

x' = x-vt

t'=t.

 

This is good. At slow speeds, we should expect that our intuition should be correct, and we have confirmed that it is.

 

So, to recap, we have mathematically developed the lorentz transformations from the first principle that the speed of light is constant in any reference frame. I'll continue when I have time. I next plan to discuss time dilation, non-simultaneity and time dilation. After that, I hope to discuss the idea of 4-vectors, on my way to developing E=mc^2 and other famous equations. I hope that if I develop 4 vectors using metric terminology, I can introduce in a purely qualitative way the ideas of general relativity.

 

Questions? Please ask.

-Will

[arkain101]

I have an important request!

 

I am looking for a link that shows with animated diagrams or drawings of how to display and explain dialations in near light speed travel.

 

Awhile ago there was a link somewhere around here that showed two boxes traveling past eachother and it had a photon bouncing between hypothetical mirrors.

 

I have been working on something related to this and I think I have finally got it. What my results show is that these dialations may not occur. It can be quite easily shown with simple geometry.

 

It would be much apreciated to help me find some good relativity diagrams that I could use in explaining what I've found.

 

Thanks, in the meantime I will keep looking.

[arkain101]

This is copy and pasted from another thread in this forum, one that I think people tired of opening.

 

Take it as a questioning, a proposition. I would like to get input and opinions.

 

.............................................................................................

Q-edit: you should know better than to do duplicate posting, you should have just given this link instead.

.............................................................................................

 

What I think is that an observer is stuck in the place of time or experience which interacts as information (light, data).

 

While time can tend to appear to vary and differentiate, everything in the grand scheme of things remains in the same time zone, nothing being less or more in the past, only the time zone that presents itself as the present to an observer. With this consideration I do beleive we can find the capability to exclude time when dealing with the univeres operations.

 

The 'past' is floating around all the time, infact its moving around right at this very moment you are reading this. Light waves showing galaxies 1000 light years away are impacting this very earth. All times are enclosed in which is now. All times change form but never traverse to the past or from a future.

[arkain101]

any thoughts?

[HIENVN]

any thoughts?

Thanks you for your attention to my thread.

I am looking for my documents to answer somebody in Science Forums, which an answer to your is my main point. Because my English skill is no good and my experience in Science Forums is lack when compare with all members in these Forums, I have spent a long time to learn English and experience of all members in these forums before I write my thread.

I currently focus on your post “Explain Mass,” with inertia is a main point for some my new threads, will coming soon in Science Forum. I wish you would help me to clear about this inertia that you have a right question and you may have a right answer.

I have viewed your Public Profile with all your post and I am surprised about your knowledge of science. I would to learn all your post to feedback for all my thread or post in the future.

 

All the best,

 

HIENVN

[arkain101]

I understand you are learning the english language.

 

I can help you learn languages with a technique related to memory.

 

It is a simple method and I am sure it will speed along your process of learning english, and other possible languages.

[HIENVN]

I understand you are learning the english language.

 

I can help you learn languages with a technique related to memory.

 

It is a simple method and I am sure it will speed along your process of learning english, and other possible languages.

Hi Arkain,

May do you show me how to learn English by anyway?

My response to you is very late because I have studied for some ideas of our members; such as TheBigDog in Law of Conservation of Energy, Erasmus00 in Rethinking…Classic Mechanics Will Be Return and you too in Relativity, an Introduction.

I think I have right answers for everybody right now, but my English skill is restricting my right responses to them.

For a general response to above everybody, science just progress if scientists known how to mixing between classical mechanics, quantum mechanics and relativity theory!

Thank you for you helps.

HIENVN

[Qfwfq]

All this is off topic.

[arkain101]

including the above :singer:

[arkain101]

Hi Arkain,

May do you show me how to learn English by anyway?

 

When you learn a word, associate a group of things with it.

 

Word: Plentiful

it means: alot of something, a large supply.

 

Associate things with this word. Plentiful it sounds like plent is full. see the Full. The word means lots, full, alot. Plentiful starts with a P. When I have to Pee I feel 'full'.

Associate what works best for you.

You could say, the word plentiful sounds like plenty full. The association you choose defines the word. If something is plentiful, then the supply of that thing is plenty full.

 

Another word.

 

Garage.

 

In the Garage you put a Car. the word Garage sounds like CAR. A car goes in a garage. It says rage. Sometimes people get road rage when they drive a car which goes in a garage.

 

 

The more you associate with the word the more you will remember it, and connect it to 'other' words. Stay consistent (the same) in how you associate each word with meaning. Use a pattern that you can follow.

 

One more example.

 

Pretned there is a word called.

 

Homvika.

 

Lets say it means the fruit on a palm tree.

 

the word homvika, it looks like home, and vodka. At "home" i have palm trees, and the fruit of those palm trees are cocanuts. I can drink vodka in those cocanuts.

 

Next time you see the word. Homvika. You will remember, home, vodka, palm tree, fruit.

Then you will say OH YAH!!! its the fruit on the tree that I can drink vodka out of at home where the palm tree's are.

 

As you do this you build a web in your mind. this is alot how Children learn a language naturally. The web will eventually connect, and connect so well the language will no longer be a 'struggle' you will 'understand' the language in a flowing connecting way, instead of a text book like way, where you need to find a page, then the word, which takes time. Where as the web has everything on one page, and you can map it. Eventually you learn the map.

[arkain101]

The more outrageous you associate things with the word, the better you will remember it.

 

Bonaville.

It is a type of motorcycle.

Bonaville, Be on a ville. In space I like to be on a bonaville, a motorcycle in space, I would die, I can die on a motorcycle on the road, the bonaville is the name of a fast road, where you can ride on a motorcycle. Be on a ville.

 

something like that. You wont forget the image of you in space on a motocycle, choking dieing, danger, bikes can kill, the word is a bike.