## INTRODUCTION

I came to know about the twin’s paradox with a Italian pocket sold with the Italian version or Scientific American “Le Scienze”

As the book did not explain the math behind it, I went to Youtube to search. I found again DrPhysicsA, who helped me create the page about EFE.

Dr PhysicsA has a BSc (physics) and PhD (nuclear physics) from King’s College, London.

Thanks to “DrPhysicsA” I make good progress in applied and advanced math as well as understanding some of the more advanced Physics.

# INDEX

- introduction
- The lecture
- Space-time – introduction of rules
- The twin paradox with constant speed
- The twin paradox with speed with accelerations
- Section 5
- Section 6
- Conclusion
- Contact me on FB

## The lecture

DrPhysicsA explain the math behind the Twins paradox in this video

When Newton sstudied Cartesius had problems moving forward, it is said that starte to read the book from the beginning. I do the same.

As I need to go back now and then to compare I use the manuscript and add screenshots do do that. See below.

OBS. These are my notations. I recommend you to write your own nottations while listening to the lecture.

## Space-time

A note: Text in Italic is taken from the manuscript

A note: Text in Italic is taken from the manuscript

*Today we’re going to look at space-time and a thing called the twins paradox and as usual we’re going to start at the very beginning so the*

*question is what is space-time.*

**what is space-time?**

*As we understand it is three-dimensional that at least is what we observe. That means to say that in any area of space you can go left and right or up and down or back and forth three dimensions and we usually represent that by three dimensions of space or three coordinates of space*

which we often label x y and z and that is space.We now need to add to that time and that’s the fourth dimension but it’s very difficult to draw a picture with four dimensions. In fact it’s not impossible and even three dimensions can be rather confusing so let’s try and keep it simple and we’re just going to think of one dimension of time and one dimension of space so it will look like this time.

which we often label x y and z and that is space.We now need to add to that time and that’s the fourth dimension but it’s very difficult to draw a picture with four dimensions. In fact it’s not impossible and even three dimensions can be rather confusing so let’s try and keep it simple and we’re just going to think of one dimension of time and one dimension of space so it will look like this time.

*We’re going to draw upwards and space we’re going to draw as the x-axis we forget about y&z for the moment just to keep things simple. So time goes up and space goes along the x-axis and that is essentially a picture of space-time now.*

*Let’s compare the two if I stand still in space then my coordinates do not*

*change. I do not move along the x-axis or the y axis or the z-axis*

I stand still and therefore that point doesn’t change.

I stand still and therefore that point doesn’t change.

**Some rules for space-time chart**

**If you standstill, time is ticking**

*But if we stand still in space-time then although my space dimension doesn’t*

*change, I stay here*on

*the x-axis (ed. points to the dot on the chart) of course time is passing by relentlessly.*

*I’m moving that direction*(ed. see arrow up)

*because although I’m not*

*moving in space I am moving in time if you standstill in the middle of a room, time is nonetheless ticking by.*

**1.**

**backwards in time is not allowed.**

Now

*there are some rules about space-time for a start. You can’t go backwards in time. So if we take our space-time chart any movement like that is forbidden because although it implies that you are moving*

*to the left in space it also suggests you’re going backwards in time and there is no known mechanism for doing that.*

*Although there are some theories that*

*suggest you might be able to find some*

*kind of hole that enables you to go back*

*to some point in the past there is no*

*mechanism for doing that and so you*

*can’t have any mote movement through*

*space time which involves going*

*backwards in time.*

*Now space-time charts are usually drawn in such a way that the speed of light is drawn at an angle of 45*

*degrees and the way you do that is you just make sure that the coordinates are appropriate.*

So for example time might be

So for example time might be

*one year and the space might be one Lightyear which is the distance light travels in a year so light travels one light-year in one year and.*

That’s how you organize the chart.

That’s how you organize the chart.

*Or you could for example have time at one*

*second and distance three times ten to*

*the eighth meters*

*because light travels*

*at three times ten to the eighth meters*

*per second*

*.*

*So we usually arrange it* (ed. the graph) just* for presentational purposes so that the **speed of light is at 45 degrees.*

**2. not allowed to exceed C in space-time**

Now here’s another rule in about our space time chart here’s the chart time versus space.

going faster than the speed of light.

It suggests it’s getting somewhere faster than light could and you therefore cannot do it so that is not allowed.

backwards in time. You’re not allowed to exceed the speed of light.

**What this means**

.

**.**

**The twin paradox – constant speed**

the twin paradox.

**They won’t celebrate their tenth birthdays**

*Imagine that this is the earth on which there are two twins and let’s imagine they just both just been born and here comes a space rocket which is traveling at speed V and V is 3/5 of the speed of light and it just so happens that this rocket has a little hook underneath it and as it goes past at 3/5 the speed of light it picks up one of these twins and carries it along at 3/5 the speed of light for a period of five years If it is traveling for five years at 3/5 of the speed of light it will travel a total distance of three lightyears now where is fit traveled that distance three light-years in five years it meets another spacecraft coming the other way also at speed V equals 3/4 C and that spacecraft also has a hook on it and as it passes this spacecraft it grabs hold of the twin and brings the twin back and that will also take five years for the return journey and as it passes the earth it releases the hook and the twin falls back down next to its brother and they are reunited after ten years and you might think that they could together celebrate their tenth birthdays each but as we will see that is not the case.*

So let’s start by drawing a very rough space-time diagram just to show what’s going on and then we’ll do it slightly more accurately so that we can understand.

So let’s start by drawing a very rough space-time diagram just to show what’s going on and then we’ll do it slightly more accurately so that we can understand.

**The first twin that remains on earth**

*what is happening is that the first twin the one that remains on earth throughout will start at the year naught and stay still basically up to year 10 and so the first twin doesn’t move along the space axis at all just stays still on the earth we’ll imagine for the moment that the earth is not moving and so the space-time direction for this twin is straight up they do not move through space they just move steadily*

through time for a period of 10 years.

through time for a period of 10 years.

**The second twin that leaves the earth**

*Now the other twin is picked up by the spacecaft and is whizzed out to this point here*(ed on tthe right side of the image above)

*then the the return spacecraft picks them up and brings them back here and the total distance that they travel is three*

lightyears.

lightyears.

**twin number one***So just to recap*

still on the earth and just travels

through time for ten years.

**twin number one**staysstill on the earth and just travels

through time for ten years.

**Twin number two***travels at 3/5 the speed of light to this point outwards three light-years and then travels back at 3/5 the speed of light and rejoins their twin on earth.*

*Now what has actually happened.*

*In my video on special relativity,*

*I explained that a moving clock appears to run slow. and I’m just going to explain that again here so that we get the full picture*

**The mirror clock**

*said that you could invent a very simple clock it would consist of two mirrors and what you do is you simply bounce*

light up and down between the two

mirrors and every time the light beam

hits a mirror it essentially causes a counter to move on and since the

distance is known and since the speed of light is invariant we know that the time to travel from the bottom mirror to the

top mirror is going to be the distance D

divided by C the speed of light.

light up and down between the two

mirrors and every time the light beam

hits a mirror it essentially causes a counter to move on and since the

distance is known and since the speed of light is invariant we know that the time to travel from the bottom mirror to the

top mirror is going to be the distance D

divided by C the speed of light.

*And that becomes in a sense our time measure one movement from the bottom mirror to the top mirror constitutes a time interval which equals D divided by C.*

**The mirror clock on a rocket, as seen from earth**

*Now suppose that apparatus is in a rocket that is going past the earth so here’s the earth with an observer on the earth and here’s the rocket*

and here’s the mirror in the rocket.

and here’s the mirror in the rocket.

*Now the rocket is moving at velocity vand for these purposes v will have to be pretty fast approaching the speed of light maybe 3/5 the speed of light.*

*So the light leaves the bottom mirror on its way up to the top mirror but before it gets there the rocket has moved so that by the time it gets to the top mirror that top mirror is now here*

*the rocket and then it’s reflected but by the time it gets to the bottom mirror the rocket has moved to here*

*and there’s the bottom mirror.*

*So to the observer on the earth what the light beam has appeared to do is not one up and down but gone like that so how do the two observers one of them inside the rocket and one of them on the earth compare these two events.*

**The person inside the rocket**says the light simply went up like that.*(ed. like the image above).*

**The person on the earth**says no it didn’t it went like that**some geometry**

*Now we can do some geometry.*

*What we’re going to do is we’re going to create a right-angled triangle*

**d***this is distance d*

that

*was the distance we had here.*(ed Between the mirrors on the rocket)

*d’**this is going to be distance we’re going to call it D Prime and of course this is also distance d.*

*vt’**What is this distance here well it’s the velocity of the rocket which is V times the time it took from to get from here to here which we’re going to call T Prime.*

**t’***it’s the time which the observer on the earth will reckon that the rocket took to get from here to here.*

*well now we can do some basic Pythagoras.*

*d’*

^{2}equals …*as we know that*

*d=ct**then we can write:*

The following operations are:

**Adding vt’ on both side and flipping side**,

OBS! (ct^{2}) should be (ct)^{2 }in the first line above.- Dividing both side with c
_{2} - Square root of both side, gives finally this equation with the
- Lorentz
**transform (sqr root (1-v**^{2}/c^{2})

** Lorentz transform
is defined as a “**lineartransformations from a coordinate frame in spacetime to another frame, that moves at a constant velocity relative to the former. (Wiki )

**The equation says that the bigger the velocity v, the bigger is the time difference between t and t’.**

*d’ = …. if*

*v= 3/5C … the relationship between the time interval for the person in the spacecraft and the time interval for the person observing*

that clock but from the earth … will be…4/5 of the

time that the person on the earth thinks

it ought to be.

that clock but from the earth … will be…4/5 of the

time that the person on the earth thinks

it ought to be.

*If five years have gone past on earth only 4 years*

will appear to have gone past in the

spacecraft.

will appear to have gone past in the

spacecraft.

two twins

*Let us suppose that there is some communication between the*

two twins.

two twins.

*The communication travels at*

the speed of light.

the speed of light.

*It cannot travel faster.*

*( ed. from left to right)*

**There can be no instantaneous****communication.***Sso let’s suppose that there is some radio waves carrying picture signals*

*but actually …*

## .

**The twin paradox – speed with accelerations.**

**Acceleration slows clock down.**

*40 years of its life in a*(ed. Accelerating)

*back on earth*

spacecraft to arrive back on earth, the clock will be

spacecraft to arrive back on earth, the clock will be

*59.000 years into the future.*

* Twins won’t see each other again*.

## .

## Section 5

## Section 6

.

## .

## conclusion

The two examples are difficult to compare as in the first constant velocity case the twin in the rocket traveled 10 years.

I the second case they talk about 40 years. How many years later would the twin be back on earth with acceleration to v=3/5C and back to v=0?

If calculation 1/4 of 59.000 years is ok, then that would equal to 14700 years on earth. Then the twins won’t met again in both cases

## .

## Contact me on Facebook

it is impossible to have a dialog on this WordPress dialog. I have tried different tools to avoid spam.

You find me on Facebook. You are welcome to contact me there. **Send me a message to present yourself** and I will be pleased to talk with you about this and other issues.