# 5.5 The lorentz transformation  (Page 6/12)

 Page 6 / 12

For any event that has a space-like separation from the event at the origin, it is possible to choose a time axis that will make the two events occur at the same time, so that the two events are simultaneous in some frame of reference. Therefore, which of the events with space-like separation comes before the other in time also depends on the frame of reference of the observer. Since space-like separations can be traversed only by exceeding the speed of light; this violation of which event can cause the other provides another argument for why particles cannot travel faster than the speed of light, as well as potential material for science fiction about time travel. Similarly for any event with time-like separation from the event at the origin, a frame of reference can be found that will make the events occur at the same location. Because the relations

$\text{Δ}{s}_{AC}^{2}={\left({x}_{A}-{x}_{B}\right)}^{2}+{\left({x}_{A}-{x}_{B}\right)}^{2}+{\left({x}_{A}-{x}_{B}\right)}^{2}-{\left(c\text{Δ}t\right)}^{2}>0$

and

$\text{Δ}{s}_{AB}^{2}={\left({x}_{A}-{x}_{B}\right)}^{2}+{\left({x}_{A}-{x}_{B}\right)}^{2}+{\left({x}_{A}-{x}_{B}\right)}^{2}-{\left(c\text{Δ}t\right)}^{2}<0.$

are Lorentz invariant, whether two events are time-like and can be made to occur at the same place or space-like and can be made to occur at the same time is the same for all observers. All observers in different inertial frames of reference agree on whether two events have a time-like or space-like separation.

## The twin paradox seen in space-time

The twin paradox discussed earlier involves an astronaut twin traveling at near light speed to a distant star system, and returning to Earth. Because of time dilation, the space twin is predicted to age much less than the earthbound twin. This seems paradoxical because we might have expected at first glance for the relative motion to be symmetrical and naively thought it possible to also argue that the earthbound twin should age less.

To analyze this in terms of a space-time diagram, assume that the origin of the axes used is fixed in Earth. The world line of the earthbound twin is then along the time axis.

The world line of the astronaut twin, who travels to the distant star and then returns, must deviate from a straight line path in order to allow a return trip. As seen in [link] , the circumstances of the two twins are not at all symmetrical. Their paths in space-time are of manifestly different length. Specifically, the world line of the earthbound twin has length $2c\text{Δ}t,$ which then gives the proper time that elapses for the earthbound twin as $2\text{Δ}t.$ The distance to the distant star system is $\text{Δ}x=v\text{Δ}t.$ The proper time that elapses for the space twin is $2\text{Δ}\tau$ where

${c}^{2}\text{Δ}{\tau }^{2}=-\text{Δ}{s}^{2}={\left(c\text{Δ}t\right)}^{2}-{\left(\text{Δ}x\right)}^{2}.$

This is considerably shorter than the proper time for the earthbound twin by the ratio

$\begin{array}{cc}\hfill \frac{c\text{Δ}\tau }{c\text{Δ}t}& =\sqrt{\frac{{\left(c\text{Δ}t\right)}^{2}-{\left(\text{Δ}x\right)}^{2}}{{\left(c\text{Δ}t\right)}^{2}}}=\sqrt{\frac{{\left(c\text{Δ}t\right)}^{2}-{\left(v\text{Δ}t\right)}^{2}}{{\left(c\text{Δ}t\right)}^{2}}}\hfill \\ & =\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}=\frac{1}{\gamma }.\hfill \end{array}$

consistent with the time dilation formula. The twin paradox is therefore seen to be no paradox at all. The situation of the two twins is not symmetrical in the space-time diagram. The only surprise is perhaps that the seemingly longer path on the space-time diagram corresponds to the smaller proper time interval, because of how $\text{Δ}\tau$ and $\text{Δ}s$ depend on $\text{Δ}x$ and $\text{Δ}t.$

#### Questions & Answers

What is photoelectric
light energy (photons) through semiconduction of N-P junction into electrical via excitation of silicon purified and cristalized into wafers with partially contaminated silicon to allow this N-P function to operate.
Michael
i.e. Solar pannel.
Michael
If you lie on a beach looking at the water with your head tipped slightly sideways, your polarized sunglasses do not work very well.Why not?
it has everything to do with the angle the UV sunlight strikes your sunglasses.
Jallal
this is known as optical physics. it describes how visible light, ultraviolet light and infrared light interact when they come into contact with physical matter. usually the photons or light upon interaction result in either reflection refraction diffraction or interference of the light.
Jallal
I hope I'm clear if I'm not please tell me to clarify further or rephrase
Jallal
what is bohrs model for hydrogen atom
hi
Tr
Hello
Youte
Hi
Nwangwu-ike
hi
Siddiquee
hi
Omar
helo
Mcjoi
what is the value of speed of light
1.79×10_¹⁹ km per hour
Swagatika
what r dwarf planet
what is energy
কাজের একক কী
Jasim
কাজের একক কী
Jasim
friction ka direction Kaise pata karte hai
friction is always in the opposite of the direction of moving object
Punia
A twin paradox in the special theory of relativity arises due to.....? a) asymmetric of time only b) symmetric of time only c) only time
b) symmetric of time only
Swagatika
fundamental note of a vibrating string
every matter made up of particles and particles are also subdivided which are themselves subdivided and so on ,and the basic and smallest smallest smallest division is energy which vibrates to become particles and thats why particles have wave nature
Alvin
what are matter waves? Give some examples
according to de Broglie any matter particles by attaining the higher velocity as compared to light'ill show the wave nature and equation of wave will applicable on it but in practical life people see it is impossible however it is practicaly true and possible while looking at the earth matter at far
Manikant
a centeral part of theory of quantum mechanics example:just like a beam of light or a water wave
Swagatika
Mathematical expression of principle of relativity
given that the velocity v of wave depends on the tension f in the spring, it's length 'I' and it's mass 'm'. derive using dimension the equation of the wave
What is the importance of de-broglie's wavelength?
he related wave to matter
Zahid
at subatomic level wave and matter are associated. this refering to mass energy equivalence
Zahid
it is key of quantum
Manikant
how those weight effect a stable motion at equilibrium
how do I differentiate this equation- A sinwt with respect to t
just use the chain rule : let u =wt , the dy/dt = dy/du × du/dt : wA × cos(wt)
Jerry
I see my message got garbled , anyway use the chain rule with u= wt , etc...
Jerry