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Δ τ 2 = Δ s 2 = ( Δ t ) 2 .

Therefore Δ τ is the time interval Δ t in the frame of reference where both events occur at the same location. It is the same interval of proper time discussed earlier. It also follows from the relation between Δ s and that Δ τ that because Δ s is Lorentz invariant, the proper time is also Lorentz invariant. All observers in all inertial frames agree on the proper time intervals between the same two events.

Check Your Understanding Show that if a time increment dt elapses for an observer who sees the particle moving with velocity v , it corresponds to a proper time particle increment for the particle of d τ = γ d t .

Start with the definition of the proper time increment:
d τ = ( d s ) 2 / c 2 = d t 2 ( d x 2 + d x 2 + d x 2 ) / c 2 .
where ( dx , dy , dx , cdt ) are measured in the inertial frame of an observer who does not necessarily see that particle at rest. This therefore becomes
d τ = ( d s ) 2 / c 2 = d t 2 [ ( d x ) 2 + ( d y ) 2 + ( d z ) 2 ] / c 2 = d t 1 [ ( d x d t ) 2 + ( d y d t ) 2 + ( d z d t ) 2 ] / c 2 = d t 1 v 2 / c 2 d t = γ d τ .

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The light cone

We can deal with the difficulty of visualizing and sketching graphs in four dimensions by imagining the three spatial coordinates to be represented collectively by a horizontal axis, and the vertical axis to be the ct- axis. Starting with a particular event in space-time as the origin of the space-time graph shown, the world line of a particle that remains at rest at the initial location of the event at the origin then is the time axis. Any plane through the time axis parallel to the spatial axes contains all the events that are simultaneous with each other and with the intersection of the plane and the time axis, as seen in the rest frame of the event at the origin.

It is useful to picture a light cone on the graph, formed by the world lines of all light beams passing through the origin event A , as shown in [link] . The light cone, according to the postulates of relativity, has sides at an angle of 45 ° if the time axis is measured in units of ct , and, according to the postulates of relativity, the light cone remains the same in all inertial frames. Because the event A is arbitrary, every point in the space-time diagram has a light cone associated with it.

A space time diagram has a space on the horizontal axis and time on the vertical axis. The light cone is a vertical cone above the origin with its vertex at the origin and sides at 45 degrees, and another vertical cone below the origin with its vertex also at the origin. Three events are shown. Event A is at the origin. Event B is inside the light cone. Event C is outside the light cone.
The light cone consists of all the world lines followed by light from the event A at the vertex of the cone.

Consider now the world line of a particle through space-time. Any world line outside of the cone, such as one passing from A through C , would involve speeds greater than c , and would therefore not be possible. Events such as C that lie outside the light cone are said to have a space-like separation from event A . They are characterized by:

Δ s A C 2 = ( x A x B ) 2 + ( x A x B ) 2 + ( x A x B ) 2 ( c Δ t ) 2 > 0 .

An event like B that lies in the upper cone is reachable without exceeding the speed of light in vacuum, and is characterized by

Δ s A B 2 = ( x A x B ) 2 + ( x A x B ) 2 + ( x A x B ) 2 ( c Δ t ) 2 < 0 .

The event is said to have a time-like separation from A . Time-like events that fall into the upper half of the light cone occur at greater values of t than the time of the event A at the vertex and are in the future relative to A . Events that have time-like separation from A and fall in the lower half of the light cone are in the past, and can affect the event at the origin. The region outside the light cone is labeled as neither past nor future, but rather as “elsewhere.”

Questions & Answers

What is the importance of de-broglie's wavelength?
Mukulika Reply
he related wave to matter
Zahid
at subatomic level wave and matter are associated. this refering to mass energy equivalence
Zahid
how those weight effect a stable motion at equilibrium
Nonso Reply
how do I differentiate this equation- A sinwt with respect to t
Evans Reply
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
de broglie wave equation
LoNE Reply
vy beautiful equation
chandrasekhar
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fitsum Reply
when you consider systems consisting of fixed charges
Sherly
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Arnulfo Reply
a unit of weight for precious stones and pearls, now equivalent to 200 milligrams.
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a science that deals with the composition, structure, and properties of substances and with the transformations that they undergo.
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Mrs Reply
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Abakar
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acids and bases
Navya
How does unpolarized light have electric vector randomly oriented in all directions.
Tanishq Reply
unpolarized light refers to a wave collection which has an equal distribution of electric field orientations for all directions
pro
In a grating, the angle of diffraction for second order maximum is 30°.When light of wavelength 5*10^-10cm is used. Calculate the number of lines per cm of the grating.
Micheal Reply
OK I can solve that for you using Bragg's equation 2dsin0over lander
ossy
state the law of gravity 6
cletus Reply
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Ebe Reply
its just a technique used for the protection of a metal from corrosion by making it cathode of an electrochemical cell.
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what is interferometer
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Practice Key Terms 4

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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