SH1690
The Consequences of Special Relativity
As a result of relativistic motion, Einstein has predicted in his postulates that moving at around the
speed of light 𝑐 has consequences. These consequences are,
1. Relative Simultaneity – It means that one (1) frame of reference views two (2) relativistic
objects to be simultaneous, while the other does not with respect to the first. Simply put, it
means the first observer observing two (2) moving objects to be moving at the same time, while
the other observer only sees one (1) moving and the other resting.
2. Time Dilation – This means that time slows down to a halt if an object is moving very fast. In
short, if an object is moving at around the speed of light (c), then time ultimately stops for that
object.
3. Length Contradiction – It means that moving very fast results in a shortening of the object’s
length.
These consequences are represented by mathematical formulas and each of them are using the
Lorentz Transformation variable (or 𝛾).
INVERSE TRANSFORMATION EQUATIONS
Consider the following variables,
𝑥 ′ = 𝛾(𝑥 − 𝑣𝑡)
𝑦′ = 𝑦
𝑧′ = 𝑧
𝑡 ′ = 𝛾 (𝑡 −
𝛾=
𝑣𝑥
)
𝑐2
1
2
√1 − 𝑣2
𝑐
These variables are equal in value against their respective S frame variables. Only the
Transformation variable remains constant.
S’
S
1. Relative Simultaneity – Given the diagram
at right, where two (2) events are shown to
have happened at the same time. This is the
instance where simultaneity is to be
observed. Using the equations clocked at
around the S’, let us determine if
t1'
x1’
t2'
x2’
x
o
x'
o'
x1
x2
t
09 Handout 3
*Property of STI
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simultaneity is apparent in the case of relativity. Given are the equations,
𝑥′
𝑦′
𝑧′
𝑡′
= 𝑥 − 𝑣𝑡
=𝑦
=𝑧
=𝑡
And the inverse transformation for static time (𝑡),
𝑡 ′ = 𝛾 (𝑡 −
𝑣𝑥
)
𝑐2
The value of each event is as follows,
𝑡 ′ = 𝛾 (𝑡 −
𝑣𝑥
)
𝑐2
𝒕𝟏 ′ = 𝜸 (𝒕 −
𝒗𝒙𝟏
) ← 𝒔𝒊𝒎𝒖𝒍𝒕𝒂𝒏𝒆𝒊𝒕𝒚 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝒇𝒐𝒓 𝒙𝟏
𝒄𝟐
𝒕𝟐 ′ = 𝜸 (𝒕 −
𝒗𝒙𝟐
) ← 𝒔𝒊𝒎𝒖𝒍𝒕𝒂𝒏𝒆𝒊𝒕𝒚 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝒇𝒐𝒓 𝒙𝟐
𝒄𝟐
The equations look rather similar, but in theory it is not. If in Classical Physics, simultaneity is
possible because of time’s invariance, in Special Relativity, simultaneity cannot be true because it
is subjective to the observer. Note that everything that is moving to one (1) observer, to the other
it remains static. In short, simultaneity really depends on who is observing the events.
2. Time Dilation – Given the diagram at left, with a swinging pendulum as the source of event.
The pendulum is swinging at a
constant velocity (v), and is being
used to determine time (t). While it
t1
was moving, the pendulum also rests
t2
on the moving frame (S’). Judging
by the way it is being measured, time
can be laid out mathematically (in
the static frame) as,
x
v →
x'
S
S’
∆𝑡 = 𝑡2 − 𝑡1 (1)
In the moving frame (S’), the time is
inversed as,
∆𝑡′ = 𝑡′2 − 𝑡′1 (2)
In the case of relativity, time is not absolute. It is subject to change because time is connected with
space and, as a result, shows a slight difference between observers. By adding the Lorentz
Transformation variable, time is said to be calculated (in the moving frame) as,
09 Handout 3
*Property of STI
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𝑡′ = 𝛾 (𝑡 −
𝑣𝑥
) (3)
𝑐2
Combining equations (2) and (3),
∆𝑡′ = 𝑡′2 − 𝑡′1
∆𝑡′ = [𝛾 (𝑡′2 −
𝑣𝑥
𝑣𝑥
)]
−
[𝛾
(𝑡′
−
)]
1
𝑐2
𝑐2
∆𝑡′ = 𝛾𝑡2′ −
𝛾𝑣𝑥
𝛾𝑣𝑥
− 𝛾𝑡1′ + 2
2
𝑐
𝑐
𝑅𝑒𝑎𝑟𝑟𝑎𝑛𝑔𝑖𝑛𝑔 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒𝑠, 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑏𝑒𝑐𝑜𝑚𝑒𝑠,
∆𝑡′ = 𝛾𝑡2′ − 𝛾𝑡1′ −
𝛾𝑣𝑥 𝛾𝑣𝑥
+ 2
𝑐2
𝑐
𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑐𝑎𝑛𝑐𝑒𝑙𝑎𝑡𝑖𝑜𝑛, 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑔𝑖𝑣𝑒𝑠,
∆𝑡 ′ = 𝛾𝑡2′ − 𝛾𝑡1′
∆𝑡′ = 𝛾(𝑡2′ − 𝑡1′ )
𝑆𝑖𝑛𝑐𝑒 (𝑡2′ − 𝑡1′ ) = ∆𝑡 ′ , (𝑡2′ − 𝑡1′ ) = (𝑡2 − 𝑡1 ), 𝑎𝑛𝑑 (𝑡2 − 𝑡1 ) = ∆𝑡,
∴ ∆𝒕′ = 𝜸∆𝒕 ← 𝑻𝒊𝒎𝒆 𝑫𝒊𝒍𝒂𝒕𝒊𝒐𝒏
By assigning new variables to replace ∆𝑡 and ∆𝑡′, we make it easier to remember. Let us assign 𝑡0
for ∆𝑡, and 𝑡𝑢 for ∆𝑡 ′ . The equation ∆𝑡 ′ = 𝛾∆𝑡 looks complicated. For us to know what will
happen, let us expand the equation, so that we might better understand time dilation.
Here is the short equation,
∆𝑡 ′ = 𝛾∆𝑡
𝑈𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑛𝑒𝑤 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠,
𝑡𝑢 = 𝛾𝑡0
𝑆𝑖𝑛𝑐𝑒 𝛾 =
09 Handout 3
1
2
√1 − 𝑣2
𝑐
, 𝑤𝑒 𝑔𝑒𝑡,
*Property of STI
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𝑡𝑢 = 𝑡0
∴ 𝒕𝒖 =
1
2
√1 − 𝑣2
(
𝑐 )
𝒕𝟎
𝟐
√𝟏 − 𝒗𝟐
𝒄
← 𝑻𝒊𝒎𝒆 𝑫𝒊𝒍𝒂𝒕𝒊𝒐𝒏
The value of 𝑡𝑢 represents the moving observer’s time as measured by the static observer. In
accordance to time dilation, by traveling at speeds close to the speed of light, time is slowing down.
Traveling at the speed of light, time stops. And, by not moving at all, time moves as normal.
3. Length Contradiction – Using the same diagram in #1 (Relative Simultaneity), given are two
(2) points of events, each are spaced apart in a certain length (𝑙) measured in a given time (𝑡).
The measurement is being done in the S frame. Plotting this mathematically,
∆𝑙 = 𝑥2 − 𝑥1 (3)
The observer in the S’ frame also performed the same measurement of length at the same time.
Plotting it mathematically,
∆𝑙′ = 𝑥′2 − 𝑥′1 (4)
In this case, let us use 𝑥 = 𝛾(𝑥′ − 𝑣𝑡). Combining it with (4), it gives us,
∆𝑙 = 𝛾(𝑥2 − 𝑥1 )
𝑥 = 𝛾(𝑥′ − 𝑣𝑡)
∆𝑙 = 𝛾[(𝑥2 − 𝑣𝑡) − (𝑥1 − 𝑣𝑡)]
∆𝑙 = 𝛾(𝑥2 − 𝑣𝑡) − 𝛾(𝑥1 − 𝑣𝑡)
∆𝑙 = 𝛾𝑥2 − 𝛾𝑣𝑡 − 𝛾𝑥1 + 𝛾𝑣𝑡
𝑅𝑒𝑎𝑟𝑟𝑎𝑛𝑔𝑖𝑛𝑔 𝑡ℎ𝑒 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠,
∆𝑙 = 𝛾𝑥2 − 𝛾𝑥1 − 𝛾𝑣𝑡 + 𝛾𝑣𝑡
𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑐𝑎𝑛𝑐𝑒𝑙𝑎𝑡𝑖𝑜𝑛,
∆𝑙 = 𝛾𝑥2 − 𝛾𝑥1
∆𝑙 = 𝛾(𝑥2 − 𝑥1 )
09 Handout 3
*Property of STI
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𝑆𝑖𝑛𝑐𝑒 ∆𝑙 𝑖𝑠 𝑎𝑙𝑠𝑜 𝑒𝑞𝑢𝑎𝑙 𝑖𝑛 𝑣𝑎𝑙𝑢𝑒 𝑡𝑜 𝑖𝑡𝑠 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 (∆𝑙 ′ ),
∆𝒍 = 𝜸∆𝒍′
Same as in Time Dilation, we have to assign new variables to easily understand this concept. For
∆𝑙, we shall assign 𝑙0 , and 𝑙𝑢 for ∆𝑙′. Expanding the equation above – and using the new assigned
variables,
𝑙0 = 𝛾𝑙𝑢
𝑆𝑖𝑛𝑐𝑒 𝛾 =
1
2
√1 − 𝑣2
𝑐
𝑙0 = 𝑙𝑢
𝑙0 =
, 𝑤𝑒 𝑔𝑒𝑡,
1
2
𝑣
√
( 1 − 𝑐2 )
𝑙𝑢
2
√1 − 𝑣2
𝑐
But, in this case, it is set in the moving observer’s point of view. To truly demonstrate the effect,
we have to rearrange this in order to get length contraction, as what the static observer will measure
against a moving object.
Transposing the Transformation variable,
𝑙0 =
𝑙𝑢
2
√1 − 𝑣2
𝑐
𝑙0 √1 −
𝑣2
= 𝑙𝑢
𝑐2
𝑅𝑒𝑎𝑟𝑟𝑎𝑛𝑔𝑖𝑛𝑔 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛,
∴ 𝒍𝒖 = 𝒍𝟎 √𝟏 −
09 Handout 3
𝒗𝟐
← 𝑳𝒆𝒏𝒈𝒕𝒉 𝑪𝒐𝒏𝒕𝒓𝒂𝒄𝒕𝒊𝒐𝒏
𝒄𝟐
*Property of STI
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Now, in the final equation, it is now evident to the static observer that the moving observer is
experiencing length contraction because, in the simplest of sense, if the static observer measures
the length of a moving object at a certain point of origin, he will get a different measurement than
the moving observer.
References:
Bauer, W., & Westfall, G. D. (2016). General physics 1 (2nd ed.). Quezon City: ABIVA Publishing House.
Boundless (2016). Boundless physics. Retrieved from
https://www.boundless.com/physics/textbooks/boundless-physics-textbook/special-relativity27/consequences-of-special-relativity-179/length-contraction-657-6319/
Freedman, R. A., Ford, A. L., & Young, H. D. (2012). Sears and zemansky's university physics (with Modern
physics) (13th ed.). Addison-Wesley.
HyperPhysics (2016). Lorentz transformation. Retrieved 2017, January 13 from the HyperPhysics Classroom:
http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/ltrans.html
09 Handout 3
*Property of STI
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