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### Special Relativity, Light Consistency, and Time Dilation

```Date: 07/30/2005 at 07:23:43
From: Kamil
Subject: Special Relativity

Is time dilation a result of light consistency or is light consistency
a result of time dilation?  Because I have accepted the second, and
from it came a paradox that is stuck in my mind.  Firstly, it says
that light moves at the same speed in every DIRECTION for every
observer because the observers have different notions of time.  And
that the time dilation is directly due to speed (not velocity since
direction doesn't matter).

So under these circumstances (if they are right) what would happen if
I traveled north and chased a light beam heading north and at the same
time saw a light beam heading south (towards me).

The first postulate of relativity states that the light beams would
move (in my perspective) at the same speed. And SR says that is
because time has dilated for me.  But if time dilation is the only
factor that makes the light beam heading north go at 'c', then it
clearly wouldn't make the light beam heading south heading 'c',
contradicting the first postulate.

I have tried to solve this problem by CAREFULLY using Microsoft
Powerpoint and including all the postulates of SR in this gedanken
(mind experiment).  But it just strengthens my belief in the
contradiction between time dilation and the first postulate of SR.
wisdom!!

```

```
Date: 08/03/2005 at 21:45:25
From: Doctor Rick
Subject: Re: Special Relativity

Hi, Kamil.

Your thoughtful questions deserve an answer, and I am sorry I haven't
been able to provide one until now.  I see that you raised some
related issues in an earlier message:

>Imagine a light beam moving to the left of a person and another light
>beam moving from the left towards the person.  Now what would happen
>if the man started moving towards the left.  Will the light beams
>still move at c?
>
>According to special relativity, they will.  But according to the
>explanation of why, is that time flows at a slower rate when moving
>at a faster speed.  So if the time moves slower for the man, than
>shouldn't the first mentioned light beam travel faster than the
>second mentioned light beam?
>
>The reason why can also be seen in the time dilation equation, where
>v squared is incorporated, this means that the direction of the
>observer shouldn't matter, so the direction of the light beam doesn't
>matter on the duration measured.
>
>My final thought is that if time flows slower in such a way that
>light travels at the same speed, than that would only make a light
>beam that travels in the same direction move at c.  But not a
>different direction.

It isn't that the speed of light is constant because time is dilated;
rather the reverse.

Constancy of the speed of light is a much simpler concept than the
time dilation formula, so it is better to see this as fundamental. It
was experiments suggesting that the speed of light is constant (along
with Maxwell's equations for propagation of light, which imply
constancy of the speed of light) that gave the impetus for
development of special relativity.

The assumption that the speed of light is constant has many
implications: not just time dilation, but also length contraction and
relativity of simultaneity. If you were told that time dilation alone
could account for constancy of the speed of light, that was an overly
simplified explanation.

You observed the distinction between a beam of light going in the
same direction as the man versus the opposite direction. This
observation shows me that the simultaneity issue is at the root of
the explanation, since it depends on the direction of motion, while
time dilation (as you have stressed) and length contraction do not.

I'm curious how you did your "gedankenexperiments" and what you
consider to be the postulates of special relativity. As I said,
constancy of the speed of light in all reference frames is the
central postulate. If you start from that postulate, there is nothing
to prove. From that postulate are derived the Lorentz transformation
equations, which are the only way I can think of to set up a computer
simulation. Using them, it is easy to show that the speed of light is
the same in the stationary ("laboratory") frame and in the man's
("rocket") frame. I'd like to know, therefore, what you did that
indicates to you that there is a contradiction.

Here is a very brief account of where the simple time-dilation
explanation fails. To measure the speed of the two light beams in the
rocket frame, we need to measure the position of each beam at two
times. The difference in the positions divided by the difference in
the times is the speed of the light beam.

To simplify the calculation, let's say that instead of coming toward
the man, they come from two flashlights carried by the man, one
pointed forward and the other pointed backward. I'll pick the two
times: t = 0, when the beams leave the flashlights (so that x=0 for
both beams), and t = 1 second. In the laboratory frame, at time 1
second, the forward beam is then at x = 300,000,000 meters (I'll call
that c meters); the backward beam is at x = -c meters; and the man
himself is at v meters (assuming his velocity is v meters per
second). So by the simplistic calculation, the forward beam is (c-v)
meters ahead of the man, and the backward beam is (c+v) meters behind
the man. The speeds of the beams relative to the man are thus (c-v)/1
meters/second and (c+v)/1 meters/second. If we adjust the time t=1
for time dilation, both speeds are changed in the same direction; but
one is less than c and the other is greater than c, so they need to
be changed in different directions in order for both to become c. The
same happens if we adjust the distances (c+v) and (c-v) for length
contraction: both speeds are adjusted in the same direction. These
adjustments cannot cause the two light beams to travel at the same
speed. This, I believe, is the argument you are making.

However, we have made a major error in talking about "the time" when
something happens. We've assumed that the two events (1: forward beam
reaches x=c in the laboratory frame, and 2: backward beam reaches x=-
c in the reference frame) happen at the same time in the man's
(rocket) frame, as they do in the stationary (laboratory) frame. They
don't! To the man, event (1) occurs first, before event (2). Since
event (1) is closer to the man than event (2), this is all consistent
with the two beams traveling at the same speed -- namely, c meters
per second. The Lorentz transformation equations confirm this
quantitatively.

I hope you'll write back, tell me more precisely what you have done
and on what postulates it was based, and discuss this matter further.
It's a challenge to grasp the concepts of relativity; it is very easy
to make assumptions that are invalid in relativity (such as the
simultaneity of the two events in my discussion above) without even
noticing the assumption, because we aren't accustomed to a
(noticeably) relativistic world.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/

```

```
Date: 08/04/2005 at 02:29:28
From: Kamil
Subject: Special Relativity

Thank you Dr. Math, I have been vigorously searching for the answer,
but yours was truly the best and clearest.  I just wanted to know 2
more things:

1) Einstein's hypothetical clock (the one where Pythagoras' theorem
was used to calculate Lorentz factor) is said to show the exact time
dilation and not some kind of mechanical flaw.  Does that mean that
if I turned the clock around 30 or 60 or even 90 degrees, it would
show the same result?

2) The Lorentz factor says that if you were to move at 0.5c time
would dilate to 20% of the previous rate.  But shouldn't it be half
in order to make the light beam go at 'c'?  I think this might have
something to do with the matter that you answered in your reply, but
I just want to be sure.

```

```
Date: 08/04/2005 at 16:16:00
From: Doctor Rick
Subject: Re: Special Relativity

Hi, Kamil.

I probably haven't read the same things you have read, so I don't
know what "hypothetical clock" you have read about, or how it was
used to calculate the Lorentz factor. However, in general, the clocks
considered in relativistic thought experiments are regarded as
theoretically perfect recorders of the time at which an event occurs
at the location of the clock. I don't understand what significance
turning the clock may have in your mind, but I wouldn't think it
would matter at all. Maybe you need to tell me more about what you

>2) The Lorentz factor says that if you were to move at 0.5c time
>would dilate to 20% of the previous rate.  But shouldn't it be half
>in order to make the light beam go at 'c'?

The time dilation effect is this: Suppose a rocket is moving with
velocity v relative to the laboratory frame. In the rocket frame two
events occur at the same place but 1 second apart. Then in the
laboratory frame, the time between the two events is

1/sqrt(1-(v/c)^2)

If v = 0.5c, this works out to

1/sqrt(1 - 0.5^2) = 1/sqrt(1 - 1/4)
= 1/sqrt(3/4)
= sqrt(4/3)
= 1.154 seconds

That's about 15% longer than the 1 second measured in the rocket
frame.

I don't know what calculations you did to conclude that it should be
10% (if that is what you mean), but as I said before, time dilation
alone does not account for the speed of light being constant. We must
consider the Lorentz transformation equations, which are more general
than time dilation, incorporating the other effects I mentioned. Have
you seen the Lorentz transformation equations?

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/

```

```
Date: 08/05/2005 at 02:27:12
From: Kamil
Subject: Special Relativity

Hello,

I learned the Lorentz equation from Einstein's method where he created
an imaginary clock which consisted of two mirrors facing each other,
then light was released from one of the mirrors and the photon kept
on reflecting from one mirror to the other.  This clock was then sent
to a velocity of 'v' perpendicular to the photon's direction.  This
created a right angled triangle from where the time dilation
equation could be derived (also length and mass).

The whole foundation of this was the fact that light consistency is
caused by time dilation.  And I was just wondering if the direction of
motion was the same as the photon (turned 90 degrees) would it still
record the same time dilation (because that clock is said to measure
time perfectly).  But please do tell me the Lorentz equations.

If you say that time dilation is the factor that causes light
consistency, then can you please explain what does?

```

```
Date: 08/05/2005 at 08:20:57
From: Doctor Rick
Subject: Re: Special Relativity

Hi, Kamil.

The clock will measure time exactly the same if it is rotated. The
clock was oriented in this particular direction only so that the
computation could be done easily from the most basic principles, so
that it could be used to derive the Lorentz transformation equations,
or part of them. If the clock is oriented differently, we could
confirm that it measures time the same way by applying the Lorentz
transformation, but we can't do that until after the transformation
has been derived.

You're still thinking backward: light consistency is not *caused* by
time dilation, it *causes* time dilation. What's special about this
thought experiment is that time dilation is the *only* effect that
light speed constancy has on the setup. The length dimensions
perpendicular to the direction of motion are not contracted,
therefore one leg of the right triangle is the same length in both
reference frames. And the two events whose time difference we
calculate are at the *same location* in the rocket frame, so that
relativity of simultaneity does not enter the picture.

The Lorentz transformation equations are as follows. Let the
coordinates measured in the laboratory frame be (x,y,z,t) while the
coordinates measured in the rocket frame are (x',y',z',t'). Set up
the coordinate systems so that an event occurring at time 0 and
location (0,0,0) in the laboratory frame is also measured at time 0
and location (0,0,0) in the rocket frame. Align the coordinate
systems so that the rocket is moving in the positive x direction, and
the x' axis is parallel to the x axis. Then

x = (x' + (v/c)t')/sqrt(1-(v/c)^2)
y = y'
z = z'
t = ((v/c)x' + t')/sqrt(1-(v/c)^2)

These transformation equations can be used to analyze any problem in
relativity by choosing the appropriate coordinates of an event. For
instance, let the events be (0,0,0,0) and (0,0,0,t') in the rocket
frame; that is, they both occur at the origin of space coordinates,
and they are separated by t seconds. Then we find that the
coordinates of the events in the laboratory frame are (0,0,0,0) and

x = (v/c)t'/sqrt(1-(v/c)^2)
y = 0
z = 0
t = t'/sqrt(1-(v/c)^2)

The difference in time between the events as measured in the
laboratory frame is increased by the factor 1/sqrt(1-(v/c)^2). That's
time dilation.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/

```

```
Date: 08/06/2005 at 00:54:46
From: Kamil
Subject: Thank you (Special Relativity)

I thank you dearly for you have cleared my misconception of
relativity.  When I first learned about relativity I thought time
dilation was a result of constancy of 'c' but then I changed my mind
to the opposite since it would just seem as though time dilation was
just an illusion.

And that raises another question, in the so called 'twin paradox' (I
know how it is not a paradox) would one twin really be younger than
the other or would they just seem younger to each other when they are
at different inertial frames?

Because if you say that time dilation is due to the speed of light,
than time dilation would only be an illusion so they would be the same
age after they are brought back to the same frame (in some books it
says that one will be younger when they are brought back to the same
frame(1) while others say they will be the same biological age(2)).

I think the reason why group (1) thinks the way they do is because
they think time dilation comes first and light consistency comes
second, whereas group (2) thinks that light consistency comes first.
If time dilation is just an illusion, then why did relativity
revolutionize physics when refraction hasn't revolutionized optics?

I must say that is the only thing that still seems confusing to me.  I
know that some might say time dilation is not an illusion since there
is no preferred inertial frame.  So, is time dilation an illusion or
not?  And if it is, what's so special about it?

```

```
Date: 08/06/2005 at 10:45:29
From: Doctor Rick
Subject: Re: Thank you (Special Relativity)

Hi, Kamil.

Time dilation is not an illusion in that it is not just a matter of
how we perceive things, but how they are physically. Often relativity
is presented in terms of how one person "sees" things, but that is a
misrepresentation. Better presentations of relativity talk in terms
of physical (though idealized) clocks set up on a grid in space, each
recording the time that an event occurs in its immediate vicinity.
This has nothing to do with a person's perception.

I haven't seen references in which an author believes that the twins
of the "paradox" will be the same age when brought together, but I
doubt that a difference is due to whether they put light constancy or
time dilation first. Everyone I know puts light consistency first--
an application of Occam's Razor (look that up), if nothing else. I
certainly do, and I believe that the twins will be different ages
when they come together.

The key to this conclusion is that in order for the twins to come
together, they *cannot* both be in inertial reference frames the
whole time. Take a look at the following page from the Dr. Math
Archives; it is long and rambling, concerning many topics of special
and general relativity, but about 2/3 of the way down I discuss

I'm sure you'll find more things to be confused about as you continue
to study, but keep going! Your curiosity will be rewarded.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/

```

```
Date: 01/01/2006 at 21:41:13
From: Kamil
Subject: Special Relativity

Thank you for your reply.  Here is an article I read:

Relativity of Simultaneity and Distance
http://www.bun.kyoto-u.ac.jp/~suchii/Einstein/rel.TS.html

Because the train is moving towards B, the observer in the train will
see B occur first if the A and B were simultaneous in the laboratory
frame (I understand why). But what if instead there was a flash bulb
in the middle of the train?

The light would reach A first, so what event does the person on the
train see happen first?  But if the light was to come from two
different sources situated on A and B.  Then B would happen first.  So
what does the observer on the train see happen first, A or B?

The problem may be solved if we say that seeing something is when the
light beam reaches us.

```

```
Date: 01/02/2006 at 10:15:40
From: Doctor Rick
Subject: Re: Special Relativity

Hi, Kamil.

... but we've already agreed that "seeing" (when a photon enters the
observer's eye) is NOT what relativity of simultaneity is about.
It's about when events (such as a flash bulb going off) occur, as
deduced by an observer in a particular reference frame. The
deduction may be based either on tracing back along a light beam,
knowing the speed of light, or on checking a time-stamped video
recording made at the location of the event, or by any other means.

You are implicitly redefining "events A and B": now they are when
the flash of light, initiated at the center of the train M, reaches
each end of the train car.

Yes, event A occurs first in the stationary reference frame, because
the rear end of the train car is closer to M by the time the light
flash reaches it.

In the reference frame that is moving with the train, events A and B
are simultaneous, because the light travels equal distances at equal
speeds to reach the two ends of the car.

Why is this a "problem" for you? If two events are simultaneous in
the stationary reference frame, the one occurring farther forward
occurs first in the moving frame. If two events are simultaneous in
the moving frame, the one occurring farther forward occurs second in
the stationary frame. These are consistent observations: always the
event that is farther forward occurs sooner, in comparison to the
event that is farther back, in the moving frame than in the
stationary frame.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
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