Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

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.
Please help me Dr. Math!  I desperately need your excellent words of
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 
read concerning this, so I can follow your thinking.

>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.

I'm glad to help you.

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 
the "clock paradox" which is your "twin paradox."

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/ 
Associated Topics:
College Physics

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/