Time Dilation And Relativity

[bezdomni]

I like relativity and wanted to make it somewhat more accesible to people. The only math I use in this is some pretty basic algebra, so you don't have to be comradered to understand it. :P

If you have any questions feel free to ask...also, if it looks like I fucked up go ahead and point it out. The only reason I wrote this is because I didn't have enough money to buy breakfast so I had a lot of time to kill this morning. :P


Relativity and Time Dilation

Time dilation, that is, the “contraction” of time, is one of the most interesting and important aspects of Einstein’s theory of special relativity. Time dilation occurs when any object moves in relation to another object…but in your every day life, time dilation is negligible. In fact, time dilation doesn’t become observable until something approaches the speed of light (which is denoted by c, since the speed of light is an absolute constant).

For an example, let’s say we have a pair of twins, where one is male and the other is female. Now, let’s say the female twin is going to travel to a star which is exactly 10 light years away in a spacecraft that can travel at 90% the speed of light, while her brother remains on Earth.

First we will have to calculate the amount of time it will take her to get to the star without calculating for time dilation. The equation is t = d/v (time = distance/velocity). So:
t = 10/0.9c
t = 11.11 (approximately).

From Earth’s frame of reference, her trip will take approximately 11 years.

Now we can calculate for time dilation, using the equation t’ = (t)sqrt{1 – v^2} ; or, dilated time = (time) x (square root of one minus velocity squared.)

Now it is a simple matter of filling in the equation:
t’ = (11.11)sqrt{1 – 0.9c^2}
t’ = (11.11)sqrt{1 – 0.81c}
t’ = (11.11)sqrt{0.19c}
t’ = (11.11)( 0.44) [approximation]
t’ = 4.8884 years

Now, as we can see; the sister only experiences approximately 5 years, while her brother on Earth experiences approximately 11 years. Now let’s say she gets to the star and doesn’t like it, so she turns around at the same velocity. Her brother is now 22 years older than he was when she left, and she is only 10 years older. This is because time becomes slower as an object approaches the speed of light. If she was going 99% the speed of light, the time would have passed even slower, because the value of v in the equation becomes closer and closer to 1. If she was traveling at 100% the speed of light (which is an “unattainable and limiting velocity” according to Einstein), then the value for v would be 1…which means time would actually stand still for the sister in the spacecraft!

Now, you might be thinking “wait, this is fucked up...if they are twins, then how can one be older than the other, since each twin could perceive the other as being at rest?” The reason for this is because the twin in the spacecraft turned around, while the other twin confined to Earth did no such thing…it is therefore not “paradoxical” for one twin to be older than the other, since only one twin actually remained at rest (or close enough). In relativistic terms, the twin on the spacecraft is no longer in the “inertial frame of reference”. This is explained in more detail by General Relativity.

EDIT: Fixed sqrt notations. The math stays the same though.

[loveme4whoiam]

Messed up... but an excellent summary. I actually managed to understand one or two lines of that, and for a guy who flunked A-Level Physics thats an achievement :D Nice one.

[ComradeRed]

Just a syntactical note, for when you write t’ = (11.11) sqrt 1 – 0.9c^2 it would be better to write t’ = (11.11) sqrt{1 – 0.9c^2} ;)

And only simple algebra, what about simple geometry :P

[bezdomni]

Originally posted by [email protected] 27 2006, 05:58 PM
Just a syntactical note, for when you write t’ = (11.11) sqrt 1 – 0.9c^2 it would be better to write t’ = (11.11) sqrt{1 – 0.9c^2} ;)

And only simple algebra, what about simple geometry :P
Ah...canonical notation will be the death of me.

[ComradeRed]

Seriously though you should try to investigate the geometric nature of time dilation and the Lorentz contraction. I got bored, so I tried using spin foam in "classical" special relativity, and it came to me backwards geometrically.

It's just hard to do constructive proofs online without diagrams :(

[RebelDog]

I think I am right in saying (according to SR) that if you were somehow to attain the speed of light then that means all energy is used up and there is none left to travel through time. Like a photon travelling at the speed of light has no time, if it slowed down then some of that energy would be used to travel through time.

[ComradeRed]

SR doesn't really say if time is an "arrow" or not, which leaves it open to debate. Granted there is the CPT symmetry, but that doesn't mean that it is "possible" (or "impossible" for that matter) to time travel (backwards at least).

Frankly, it may or may not work out in theory, practicalities is another matter completely!

[bezdomni]

Originally posted by The [email protected] 28 2006, 04:16 PM
I think I am right in saying (according to SR) that if you were somehow to attain the speed of light then that means all energy is used up and there is none left to travel through time. Like a photon travelling at the speed of light has no time, if it slowed down then some of that energy would be used to travel through time.
Yes, a photon (or any wave/particle) travelling at the speed of light does not "experience" time, because it ends up as:
t'=(t)sqrt{1-1}
t'=(t)0
so
t'=0

So, if you were to break the laws of physics and create a spacecraft that can travel at the speed of light, then you would be able to go anywhere instantly (from your frame of reference, at least). You'd be able to travel millions of lightyears and experience no change in time whatsoever.

One of the postulates of Special Relativity is that the speed of light is an absolute velocity. To illustrate what this means with an example, let's say you are on a train situated next to a beam of light. No matter how fast you go (even if it is 99% speed of light), the beam of light will always be moving much faster than you. You will never be able to catch up with the beam of light, since the speed of light is an absolute constant, regardless of your own velocity and frame of reference.

Forward time travel does seem to be theoretically possible. If you take the example in my original post and use bigger numbers, you'd be coming back to Earth thousands of years into the future and percieving only a few days or months (depending on your speed in relation to the speed of light).

Special Relativity is intense stuff!


Seriously though you should try to investigate the geometric nature of time dilation and the Lorentz contraction.
I read Relativity: The Special and the General Theory by Einstein, so I am reasonably familiar with the Lorentz Transformation. I'm using the algebraic model here just because it is easier to understand.

Actually, that particular book by Einstein doesn't go beyond Algebra. As long as you have a basic working knowledge of matricies it's a pretty simple read.

The technical papers are a different story though.

[ComradeRed]

Yes, a photon (or any wave/particle) travelling at the speed of light does not "experience" time, because it ends up as:
t'=(t)sqrt{1-1}
t'=(t)0
so
t'=0 Yes, but have you gotten in to light cones? They're really keen stuff (I'll keep saying keen as an affront to CyM :P).

You see, if you use the mathematical structure of light cones, there is a way to determine if there is any causal relations between events (events represented by points).

You see, if we set time to be complex and we use Pythagora's theorem to figure the length of the 4-vector, we get a small number (depending on velocity and/or position in the coordinate system). IT would be:

x^2 + y^2 + z^2 + (ict)^2 = R^2

but if we assume it's a photon, R^2 = 0. Then it follows that:

x^2 + y^2 + z^2 = r^2

that

x^2 + y^2 + z^2 - (ct)^2 = 0

so

(ct)^2 = r^2 = x^2 + y^2 + z^2

which is the "null" lightcone. That is to say, that events that lie on this path may have causal relations to the origin of this light cone.

It gets really keen if we use tensors, because the length of the 4-vector is R^2. If R^2<0, the velocity is faster than the speed of light or it is highly likely to be a causal effect of the event that is the basis of the coordinate system; if R^2 == 0, then it is likely to be light but it may have some causal relation to the event, and if R^2>0, there is little chance to have any causal relation because nothing travels faster than light.

It is noteworthy to state it is only mathematically possible to have R^2<0 in General Relativity, and only under rare circumstances.

[RebelDog]

The same can be said for us right now, yes? I sit at this computer and I am travelling at about 65,000 mph through space with the earth. This must mean most of my energy is used up travelling through time. The faster I go the slower the time, relative to an observer in another frame of reference. The photon however does the same thing no matter who observes it.
Have I got this correct?

[ComradeRed]

The reference frame I like to use for measuring relative time dilation is one in a vacuum.

The faster you go, time moves relatively slower.

And actually, the properties of the photon changes depending on your reference frame. In an inertial reference frame, an observer may observe an electron emit a photon; whereas in an accelerated reference frame (cough*General*Relativistic), one observes the opposite occurring.

[bezdomni]

If anybody is actually interested in reading about relativity theory in Einstein&#39;s own words, but don&#39;t have a very strong background in mathematics, I strongly recommend Relativity: The Speical and the General Theory by Albert Einstein.


Yes, but have you gotten in to light cones?
I think it was in a taped lecture I watched once. The guy was really goofy looking and he had a big moustache.


The faster I go the slower the time, relative to an observer in another frame of reference.
Exactly right.


The photon however does the same thing no matter who observes it.
In special relativity, yes; in general relativity, not necessarily.

The difference between Special and General relativity is that SR assumes constant uniform motion between two objects, while GR focuses on objects that are acclerating.

That&#39;s really a bit of an oversimplification, but it&#39;ll have to do for now.

[ComradeRed]

SR = relativity focusing on velocities
GR = relativity focusing on acceleration

GR actually tweaks Newtonian Gravity to something far more interesting: a distortion of spacetime. This naturally comes at a great expense of nearly impossible mathematics.

[RebelDog]

A question that bothers me sometimes which is actually a bit off topic but I&#39;m going to ask it anyway. We basically know how heavy the universe is, right? We feel the need to add in things like dark matter because what we see doesn&#39;t add up to what it says on the scales. OK now what I am wondering is basically, how can we ever know the extent of the universe if there are galaxies that lie outside our &#39;cosmic horizon&#39; as its called. The reason they do lie outwith our senses is to do with their space/time travelling (relative to us) beyond the velocity of light (not forbidden by SR)
This means that we cannot see them or feel them as GR says gravity also travels at the speed of light and doesn&#39;t have an instant effect as Newton believed. How can we ever know the mass of the universe and how do we know its 14 billion years old? It is a strange concept that space/time has no speed limit.

[bezdomni]

Sorry for replying so late to this. I had to some serious drinking...er...thinking to do.


We basically know how heavy the universe is, right?
No. We know very little about the universe as a whole. We don&#39;t really even know how old the universe is. In fact, the calculations for the age of the universe has been lower than the age of some stars. At the moment, there is no incredibly accurate means to calculate the actual age of the universe. Therefore, we don&#39;t know the actual size of the universe (we have only observed a very small section of it, even with our most powerful telescopes).

And it is really difficult to know how much mass something has without knowing how big it is. :P


OK now what I am wondering is basically, how can we ever know the extent of the universe if there are galaxies that lie outside our &#39;cosmic horizon&#39; as its called. The reason they do lie outwith our senses is to do with their space/time travelling (relative to us) beyond the velocity of light (not forbidden by SR)
I&#39;d say the reason they lie outside of our senses is because we don&#39;t have good enough telescopes. ;)


This means that we cannot see them or feel them as GR says gravity also travels at the speed of light and doesn&#39;t have an instant effect as Newton believed. How can we ever know the mass of the universe and how do we know its 14 billion years old?
Like I said earlier, we don&#39;t actually know the age or mass of the universe.

It would be cool if we did though. Someday....someday.

[RebelDog]

I guess it will always be a case of the universe being what we can detect. I suppose space/time travelling beyond c is actually undetectable and cannot be proven either way. If there are parts of the universe whose light cannot ever reach us then that part of the universe is irrelevant to us (our universe).

[ComradeRed]

I suppose space/time travelling beyond c is actually undetectable and cannot be proven either way. Well, in theory according to General Relativity, it is feasible to travel faster than the speed of light.

DISCLAIMER: it depends on the reference frame, whether it is inertial, moving, or accelerating. The reason is that we can never be in a vacuum, General Relativity explains that matter "generates" spacetime.

So even a lonely photon could never be in a vacuum. Thus it&#39;s speed in a vacuum is never knowable...only approximate.

But if you, say, get caught in the gravitational pull of a black hole, and you are accelerating towards it, it is feasible to accelerate so greatly that if you calculate out the velocity in an inertial reference frame that the velocity is relatively greater than the speed of light.

Curiously, because you would be moving faster than the speed of light, you would move faster than the photons of clocks, which means the clocks will tell you that you are going back in time. :D

[RebelDog]

But if you, say, get caught in the gravitational pull of a black hole, and you are accelerating towards it, it is feasible to accelerate so greatly that if you calculate out the velocity in an inertial reference frame that the velocity is relatively greater than the speed of light.

Curiously, because you would be moving faster than the speed of light, you would move faster than the photons of clocks, which means the clocks will tell you that you are going back in time.

Isn&#39;t that a paradox? If you are trapped in the gravitational pull of a black hole and going back in time you could theoretically go back to a time when you were not caught in the gravitational pull of the black hole and thus escape the inescapable.
Or would going back in time make the black holes gravitation repulsive?

[ComradeRed]

Originally posted by The [email protected] 11 2006, 02:02 AM

But if you, say, get caught in the gravitational pull of a black hole, and you are accelerating towards it, it is feasible to accelerate so greatly that if you calculate out the velocity in an inertial reference frame that the velocity is relatively greater than the speed of light.

Curiously, because you would be moving faster than the speed of light, you would move faster than the photons of clocks, which means the clocks will tell you that you are going back in time.

Isn&#39;t that a paradox? If you are trapped in the gravitational pull of a black hole and going back in time you could theoretically go back to a time when you were not caught in the gravitational pull of the black hole and thus escape the inescapable.
Or would going back in time make the black holes gravitation repulsive?
It would appear so, which is one of the interesting things of General Relativity.

Just remember this when you&#39;re out cruising in your spaceship: when caught in the gravitational pull of a black hole, go as fast as you can into it :D

[RebelDog]

The 2nd law of thermodynamics;


The entropy of an isolated system not at equilibrium will tend to increase over time, approaching a maximum value.

Does our &#39;isolated system&#39; comprise of only that which is within the cosmic horizon. When calculating the entropy of our universe would it be fair to say that different observers will come up with different values due to them having different cosmic horizons? Or do they have different cosmic horizons?

[Severian]

Originally posted by The [email protected] 12 2006, 11:58 PM
The 2nd law of thermodynamics;


The entropy of an isolated system not at equilibrium will tend to increase over time, approaching a maximum value.

Does our &#39;isolated system&#39; comprise of only that which is within the cosmic horizon.
Physics ain&#39;t specially my thing, but I think I can take a stab at this one:

That isn&#39;t an isolated system. Energy and mass can enter or leave the observed part of the universe - and constantly does.

The universe as a whole is an isolated system, since (as far as anyone knows) nothing enters or leaves it.

[RebelDog]

Originally posted by Severian+Sep 13 2006, 07:30 AM--> (Severian @ Sep 13 2006, 07:30 AM)
The [email protected] 12 2006, 11:58 PM
The 2nd law of thermodynamics;


The entropy of an isolated system not at equilibrium will tend to increase over time, approaching a maximum value.

Does our &#39;isolated system&#39; comprise of only that which is within the cosmic horizon.
Physics ain&#39;t specially my thing, but I think I can take a stab at this one:

That isn&#39;t an isolated system. Energy and mass can enter or leave the observed part of the universe - and constantly does.

The universe as a whole is an isolated system, since (as far as anyone knows) nothing enters or leaves it. [/b]
Energy and mass can of course enter our observed universe. Energy and mass that lies beyond our cosmic horizon and thus travelling away on space time at beyond c can never interact with us. This is not a case of making more powerful telescopes, more powerful telescopes cannot make photons travel at a velocity beyond c. The term "the universe as a whole" is surely an ambiguous statement. Our universe must comprise of that which can manifest itself and not that which can only mean something that can only ever be utterly specualation based on 0% proof. The speed of c determines our universe.

[Severian]

Originally posted by The [email protected] 13 2006, 03:26 PM
Energy and mass can of course enter our observed universe. Energy and mass that lies beyond our cosmic horizon and thus travelling away on space time at beyond c can never interact with us.
OK, I didn&#39;t understand that distinction.

But if objects just beyond our cosmic horizon can affect objects just within it....then it&#39;s not an isolated sytem.

[RebelDog]

But if objects just beyond our cosmic horizon can affect objects just within it....then it&#39;s not an isolated sytem.

If an object &#39;just beyond&#39; our cosmic horizon could effect things &#39;just within&#39; it, then the object that is just within could relate to us information about the object beyond our cosmic horizon and violate special relativity. So this must be impssible I would think.
I think you and I both need to understand s/r and g/r better. Its very confusing.

[bezdomni]

Originally posted by The [email protected] 13 2006, 05:58 AM
The 2nd law of thermodynamics;


The entropy of an isolated system not at equilibrium will tend to increase over time, approaching a maximum value.

Does our &#39;isolated system&#39; comprise of only that which is within the cosmic horizon. When calculating the entropy of our universe would it be fair to say that different observers will come up with different values due to them having different cosmic horizons? Or do they have different cosmic horizons?
Good fucking question.

I am pretty sure this is a "controversial" issue; meaning we haven&#39;t really got the answers yet. Black hole thermodynamics is a pretty new and groundbreaking field.

I don&#39;t feel like I know enough about the topic to answer this question in a satisfying way, as I severely doubt there even is a definitive answer yet. However, I know Stephen Hawkings does a lot of work with black hole thermodynamics. If you have time, I&#39;d recommend picking up some of his work (it&#39;s pretty accessible to a person with little knowledge about physics).

It is safe to say that our universe is not in equilibrium, and according to the 2nd law of thermodynamics, the universe will reach a "maximum value".

As far as cosmic horizons and their effect on entropy, I can&#39;t really say.

[ComradeRed]

Originally posted by The [email protected] 12 2006, 09:58 PM
The 2nd law of thermodynamics;


The entropy of an isolated system not at equilibrium will tend to increase over time, approaching a maximum value.

Does our &#39;isolated system&#39; comprise of only that which is within the cosmic horizon. When calculating the entropy of our universe would it be fair to say that different observers will come up with different values due to them having different cosmic horizons? Or do they have different cosmic horizons?
This is a rather contraversial question, since there are two big camps involved here that are contesting for the view of the universe: is the universe finite or infinite?

I myself am partial to the former view, as I am a general relativist (the cosmological constant is equal to r^-2 and a function of the entire universe -- two very important things&#33;).

But if you take the former view, what happens is that entropy is counter acted by gravity. Entropy has to do with a sort of confusion about the possible states of a given system, whereas gravity reduces the possible states (because the bodies are attracted to be closely together).

Misner, Thorne, and Wheeler explain it better in Gravitation...due to current circumstances, that is in my possession but in Storage.

But if you assume the universe is infinite, then it doesn&#39;t matter because the energy in the universe is also infinite. And the cosmological constant is also infinite.

Entropy is taken in special consideration due to its peculiar relation in information theory, which is the crux of relativity (the whole light cone ordeal). So, there is work needing to be done in this area.

But Quantum Gravity should answer the question. The problem is that classical General Relativity (as any classical pre-quantized field is) its fairly vague and has deep questions; these questions are answered with such clarity in a quantized version that makes intuitive sense (e.g. the role of photons in electromagnetism).

Quantum Gravity should have the answer we all are looking for.