How much is Special Relativity a needed foundation of General Relativity

[Aether]

In what sense would they be "equivalent"?!

In the same sense as J.D. Bekenstein intends here:

An astonishing theory called the holographic principle holds that the universe is like a hologram: just as a trick of light allows a fully three dimensional image to be recorded on a flat piece of film, our seemingly three-dimensional universe could be completely equivalent to alternative quantum fields and physical laws "painted" on a distant, vast surface.The physics of black holes--immensely dense concentrations of mass--provides a hint that the principle might be true. -- J.D. Beckenstein, Information in the Holographic Universe, Scientific American:p59, (August 2003).

Here's a quick insanity check:
If a 5-dimensional manifold is "equivalent" to a 4-dimensional manifold via the holographic principle,
and if a 4-dimensional manifold is "equivalent" to a 3-dimensional manifold via the holographic principle,
and if a 3-dimensional manifold is "equivalent" to a 2-dimensional manifold via the holographic principle,
and if a 2-dimensional manifold is "equivalent" to a 1-dimensional manifold via the holographic principle,
and if a 1-dimensional manifold is "equivalent" to a 0-dimensional manifold via the holographic principle,

then why would we ever study anything but 0-dimensional manifolds? They very easy things to understand!

According to P.S. Wesson the [+(---)+] signature manifold has "good physical properties", and according to J.D. Bekenstein "our seemingly three-dimensional universe could be completely equivalent to alternative quantum fields and physical laws "painted" on a distant, vast surface" so that the (---) part of the [+(---)+] signature manifold might be completely equivalent to the (--) part of a [+(--)+] signature manifold. I am assuming that a [+(--)+] signature manifold having the same "good physical properties" as the [+(---)+] would be something worth studying; what do you think?

 

[lalbatros]

(multiple quotation) ... holographic principle ...

The holographic principle indeed questions the notion of space and dimensions.

I think that the origin of the holographic principles lies in the point of view that physics is about information and that the universe can hold a large but a finite amount of information that are "processed" at a finite rate. (see http://www.phy.duke.edu/~hsg/einstein/seth-lloyd-ultimate-computer.pdf")

Therefore, I think the holographic principle does not suggest us simply to drop one dimension in physics and makes thinks a bit simpler. Actually, it illustrates -for me- the possibility that dimensions themselves might be the simplification while the reality might very well be space and dimension-free.

The ultimate physical description of the universe might very well be a huge amount of qbits and the existence of space and the 4 dimensions might only be a very happy opportunity to make physics simpler.

I am very curious to see if such a "reverse" point of view, from qomputers to the universe, could bring us something useful.

Michel

 

[Aether]

Thanks for the interesting article Michel.

...black holes could in principle be 'programmed': one forms a black hole whose initial conditions encode the information to be processed, let's that information be processed by the Planckian dynamics at the hole's horizon, and extracts the answer of the computation...

At the black-hole limit, computation is fully serial: the time it takes to flip a bit and the time it takes a signal to communicate around the horizon of the hole are the same.

Therefore, I think the holographic principle does not suggest us simply to drop one dimension in physics and makes thinks a bit simpler.

I'm not looking "simply to drop one dimension in physics and make things a bit simpler", but rather to unify physics in terms of "planckian dynamics at the hole's horizon" (or rather at the universe's horizon); either that, or to rule out the possibility of such a thing. The holographic principle doesn't help do that, but rather it may help explain how such a model might appear to us as a projection in four-dimensional space-time.

 

[Sammy k-space]

Special Relativity is just Linear Doppler effect: C is constant by definition; the mass/energy of the photon is conserved for a stationary observer relative to the source, but for a source moving away from the observer there is a decrease in frequency as
v^2/2C^2 ie. vv/2CC where v is the velocity of the source away from the observer. It is a vector quanity; the quantity is added for a source moving toward the observer. This is easily calculated as E=mCC=hn; CC=E/m ie. C squared is the constant of proportionaliy between a mass and it's energy.
For general relativity we can look at two cases: first let a photon move toward the center of mass of an observer, since C is constant the gravitational increase is seen as an increase in frequency; for the other case let the photon be traveling close to a mass but toward an observer stationary to the source; the photon will curve toward the mass, but since C is defined as constant the velocity along the curve is C but the observer will see the curve and the shift in frequency. The change will be as if m=hn/CC.
For a very special case take twin photons with the same frequency and moving in the opposite direction, the observer is stationary relative to the source and "sees" one photon coming and one leaving, but the photons overlap in the view, so E1=hn and E2=hn but sum of the vectors is zero.

 

[jimmy neutron]

I'm not sure what you're trying to say. I meant that SR is just a special case of general relativity, so everything in SR is contained in GR.

Within both special and general relativity, there is an unavoidable constant we call c. Of course it isn't necessary that that parameter has anything to do with electromagnetic phenomena, but experimentally, it does.

By that I assume that you mean that, otherwise, light signals would have to travel at speeds less than c?

 

[Deepak Kapur]

If one had to built an invariant theory for gravitation, applicable in any system of coordinate, could it not be possible to create one without knowing about SR (constancy of c, EM, ...).

Could such an off-road journey teach us something, and couldn't SR pop up in some other way?

Thanks for your ideas,

Michel

I doubt the predictions of relativity. To take a simple example, 'Time Slows With Increasing Speed'. If time slows down, so will all the processes like the functioning of our hearts, brains, the solar system, the atomic system and so on. So, the universe will just tend to come to a stand still at extremely high speeds. This is absurd!

 

[Fredrik]

I doubt the predictions of relativity. To take a simple example, 'Time Slows With Increasing Speed'. If time slows down, so will all the processes like the functioning of our hearts, brains, the solar system, the atomic system and so on. So, the universe will just tend to come to a stand still at extremely high speeds. This is absurd!

The post you're quoting was written in 2006.

Do you really think that you can refute the most well-understood and most thoroughly tested theory in the history of science with an argument that any kid can come up with, and without making an effort to find out what the theory says?

 

[Deepak Kapur]

The post you're quoting was written in 2006.

Do you really think that you can refute the most well-understood and most thoroughly tested theory in the history of science with an argument that any kid can come up with, and without making an effort to find out what the theory says?

Yes, you are right. It's a childish question. But don't underestimate the importance of such questions as they often turn out to be the germs of excellent theories in case of gifted individuals like Einstein ( he himself was used to such questions).

A few more childish questions.

1. Can we ever understand the mystery of nature. Suppose the String Theory (which in some cases even refutes Einstein's Theories) is able to find the fundamental particle. The immediate question would be 'What is this 'fundamental' composed of?

2. The only saving grace is the adage 'Something is Better than Nothing'. Even scientists are aware of this and never give up their scientific enquiries even if faced with bizarre contradictions like wave-particle duality and all the other paradoxes. Mind you, many scientist dealling with quantum mechanics are still skeptical of it, but they don't want to undermine the superstructure of science and have learned to live with it. Different kinds of politics is also involved in such an attitude. After all only a child can have the audacity to ask God 'Who made you?'

3. How can uniform motion be ever possible, when galaxies are moving away from each other at incredible speeds and time is continuously slowing down. Doesn't science feel it 'convenient' to deal with appoximations and simplifications rather than reality (whatever that is).

4.You would agree that in the laws of mechanics no mention is made of the shape of the body undergoing motion, whereas in real life it makes hell of a difference. Similarly it's extremely difficult to solve three-body problem (what to talk of greater number of bodies), because of the feedback effect. But scientists always tend to avoid such feedbacks and proceed with the simplest of cases. Can it lead us further or enmesh us in a labyrinth of mathematics and incomplete laws? (O! Lord, at least give us a single universal law that is proof against any kind of further enquiry!)

More will follow in case you reply.

 

[Mentz114]

I doubt the predictions of relativity. To take a simple example, 'Time Slows With Increasing Speed'. If time slows down, so will all the processes like the functioning of our hearts, brains, the solar system, the atomic system and so on. So, the universe will just tend to come to a stand still at extremely high speeds. This is absurd!

'Time slows with increasing speed' is not what relativity says - so your argument is based on a complete misunderstanding of relativity. Also, you have no idea what science is about and for.

 

[jtbell]

I doubt the predictions of relativity.

Check out the experimental evidence:

Experimental Basis of Special Relativity

 

[Deepak Kapur]

'Time slows with increasing speed' is not what relativity says - so your argument is based on a complete misunderstanding of relativity. Also, you have no idea what science is about and for.

Plz don't be impatient. Impatience may also signal absence of logic.

There are many interpretations of relativity ( some even contradictory).

To go by your interpretation a very-2 high gravitational field would in theory make a clock stop functioning (or make it extremely-2 slow) for another observer who is far away from the gravity source.

This again amounts to what I have said above. Processes can't come to a stand still just by the presence of super high gravity.

 

[Deepak Kapur]

Check out the experimental evidence:

Experimental Basis of Special Relativity

Thanks. I have already read that.

 

[Mentz114]

Nothing you've said is worth refuting because you don't understand what you are talking about.

For instance

Processes can't come to a stand still just by the presence of super high gravity.

That is not what GR predicts. Again you base your remarks on misunderstandings.

 

[Fredrik]

More will follow in case you reply.

I have answered questions like these many times in the past, but I think I'll pass this time. I don't want to spend 10-15 hours explaining physics (starting with an explanation of what a theory is) to someone who probably would ignore everything I say anyway. This is a forum for people who want to learn stuff, not for people who want to criticize things they don't understand.

Plz don't be impatient. Impatience may also signal absence of logic.

And criticizing the best understood and most thoroughly tested theory in the history of science without making an effort to understand what it says, signifies what exactly? You can't demand that others be patient with you when you show up here with this attitude.

To go by your interpretation a very-2 high gravitational field would in theory make a clock stop functioning (or make it extremely-2 slow) for another observer who is far away from the gravity source.

This again amounts to what I have said above. Processes can't come to a stand still just by the presence of super high gravity.

The problem with your posting this "argument" against relativity (twice!?) isn't that it's extremely naive. It's perfectly OK to ask uneducated questions. The problem is that you clearly know that your argument can't be right, and still talk to us as if you have disproved relativity. If it had been possible to disprove relativity with an argument that any kid can come up with, it would have been done a hundred years ago, and we wouldn't need your help with it. If you continue with this nonsense, you might get banned from the forum.

 

[Deepak Kapur]

I have answered questions like these many times in the past, but I think I'll pass this time. I don't want to spend 10-15 hours explaining physics (starting with an explanation of what a theory is) to someone who probably would ignore everything I say anyway. This is a forum for people who want to learn stuff, not for people who want to criticize things they don't understand.


And criticizing the best understood and most thoroughly tested theory in the history of science without making an effort to understand what it says, signifies what exactly? You can't demand that others be patient with you when you show up here with this attitude.


The problem with your posting this "argument" against relativity (twice!?) isn't that it's extremely naive. It's perfectly OK to ask uneducated questions. The problem is that you clearly know that your argument can't be right, and still talk to us as if you have disproved relativity. If it had been possible to disprove relativity with an argument that any kid can come up with, it would have been done a hundred years ago, and we wouldn't need your help with it. If you continue with this nonsense, you might get banned from the forum.

I am not trying to refute anything but am trying to satisy my curiosity. As far as this forum goes, I haven't got any logical answer till now apart from accusations and non-sensical arguments.

It's a usual strategy of such forums.

1. Goad someone so that he indulges in impatient remarks.

2. If someone is not provoked, dismiss him as being non-sensical.

Plz refrain from 'blind faith' in anything and try to give logical answers to the points (however uneducated they might be) I have raised.

 

[Fredrik]

I am not trying to refute anything but am trying to satisy my curiosity.

I might have believed you if you had said something to show us that you understand that these are the only possible reasons why your understanding of relativity disagrees with your intuition about the real world:

1. You don't actually understand what these theories (SR and GR) say.

2. Your intuition is wrong.

Instead you have been strongly suggesting that the problem is with relativity. That attitude is very inappropriate in this forum.

It's a usual strategy of such forums.

1. Goad someone so that he indulges in impatient remarks.

2. If someone is not provoked, dismiss him as being non-sensical.

Plz refrain from 'blind faith' in anything...

That's definitely not true, but it's a usual strategy of science-haters to make accusations like this one, where they describe their own behavior and claim it's how skeptics and scientists behave. This is also inappropriate here.

...and try to give logical answers to the points (however uneducated they might be) I have raised.

You haven't raised any points. You haven't asked any questions. All you've done is to suggest that your "argument" means that relativity is wrong. (This is also an implicit suggestion that every scientist in the last 100 years was a complete idiot).

 

[atyy]

I doubt the predictions of relativity. To take a simple example, 'Time Slows With Increasing Speed'. If time slows down, so will all the processes like the functioning of our hearts, brains, the solar system, the atomic system and so on. So, the universe will just tend to come to a stand still at extremely high speeds. This is absurd!

Yes, it is absurd. The statements you make are pieced together from popular explanations of relativity. In relativity there are several times - proper time and coordinate time, and there are many coordinate times. If you take a statement about proper time and another statement about coordinate time and draw logical conclusions from them, you will be all mixed up.

So things to distinguish: proper time vs coordinate time, inertial frame versus noninertial frame, local reference frame versus global reference frames, special relativistic time dilation vs general relativistic time dilation. Only predictions about experiments really matter.

 

[Deepak Kapur]

I might have believed you if you had said something to show us that you understand that these are the only possible reasons why your understanding of relativity disagrees with your intuition about the real world:

1. You don't actually understand what these theories (SR and GR) say.

2. Your intuition is wrong.

Instead you have been strongly suggesting that the problem is with relativity. That attitude is very inappropriate in this forum.


That's definitely not true, but it's a usual strategy of science-haters to make accusations like this one, where they describe their own behavior and claim it's how skeptics and scientists behave. This is also inappropriate here.


You haven't raised any points. You haven't asked any questions. All you've done is to suggest that your "argument" means that relativity is wrong. (This is also an implicit suggestion that every scientist in the last 100 years was a complete idiot).

I didn't think 'political correctness' is also required in 'public forums'.

Anyhow, answer my next question (except saying that the question itself is wrong).

When there is no absolute concept of time and distance (as stated by general relativity), how can we talk about an absolute entity like the Speed of light?

 

[Fredrik]

Now that's a real question.

There's just one little problem. The complete answer is long and mathematical. It would take a long time to write it down, and I don't even know if you'd be interested in a mathematical answer. The very short answer is that the speed of light isn't absolute. You can make it whatever you want by choosing an appropriate coordinate system. But there's a class of coordinate systems that are particularly important. They're called inertial frames. The claim that the speed of light is "invariant" actually means that it's the same in all inertial frames, not that it's the same in all coordinate systems.

Why is it the same in all inertial frames? That's just a mathematical property of inertial frames on Minkowski spacetime and null geodesics, the curves that we use to represent the motion of massless particles mathematically.

Why do we use this particular model of space and time? Because the theory based on it makes better predictions about results of experiments than theories based on other models. There is actually only one other model that's consistent with the requirement that inertial observers would describe each other as moving as described by straight lines, and that's the Galilean spacetime, which is used in Newtonian mechanics.

 

[Deepak Kapur]

Now that's a real question.

There's just one little problem. The complete answer is long and mathematical. It would take a long time to write it down, and I don't even know if you'd be interested in a mathematical answer. The very short answer is that the speed of light isn't absolute. You can make it whatever you want by choosing an appropriate coordinate system. But there's a class of coordinate systems that are particularly important. They're called inertial frames. The claim that the speed of light is "invariant" actually means that it's the same in all inertial frames, not that it's the same in all coordinate systems.

Why is it the same in all inertial frames? That's just a mathematical property of inertial frames on Minkowski spacetime and null geodesics, the curves that we use to represent the motion of massless particles mathematically.

Why do we use this particular model of space and time? Because the theory based on it makes better predictions about results of experiments than theories based on other models. There is actually only one other model that's consistent with the requirement that inertial observers would describe each other as moving as described by straight lines, and that's the Galilean spacetime, which is used in Newtonian mechanics.

Why is it same in all the inertial frames when there is no absolute definition of time and distance (time dialtion, length contraction) even in the inertial frames. As a matter of fact how can (or should) speed be determined with so much of relativity around (even in inertial frames of reference).

 

[Deepak Kapur]

Nothing you've said is worth refuting because you don't understand what you are talking about.

For instance



That is not what GR predicts. Again you base your remarks on misunderstandings.

Not be able to understand the other person is indeed lack of understanding.

I never said 'it' predicts. It can be one of the implications. A clock so slow (as it apperas to an observer) that all the processes virtually coming to a stand still.

 

[Rasalhague]

Why is it same in all the inertial frames when there is no absolute definition of time and distance (time dialtion, length contraction) even in the inertial frames. As a matter of fact how can (or should) speed be determined with so much of relativity around (even in inertial frames of reference).

Speed is a ratio between a distance (how far travelled) and a time (how long it takes). Distances and times change when we switch from measuring them in one inertial reference frame to another, but the ratio between the distance light travels and the time it takes stays the same.

Suppose we measure the speed of light through a vacuum (I'll just call it "the speed of light" from now on) in one inertial reference frame (a non-accelerating spacetime coordinate system, with three coordinates for space and one for time, covering a region small enough and brief enough that the effects of gravity are negligible) and find it to be a certain value c.

I'll use the letter v to stand for the speed of some other inertial reference frame moving parallel to the pulse of light, as measured in our original reference frame. I'll use another variable, defined like this:

$$\gamma = \frac{1}{\sqrt{1-\left ( \frac{v}{c} \right )^2}}.$$

The Greek letter $\gamma$, called "gamma", is just a handy symbol, conventionally used to simplify the equations below. Relativity predicts that intervals of space and time will differ in our new inertial reference frame according to the following equations,

$$\Delta x' = \gamma \left ( \Delta x - v \; \Delta t \right )$$

and

$$\Delta t' = \gamma \left ( \Delta t - \frac{v}{c^2} \; \Delta x \right )$$

where $\Delta x$ is some distance, for example the distance between the place at which a pulse of light is emitted and the place where it's received, measured according to our original reference frame (that is, as measured by rulers at rest in that reference frame), and $\Delta x'$ is the distance between those same points measured according to our new reference frame. Similarly, $\Delta t$ is an interval of time, for example the time between emission and reception of a pulse of light according to our original reference frame (that is, as measured by clocks at rest in that reference frame), and $\Delta t'$ the time between these events according to our new reference frame.

The speed of light in our original reference frame is

$$c = \frac{\Delta x}{\Delta t},$$

the distance the light travels divided by the time it takes to travel that distance. The speed of light in our new reference frame is therefore

$$\frac{\Delta x'}{\Delta t'}=\frac{\gamma \left ( \Delta x - v \; \Delta t \right )}{\gamma \left ( \Delta t - \frac{v}{c^2} \; \Delta x \right )}=\frac{\frac{\Delta x}{\Delta t}-v}{1-\frac{v}{c^2}\frac{\Delta x}{\Delta t}}=\frac{c-v}{1-\frac{v}{c}}=\frac{c(c-v)}{c-v}=c$$

So, without paradox, the speed of light is the same in any inertial reference frame moving at some non-zero speed relative to our original inertial reference frame, even though neither the distance the light traveled nor the time it took are the same as they were in the original inertial reference frame.

 

[Fredrik]

Why is it same in all the inertial frames when there is no absolute definition of time and distance (time dialtion, length contraction) even in the inertial frames. As a matter of fact how can (or should) speed be determined with so much of relativity around (even in inertial frames of reference).

You should think of spacetime as an abstract set of points. Those points are called events. Inertial frames are functions that assign 4-tuples of real numbers (t,x,y,z) to events. Those numbers (the coordinates of events) are used to define velocity the same way as in pre-relativistic physics:

$$\vec r=(x,y,z)$$

$$\vec v=\frac{d\vec r}{dt}$$

Speed is the magnitude of the velocity:

$$v=|\vec v|=\sqrt{v_1^2+v_2^2+v_3^3}$$

It's hard to explain why this formula applied to a null geodesic gives us the result =1 in every inertial frame. To really understand it, you'd have to understand what a geodesic is, which requires that you know some differential geometry. So let me skip most of those technicalities and go directly to what we'd end up with if we went through with all those mathematical details. But before we get started, I should tell you that there's a very natural way to associate an inertial frame with the motion and spatial orientation of an observer that's moving with constant velocity forever. This allows us to identify "observers" with "frames".

We can pick an inertial frame (any inertial frame will do) and use it to identify Minkowski spacetime with $\mathbb R^4$. Because an inertial frame was used in this identification, we can think of this copy of $\mathbb R^4$, as representing the point of view of the observer associated with that inertial frame. We clearly need a formula that tells us how to calculate the coordinates that one observer assigns to an event, given the coordinates that another observer assigns to the same event. In 1+1 dimensions (let's keep it as simple as possible), it's a function

$$x\mapsto \Lambda x+a$$

where x and a are 2×1 matrices and

$$\Lambda=\gamma\begin{pmatrix}1 & -v\\ -v & 1\end{pmatrix}$$

$$\gamma=\frac{1}{\sqrt{1-v^2}}$$

This is in units such that c=1. Such a function is called a Poincaré transformation or a Lorentz transformation. (The term Lorentz transformation is often reserved for the case a=0). We can use this to calculate the speed of light in another inertial frame. The simplest possible scenario we can consider is when both inertial frames have the same origin, and the light we're considering is at x=0 when t=0 (that's in both frames, since they have the same origin). Now it's a really easy excercise to show that the curve that represents the motion of the light looks the same in both coordinate systems (a straight line through the origin with slope 1).

This may not be very satisfying unless you know why we're working with Lorentz transformations. As I said before, the requirement that inertial observers must describe each other's motion as straight lines is much stronger than it looks, and more or less forces us to only consider a spacetime where a coordinate change between inertial frames is done using a Lorentz transformation or a Galilei transformation. But we can actually ignore that completely and just say that using Minkowski spacetime (and therefore the Lorentz transformation) is an axiom of the theory, and is ultimately justified by the fact that theories of matter and interactions in Minkowski spacetime predict the results of experiments so well.

And anyway, you weren't really asking why the speed of light is the same in all inertial frames. You were just trying to find out how it's possible at all. I hope the above shed some light on it, even though I left out a lot. You might also want to have a look at pages 8-9 in Schutz. Light has speed 1 in all inertial frames because a Lorentz transformation tilts the x-axis by the same amount as the t axis. (Schutz's argument on those pages is actually that the invariance of the speed of light implies that tilting of the x axis, but the diagrams would have been the same even if he had been arguing for the converse statement).

 

[Deepak Kapur]

Speed is a ratio between a distance (how far travelled) and a time (how long it takes). Distances and times change when we switch from measuring them in one inertial reference frame to another, but the ratio between the distance light travels and the time it takes stays the same.

Suppose we measure the speed of light through a vacuum (I'll just call it "the speed of light" from now on) in one inertial reference frame (a non-accelerating spacetime coordinate system, with three coordinates for space and one for time, covering a region small enough and brief enough that the effects of gravity are negligible) and find it to be a certain value c.

I'll use the letter v to stand for the speed of some other inertial reference frame moving parallel to the pulse of light, as measured in our original reference frame. I'll use another variable, defined like this:

$$\gamma = \frac{1}{\sqrt{1-\left ( \frac{v}{c} \right )^2}}.$$

The Greek letter $\gamma$, called "gamma", is just a handy symbol, conventionally used to simplify the equations below. Relativity predicts that intervals of space and time will differ in our new inertial reference frame according to the following equations,

$$\Delta x' = \gamma \left ( \Delta x - v \; \Delta t \right )$$

and

$$\Delta t' = \gamma \left ( \Delta t - \frac{v}{c^2} \; \Delta x \right )$$

where $\Delta x$ is some distance, for example the distance between the place at which a pulse of light is emitted and the place where it's received, measured according to our original reference frame (that is, as measured by rulers at rest in that reference frame), and $\Delta x'$ is the distance between those same points measured according to our new reference frame. Similarly, $\Delta t$ is an interval of time, for example the time between emission and reception of a pulse of light according to our original reference frame (that is, as measured by clocks at rest in that reference frame), and $\Delta t'$ the time between these events according to our new reference frame.

The speed of light in our original reference frame is

$$c = \frac{\Delta x}{\Delta t},$$

the distance the light travels divided by the time it takes to travel that distance. The speed of light in our new reference frame is therefore

$$\frac{\Delta x'}{\Delta t'}=\frac{\gamma \left ( \Delta x - v \; \Delta t \right )}{\gamma \left ( \Delta t - \frac{v}{c^2} \; \Delta x \right )}=\frac{\frac{\Delta x}{\Delta t}-v}{1-\frac{v}{c^2}\frac{\Delta x}{\Delta t}}=\frac{c-v}{1-\frac{v}{c}}=\frac{c(c-v)}{c-v}=c$$

So, without paradox, the speed of light is the same in any inertial reference frame moving at some non-zero speed relative to our original inertial reference frame, even though neither the distance the light traveled nor the time it took are the same as they were in the original inertial reference frame.

Quite helpful! (though equations were not visible)

Since it's a forum, I presume that even if I ask (relply to) extremely large number of questions, I am not going to pester anybody.

1. Now coming to our galaxy (which is accelerating at tremendous speed). Is the speed of light same at every point in the galaxy (between galaxies for that matter). Is this speed equal to the speed of light in an inertial frame. Why?

2. Why does light itself not experience time-dialation. If you say its massless, it's at least energy that has probably resulted from mass annihilation. So, it's "something" after all and don't forget about the 'photons' of light that act like particles.

 

[Fredrik]

(though equations were not visible)

Search the feedback forum. I think others have had the same problem in the past. Maybe you're just using a really old browser and need to upgrade.

1. Now coming to our galaxy (which is accelerating at tremendous speed). Is the speed of light same at every point in the galaxy (between galaxies for that matter). Is this speed equal to the speed of light in an inertial frame. Why?

Our galaxy isn't accelerating significantly. Special relativity holds "locally", in small regions of spacetime. When we're talking about far away galaxies, we need general relativity. At any location in spacetime (in this galaxy or any other), we can consider coordinate systems that I'll call "local inertial frames" (unfortunately there doesn't seem to be a standard name for them). They are coordinate systems that can be associated with the motion of massive particles in a natural way, and the speed of light at the origin of the coordinate system is the same in any of them. However, things are not so simple when you compare things that are far away from each other.

Edit: Apparently it's complicated enough to confuse me too. I had to edit my post to rewrite this part.

The solutions of GR that describe homogeneous and isotropic universes are called FLRW solutions. When we're working with one of them, it's convenient to use a coordinate system in which the galaxies are more or less stationary (they'll have speeds of a few hundred km/s relative to a nearby objects that stay at constant position coordinates). In this coordinate system, it's convenient to define another kind of "speed" to be a measure of how fast distant objects are moving away from each other. Define d(t) to be the proper distance between the two objects in a hypersurface of constant coordinate time t, and define the new kind of speed to be d'(t). The expansion of space ensures that the speed is non-zero for two objects that stay at constant position coordinates, and for distant galaxies, it can even be much higher than c.

This doesn't contradict the invariance of the speed of light, because that's a statement about a different kind of speed in a different kind of coordinate system. If we apply that same definition of speed to the light emitted from a star in a distant galaxy, then it's speed clearly isn't going going to be c. But that light does move at c in a local inertial frame associated with the motion of the star that emits it, no matter how far away it is.

2. Why does light itself not experience time-dialation. If you say its massless, it's at least energy that has probably resulted from mass annihilation. So, it's "something" after all and don't forget about the 'photons' of light that act like particles.

One answer is that particles simply don't have experiences, but that's not the whole story, because we sometimes talk about a massive particle's point of view. When we do, we're specifically referring to the description of events in spacetime using the inertial frame we'd associate with the massive particle's motion. The concept of "the photon's point of view" doesn't quite make sense because there's no natural way to associate an inertial frame with its motion. The problem is that the method we'd like to use to determine which subset of spacetime to call "space, at time t" doesn't work for photons. (This method for massive particles is described in the part of Schutz's book that I linked to in my previous post).

 

[Rasalhague]

A quick fix if you're still having trouble with seeing the equations in earlier posts: you could try copying the LaTeX code (try left clicking on the equation; that should open a window with the code) and pasting it into an online LaTeX editor such as this one:

http://www.codecogs.com/components/equationeditor/equationeditor.php

Or if that doesn't work, there are lots of others.

 

[Deepak Kapur]

Search the feedback forum. I think others have had the same problem in the past. Maybe you're just using a really old browser and need to upgrade.


Our galaxy isn't accelerating significantly. Special relativity holds "locally", in small regions of spacetime. When we're talking about far away galaxies, we need general relativity. At any location in spacetime (in this galaxy or any other), we can consider coordinate systems that I'll call "local inertial frames" (unfortunately there doesn't seem to be a standard name for them). They are coordinate systems that can be associated with the motion of massive particles in a natural way, and the speed of light at the origin of the coordinate system is the same in any of them. However, things are not so simple when you compare things that are far away from each other.

Edit: Apparently it's complicated enough to confuse me too. I had to edit my post to rewrite this part.

The solutions of GR that describe homogeneous and isotropic universes are called FLRW solutions. When we're working with one of them, it's convenient to use a coordinate system in which the galaxies are more or less stationary (they'll have speeds of a few hundred km/s relative to a nearby objects that stay at constant position coordinates). In this coordinate system, it's convenient to define another kind of "speed" to be a measure of how fast distant objects are moving away from each other. Define d(t) to be the proper distance between the two objects in a hypersurface of constant coordinate time t, and define the new kind of speed to be d'(t). The expansion of space ensures that the speed is non-zero for two objects that stay at constant position coordinates, and for distant galaxies, it can even be much higher than c.

This doesn't contradict the invariance of the speed of light, because that's a statement about a different kind of speed in a different kind of coordinate system. If we apply that same definition of speed to the light emitted from a star in a distant galaxy, then it's speed clearly isn't going going to be c. But that light does move at c in a local inertial frame associated with the motion of the star that emits it, no matter how far away it is.


One answer is that particles simply don't have experiences, but that's not the whole story, because we sometimes talk about a massive particle's point of view. When we do, we're specifically referring to the description of events in spacetime using the inertial frame we'd associate with the massive particle's motion. The concept of "the photon's point of view" doesn't quite make sense because there's no natural way to associate an inertial frame with its motion. The problem is that the method we'd like to use to determine which subset of spacetime to call "space, at time t" doesn't work for photons. (This method for massive particles is described in the part of Schutz's book that I linked to in my previous post).

What would be the scenario if we consider that the galaxies are not moving at all but the space between them is expanding.

Mind you, this is not the opposite of what we have discussed before.

 

[Fredrik]

I'm not sure I understand the question (because it seems to me that the answer is in the text you quoted).