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Locally Lorentzby JDoolin
Tags: locally lorentz 
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#19
Dec1510, 12:09 PM

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JDoolin you are misinformed, global Lorentz invariance only occurs in a flat spacetime.
In curved spacetime there is no global Lorentz invariance, there is only Lorentz invariance at the local level. 


#20
Dec1510, 12:43 PM

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You are using what I would call the common definition of distance and time, which are observer dependent. I am perfectly fine with that, because it is what any ordinary person thinks of when he hears the word "distance." By contrast, the metric distance between the source and destination events here is zero. This is the quantity that is unchanged by Lorentz Transformation. What the "metric distance" represents is an invariant quantity relating the distance and time between two events. As far as semantics are concerned, I actually object to even calling this quantity a "distance", preferring the more abstract term, "spacetimeinterval" and your objection makes it clearer why. Because there is a difference between these two situations, where the receiver is 2R away, and where the receiver is R away. Yet the spacetimeinterval for the photons traveling the different distances is the same in both cases. (Edit: By the way, I don't think there is any photon that is detected by both receivers. If it is detected by the first receiver, it's energy is absrobed by the first receiver. That's part of the reason why I think it is appropriate to say that a photon (a single quantum photon) is a direct interaction between particles at the source, and particles at the destination. I wasn't sure if this was a subtlety or just obvious. The subtlety comes in when you have interference, and though the light goes through both slits and there is selfinterference, you still don't actually have an event. The photon is still a direct interaction with the source and destination, but somehow modified by the interference device in a geometric way, but not an "event"ful way.) 


#21
Dec1510, 01:19 PM

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On the other hand, it is easy to deal with a local slowing speed of light within the context of a globally lorentz spacetime. I think the trouble occurs when distance and time are confused with the proper time and proper distance (what you call the "metric distance.") You'll notice that within the lorentz transformation equations, the metric distance is not to be found. Neither proper time nor proper distance of events appears in the operator or the inputs or the outputs of the equation. These notions are irrelevant to the geometry of spacetime at large. However, in MTW's description of Locally Lorentz, the defintion of proper space and proper time have a central relevance. They are following geodesics where distance is used from the planetary frame, while time is used from the falling object's frame, as modified by the planetary gravity. And it is within these coordinates that somehow some principal of least action is calculated, and in some sense, if you use the right variables, the particle's trajectory is straight. Is this straightness an application of some "local lorentzian" property? I'm not entirely sure, but I'd rather see the variables defined, and the reasoning clearly outlined. 


#22
Dec1510, 01:20 PM

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AB 


#23
Dec1510, 01:31 PM

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#24
Dec1610, 09:57 AM

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Alright, I'm coming around to a legitimate meaning of "locally lorentz" but it involves defining some extra variables, and it preserves my meaning of "globally lorentz." In fact, I think, if we clearly define our variables, both are true in different contexts, and both are false in the opposite context!
We can define t_{local} by imagining a clock at a particular position in a gravitational field, suspended by a wall or a pole. This local time is a function of the clock's position in the gravitational field, but it is also a function of some external global observerdependent time t_{global}. The main point is we have three different sets of variables. The third set of variables is, of course, related to the geodesic of the object falling through the space. The "locally lorentz" behavior can then be described as [tex]\Delta \tau^2=\Delta t_{local}^2\Delta x_{local}^2[/tex] While the "globally lorentz" behavior applies to the global coordinates [tex]\begin{pmatrix}
c t_{global}' \\ x_{global}' \end{pmatrix}= \begin{pmatrix} \gamma & \beta \gamma \\ \beta \gamma & \gamma \end{pmatrix} \begin{pmatrix} c t_{global} \\ x_{global} \end{pmatrix}[/tex] 


#25
Dec1610, 11:27 AM

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Consider that the receiver is either chasing the emitter or vice versa. Whatever effects of relativistic doppler happens to the emitter, you are assured that exactly the inverse effect will apply to the receivers. Even though the doppler effect itself is frame dependent, the coupling of two inverse effects means there is a sort of invariance in the events themselves. Naturally, since the same events happen, regardless of what reference frame, the digital readouts on the intensity probes will read the same. 


#26
Dec1610, 09:32 PM

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#27
Dec1610, 09:43 PM

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In other words, there must be *some* physical difference between the cases, and it can't be the spacetime interval since that's zero in both cases. Therefore, I don't think you can use the zero spacetime interval to justify the term "direct interaction", since the interval can't be the primary causal factor involved; there must be something else involved that is *not* accounted for by the interval, and which differs in the two cases, and which therefore makes them something different than just a "direct interaction". 


#28
Dec1810, 12:43 PM

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But I think, yes, now that I think of it, there is another invariant in the two cases. The number of wavelengths in the photon between the two events must be the same, regardless of what reference frame you are in. There are all kinds of differences that are variant, but the count of wavelengths would be an invariant.



#29
Dec1810, 05:42 PM

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That said, the amplitude that is determined experimentally as I just described is *also* what we would use to determine the probability of detecting a single photon emitted from the source, at a given detector (the probability is just the square of the amplitudemore precisely the squared modulus of the amplitude, since the latter is a complex number). I don't know if the precise experiment I described has been done at the single photon level, but plenty of other single photon experiments have been done (i.e., experiments where, with very high probability, there is never more than one photon in the experiment at a time), and their results are exactly what is predicted by calculating the probability from the amplitude as I've described. 


#30
Dec1910, 09:50 AM

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This means that the number of wavelengths between the emitter and receiver cannot be an invariant after all... or we have to throw away special relativity. Now that I look back at it, I believe it was this very issue that made me begin to say "The photon represents a direct interaction between the source and destination." (Because there is an observer dependent variation in the number of wavelengths between the events. (sometimes the photon is represented as a "packet" often a sinc wave: sin(x)/x ; it moves at the speed of light but doesn't necessarily exactly oscillate in space.)) The attached diagrams helped me to see my mistake. There are two emitters in the diagram, and one receiver between them. In the first diagram the two emitters are stationary and the number of waves is the same. In the second diagram the two emitters are traveling to the left. Because of the relativity of simultaneity, the emitter on the right began broadcasting long before the emitter on the left. This allows there to be a larger number of wavelengths present in the approaching emitter than there are in the receding emitter. Edit: With the receding emitter and the chasing receiver, eventually the wavelength of the photon is going to be longer than the lorentz contracted distance between the emitter and receiver. This is another reason that I say "the photon is a direct interaction between the emitter and receiver." 


#31
Dec1910, 11:57 AM

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Try it with one emitter, at x = 0, and two receivers, one at x = R and one at x = 2R (i.e., both lying in the same direction from the emitter, but one twice as far away as the other), in the frame in which they are all mutually at rest. Then transform to a frame in which all three are moving with the same velocity v. That's the case I was talking about. 


#32
Dec1910, 07:38 PM

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#33
Dec2110, 10:22 AM

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You mischaracterized what I said. The two possibilities are (a) Both the frequency and the distance go up, or (b) both the frequency and the distance go down. It may go against your intuition (I know it goes against mine), but you need to think about it some more... I'll try to make this argument as simple as I can, but I think it comes down to two very important key ideas, that so far, you have not acknowledged. The first is that the distance between the emission event and absorption event is NOT an invariant. I'm not sure how to explain it any better, except tell you to go back and look at the diagrams that I've given you and admit it. Put it in your own words, explaining why the distance is larger when the emitter is chasing, and the distance is smaller when the receiver is chasing. It's not really difficult enough to show you a bunch of intricate math. Seeing is believeing. (the ascii diagram may have failed you? Do you need me to draw six lines in a jpeg to show you?) The second issue is whether the frequency is going up or down. With this, you have to put yourself in the position where the receiver passes you. If the emitter is chasing the receiver, then the frequency goes up. If the emitter is running away, then the frequency will be lower. Again, this isn't something I can explain with a bunch of intricate math. You just have to think about it, and admit that it's true. The two possibilities are (a) the frequency goes up and the distance goes up. or (b) the frequency goes down and the distance goes down. I'm sorry, but there is no option, as you claimed, for the frequency to go up while the distance goes down, or vice versa Edit: Do I need to mention that the number of wavelengths along the actual nullpath of the photon is zero? It is akin to watching a surfer on a wave; who doesn't move forward or backward or up or down so long as the camera keeps up with the wave. So yes, in fact that is an invariant. 


#34
Dec2110, 06:58 PM

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If what you meant to say is that the distance, meaning spatial distance, between the *emitter* and the *receiver* can be different in different frames (so it is not an invariant), then I agree, but there is no frame where the events on the emitter and receiver worldlines between which "distance" is measured (i.e., which are crossed by a single line of simultaneity in the frame) will be the emission and absorption *events* (again, because those events are nullseparated, not spacelikeseparated). This meaning of "distance"i.e., distance between emitter and receiveris what I was using the term "distance," without qualification, to refer to. However, I may not have been clear enough about that, since I was also talking about an observer that sees both emitter and receiver as moving, and "distance" could have been taken to mean the distance to that observer, at some particular time, which is *not* the way I meant it. See below. The above is what I've been talking about when I talk about "distance", and that's why I said that it was possible for the frequency to go up while the distance was going down: if the emitter is moving towards the observer, then the frequency goes up, but the distance between emitter and receiver goes down by Lorentz contraction. My quick back of the envelope check seems to indicate that in this case, the Doppler shift in frequency exactly compensates for the Lorentz contraction, so the product (frequency / c) * (distance between emitter and receiver) does in fact stay constant. (Note the way I stated the product this timethis is what I meant last time, but the term "distance to source" was ambiguous, and that may have caused confusion. Sorry about that.) The subtlety, of course, is that the above nice compensation only works for a Doppler *blue* shift! For the case where the emitter (and hence also the receiver) is moving *away* from the observer, so there is a Doppler red shift, the nice compensation doesn't work; distance between emitter and receiver goes down (by Lorentz contraction), but frequency *also* goes down! So this product as it stands won't work as an invariant. However, there is something that will; see below. You'll note that, with reference to this particular point, I did not just use the term "distance," unqualified; I used the phrase "the number of wave crests counted along a given line of simultaneity, in a given proper distance along that line." 


#35
Dec2110, 07:03 PM

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#36
Dec2210, 01:04 PM

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You should distinguish between "distance between objects" and "distance between events." The "unqualified distance" term you are using, is the distance between the emitter and receiver objects. This is the more familiar distance discussed under the heading of "length contraction" or "lorentz contraction", where solid bodies appear to contract in the direction of motion. But distances can and should also be measured between nonsimultaneous events. For instance in this emission and absorption example, you shouldn't pretend that the distance traveled by the photon is equal to the distance between the emitter and receiver. Why? Because while the photon is under way, the receiver either moves toward or away from the emitter. This means that the distance between the receiver and emitter is irrelevant. It is the distance between the emission EVENT and absorption EVENT which is relevant. You claimed the "distance between events" is not welldefined, but I disagree. Whenever you do a lorentz transformation, the x, y, and z terms refer to the coordinates of events; not objects. The distance between the emission and absorption events in any given frame is another distance [tex]\sqrt{\Delta x^2+\Delta y^2+\Delta z^2}[/tex]. The LT equations themselves do not distinguish to whom or to what the events occurred. All that is of concern is where and when, relative to a stationary origin. Now for the snowflake example: Let's note that this distinction (distance between events vs. distance between objects) should show up in basic "Galilean" relativity as well. We could talk about snowflakes coming down with a driver passing through them. The question is, are the snowflakes really coming straight down, or are they going almost 55 miles per hour horizontally? The fact is, you could arrange two rulers, one long, nearly horizontal one traveling along with the car, and another short vertical one stationary with the ground, and the snowflake follows the path of both rulers. The top end of the two rulers meet when the snowflake passes the top, and the bottom end of the two rulers meet when the snowflake passes the bottom. (but at every moment, the two rulers and the snowflake align.) My premise is that both the driver of the car, and the person on the ground use their own rulers and give correct (but different) answers for the distance between those two events. That is the premise behind the galilean transformation, and the lorentz trasformation. First let me point out my snowflake example above. Who is correct? The guy in the car who thinks the snowflakes are coming right at him, or the person on the ground who thinks the snowflakes are coming straight down? You can pick out a frame of reference that meets some arbitrary criteria, but I think the most sensible criteria is "observer dependence." Use the distance as measured by a ruler in that reference frame, where your observer is. The driver and the person on the ground are equally correct, so long as they make it clear who and where they are. You may still feel like one of the drivers is incorrect, but remove the planet, and just have the car and the snowflake flying through space. The snowflake is coming at the car at 55 miles per hour and travels a whole second at that speed. That distance is welldefined, even though the events did not happen at the same time. 


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