Iggy

Perhaps. Let me think about it... If this is something that has never been explored then I think it would make an interesting problem to solve. For example, if fish under an amber or red light don't school together by color then we'd know it is a visual cue. Otherwise it may be a behavioral cue and fish may not actually know what color they are. I wonder if it is a question that has been answered.

I have heard that genetic studies and other research has disproved the idea that physical and behavioral qualities are a function of breeding populations. Here are some applicable quotes: Keita, S O Y et al. 2004. "Conceptualizing human variation". Nature 36 (11s):"Modern human biological variation is not structured into phylogenetic subspecies ('races'), nor are the taxa of the standard anthropological 'racial' classifications breeding populations. The 'racial taxa' do not meet the phylogenetic criteria. 'Race' denotes socially constructed units as a function of the incorrect usage of the term." AAA 1998:"Evidence from the analysis of genetics (e.g., DNA) indicates that most physical variation, about 94%, lies within so-called racial groups. Conventional geographic 'racial' groupings differ from one another only in about 6% of their genes. This means that there is greater variation within 'racial' groups than between them." Lee et al. 2008:"We caution against making the naive leap to a genetic explanation for group differences in complex traits, especially for human behavioral traits such as IQ scores" Harrison, Guy (2010). Race and Reality. Amherst, Prometheus Books:"Race is a poor empirical description of the patterns of difference that we encounter within our species. The billions of humans alive today simply do not fit into neat and tidy biological boxes called races. Science has proven this conclusively. The concept of race (...) is not scientific and goes against what is known about our ever-changing and complex biological diversity." Roberts, Dorothy (2011). Fatal Invention. London, New York, The New Press:"The genetic differences that exist among populations are characterized by gradual changes across geographic regions, not sharp, categorical distinctions. Groups of people across the globe have varying frequencies of polymorphic genes, which are genes with any of several differing nucleotide sequences. There is no such thing as a set of genes that belongs exclusively to one group and not to another. The clinal, gradually changing nature of geographic genetic difference is complicated further by the migration and mixing that human groups have engaged in since prehistory. Human beings do not fit the zoological definition of race. A mountain of evidence assembled by historians, anthropologists, and biologists proves that race is not and cannot be a natural division of human beings." Linking physical or behavioral standards to breeding groups or 'races' looks like something rather rejected by modern science.

A couple years ago 6 goldfish were born from the same batch of eggs and from the same female in my outdoor pond. 4 are black, 2 are orange, and one is yellow. They are all the same size. As they swim, the four black ones always stick together (usually touching or no more than a couple inches apart)—same for the 2 orange. The yellow one is usually by itself. My question is, how does a goldfish know what color it is? I couldn't find a quick answer online.

I do. It's quite approximate and weak-field-limit of me, but that's just the way I talk. Who can't talk in terms over c squared? Quite dielectric.

So... you aren't just mistaking my meaning, you are literally mistaking my words? How fun! If we were involved in something dialectic then it would suggest that there were a difference of opinion, but I assure you that is not the case. It's just that you said two wildly different things, and that is *dielectric*. Perhaps you could look the word up, and consider your position, and get back to me, because either "Schwarzschild coordinates are the standard", or "the coordinate speed of light isn't physical". You'll have to pick one or the other, and I guarantee you the dictionary says that the two are dielectric. It's good to terms of -c^2. If you're good to salt then you know exactly from whence it came, and I'm a little embarrassed you ask.

This sentence is meaningless. Dielectric means that you can't get from one to the other. My hope for you faded fast. We don't understand each other. I'll welcome your solution to Shapiro delay as proffered in post 3, or I'll welcome your insistence that nothing non-local is physical. I'll welcome either thing you've said, but they are fiercely dielectric. You can't get from one to the other, you know? The forces of the universe will keep those assertions apart despite your best efforts.

I apologize profusely. Is the variable speed of light the unphysical entity, or the "standard answer", both of which you'd said. Could you please just say one is right and one is wrong so I can understand what you mean? You were right before and wrong now, or right now and wrong before? I can't tell until you say it explicitly. Is it that the "variability is applied to an unphysical entity" or is it that "Schwarzschild coordinates are standard"? I'm happy with either. They are physically dielectric. Can't you choose one thing of which you've said?

I'm sorry... the "coordinate speed of light" is unphysical, or "Schoolchild coordinates are the standard". Which were you saying? It looks above like you just negated something you said earlier. Were you ok with that as long as you're interjecting with me now? It's a bit difficult to tell. Can you just quickly say "the coordinate speed of light is unphysical, and therefore the earlier thing I said about it being the standard answer is just my best attempt at a pail stain on the carpet". Can't you just say that? Or some humerus variation of it? It would end our confrontation immediately.

Correction: the coordinate speed of light is variable in Schwarzschild coordinates. That isn't a correction. That is something logically equivalent to what I said. I'm sure Euclid made some point about things logically equivalent to a mutual party are logically equivalent to themselves. The speed of light in a certain set of coordinates doesn't need corrected as the certain coordinate speed of light in coordinates. That is just you skipping the issue as always, I hope. I hope for you as always. Please derive Shapiro delay in a way that reflects nothing of Schwarzschild coordinates. While you're at it, please say that post 3 has nothing to do with anything real or physical. Thank you.

Tell me about it. The speed of light is variable in Schwarzschild coordinates. How wrong can one person be? Which of the two above quotes would you like to retract? I'll be happy with either, but you still haven't answered.

Yes, I'm perfectly capable of doing that. If, then the radial coordinate velocity of light is, [math]v_c = \pm \left(1- \frac{r_s}{r} \right)[/math] and in terms of acceleration of gravity, g, and two radial points, A and B... [math]v_{BA} = \left(1- \frac{gr}{2c^2} \right)[/math] (where A is further than B and [math]v_{BA}[/math] is the average velocity of light between the two as measured by A) If true then we get Shapiro delay for a round trip A->B->A... [math]\Delta \tau = \Delta t_{flat} \left(1 + \frac{gr}{2c^2} \right)[/math] and then light is slowed relative to flat spacetime, and many experiments have confirmed this, therefore, You have to decide if you're the sort of person that says "the answer in terms of Schwarzschild coordinates is the standard" or decide if you're the sort of person that denies that Shapiro delay originates in a variable speed of light. Declaring both is incongruent. Are you more interested in proving yourself (and post #3) correct, or proving me wrong?

Distance is constant in the Rindler metric a priori. A rindler observer at x=1 is always at x=1, and the same with an observer at x=2. The distance between them is constant one by definition (it's assumed by the metric). Constant distance is proven in the derivation of the metric, not with the metric. It's like asking someone to use a meter stick to prove that the meter stick is one meter. It's a confused question.

I appreciate your feedback, unfortunately we are no closer to a common understanding. Both your metric and mine are Rindler, and they are physically equivalent. I will explain in a moment. I should first say that the t variable on the right hand side of your equation is not proper time, but rather coordinate time. This is common to all metrics (the t and r on the right hand side of the Schwarzschild metric, for example, are coordinate time and radial distance). In fact, you rightly solve the metric above to get the following: If [math]d \zeta[/math] and [math]d \tau[/math] were proper distance and time then that wouldn't be a coordinate speed (which it should indeed be). It would be a proper speed. More important to your point, however... The Rindler metric I gave is more common than yours (it is found on wiki you'll notice). Yours can be found on the last link I gave to JVNY (here). Mine: [math]ds^2 = (ax)^2(dt)^2 - dx^2[/math] Yours: [math]ds^2 = (1+ax_i)^2 dt^2 - {dx_i}^2[/math] The difference between them is nothing more than a shifting of the x-axis origin by 1/a. In geometry it is called a coordinate translation. In other words, plug [math]x = x_i + 1/a[/math] into my line element and you will get yours. The x axis has been relabeled -- it has been shifted (which makes no physical difference). The rindler horizon in my metric is at x = 0 and proper acceleration, a, starts at x = 1/a The rindler horizon in your metric is at xi = -(1/a) and proper acceleration, a, starts at xi = 0 Of course, this means that the observer at x in my line element has a different speed of light from the observer at xi in yours, because we are talking about two different observers with a distance 1/a between them. As an analogy, if I said "Mt. Everest is 4,650 meters tall from its base", it makes no sense for you to object by saying "No, Mt. Everest is 8,848 meters tall from sea level". It is an obfuscation, not a disagreement. My speed of light: [math]v_c = ax[/math] where my x is the same as your [math]x_i + 1/a[/math]... therefore my speed of light rewritten in your relabeled x-axis: [math]v_c = a(x_i + \frac{1}{a})[/math] [math]v_c = ax_i + 1[/math] My speed of light is the same as your speed of light. You've just found a line element where the x axis is shifted by 1/a. Mt. Everest can be both one height from its base and another from sea level. This absolutely does not change the conclusion I made -- that the coordinate speed of light is 100 times greater 100 times further from the horizon. No, that's either a typo or you're misunderstanding how to use the metric. The observer at x = 1/a will measure the speed of light to be zero at x = 0. The observer at x = 1/a does all of the measuring to determine the coordinate speed [math]v_c = xa[/math]. In my metric, the Rindler horizon is at x=0 which is why the coordinate speed of light is zero there (just like it would be zero at the Schwarzschild horizon). In your metric the Rindler horizon is at x = -1/a (remember, the x axis has been shifted by 1/a between them), and the observer at x = 0 does all the measuring. That is why plugging -1/a into your speed of light equation gives zero. The observer at x=0 measures the coordinate speed of light to be zero at the horizon. I understand that you are trying to show that coordinate speeds have no physical meaning, but if you understand what the coordinates mean I think there is a good amount of meaning. There is, after all, no way to discuss non-local speeds in GR except via coordinate speeds. Denying physical meaning because GR works with any coordinate system is a bit like saying "this building could be measured in feet or meters and they give different values therefore the height of the building has no physical meaning". Ok, I think I gotcha.

I forgot to answer this part, sorry. It seems you're finding the coordinate speed of light where the t axis is made by the worldlines of the accelerating ships, and the x axis is made by the lines of simultaneity in the inertial frame. The problem I tried to touch on in a rush yesterday is that time dilation isn't accounted for in that case. As an analogy, if someone sat near the event horizon of a black hole with rockets firing at enormous acceleration in order to hold their position then they would see the universe above them unfold at an extremely rapid rate. Light would move from one side of a galaxy to the other in an instant (from their perspective). If they set themselves up as the origin of a coordinate system then they'd need to account for time dilation to account for the increased coordinate speed of light above them. The natural way to do this in the case of acceleration is rindler coordinates in which the speed of light is [math]v_x = x \cdot g / c[/math]. In your example, an observer at x=1 is considering the speed of light of someone at x=100. The speed of light in that case is, [math]v_x = 100 \cdot 1 = 100[/math] In other words, the speed of light is one hundred times less for someone one hundred times closer to the rindler horizon. Let's do that in real-world numbers... Let's say I am in a rocket accelerating at 100,000 km/s2. To my right is a line of rockets with less and less proper acceleration, and to my left is a line with greater and greater such that we maintain proper distance. I calculate my distance to the Rindler horizon... [math]x_0 = \frac{c^2}{g_0} = \frac{(299792 \ km/s)^2}{100000 \ km/s^2} = 898752.4 \ km[/math] 100 times this distance is 89,875,243 km, and the coordinate speed of light there would be... [math]v_x = \frac{xg}{c} = \frac{89,875,243 \cdot 100,000}{299792} = 29,979,200 \ km/s[/math] one hundred times the local speed of a ray of light.

I think the thing is that accelerated coordinates usually transform away from the horizontal Cartesian-like time coordinate. My understanding is that they should look more like... from this site. Or... From wiki. I think everything you're saying about the spatial coordinate is good, but the time coordinate would spread out as it fans right to reflect how clocks tick faster further from the origin. A light ray at 45 degrees around the middle of these diagrams would cross about one unit of t for every unit of x, but if you imagine traveling far from the origin where the temporal lines are really spread out, a light ray could cross a lot of hyperbolic spatial lines between each temporal coordinate line. Hopefully I'm making sense. I found, also, an interesting site comparing the speed of light in a uniformly accelerating frame, and a uniform gravitational field... Speed of Light in a Gravitational Field Actually, this site: http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_pathway/uniform_acceleration.html about the diagrams, for sure