I am merely trying to point out here that the application of the velocity addition formula for a non-vacuum appears to be inconsistent with the assumptions it was derived from. I am not suggesting that the velocity addition formula (or indeed the Lorentz transformation) is otherwise consistent (in fact, as shown on my page http://www.physicsmyths.org.uk/lorentz.htm it isn't, but it would probably go too far to discuss this in this thread). Where did Einstein claim that it is the ' idealized x=ct ' he considers? Nowhere. According to his definitions, x=ct is the 'equation of motion' of a light signal, and this holds quite obviously only in a vacuum (which Einstein himself points out repeatedly). An 'idealized' speed of light would be metaphysical speculation and indeed logically inconsistent. It would be equivalent to saying that the equation 1=2 is true in an idealized sense when you consider the 2 as an idealized 1. Thomas
Which assumption? I still don't understand why you think SRT should be restricted to vacuum conditions. On your site, do you actually conclude that the Lorentz transform is inconsistent, or just that the derivation presented was inconsistent (which wouldn't tell you anything about the transform itself)? He does claim that the speed of light in a vacuum is invariant. In a sense the propagation of light in a vacuum is in ideal conditions, ie. unhindered. I agree it wasn't a clear remark, and I'll rephrase it if necessary. x=ct is the equation of anything that happens to be travelling at speed c, including (but not limited to) light under vacuum conditions. Einstein was claiming that it was the speed c that was invariant. Because light travels at this speed in a vacuum, he referenced the speed of light in a vacuum in his original postulate. I still don't see why this should restrict the theory to vacuum conditions only. I don't see the logic here, though I'll agree that "idealized" was not the best adjective to use in this context.
Why would you think that, as an analogy, the earth's gravitational acceleration of 9.81 m/sec^2 is an invariant quantity which holds under any circumstances? It doesn't. It is only invariant for all bodies in a vacuum. If your assumption is that the acceleration is 9.81 m/sec^2 , then this means that the situation applies to a vacuum but not a medium. How can something be consistent that has been derived inconsistently? In this sense it is not further surprising that the Lorentz transformation results in logical paradoxes like the Twin Paradox (or indeed the problem with the Fresnel drag we have been discussing here). Thomas
I don't. Only approximately. It also varies with latitude and altitude. My assumtion is exactly what I assume - nothing more and nothing less. If I assume that the acceleration due to gravity is constant everywhere without exception, nothing is preventing me from building a perfectly consistent theory based on this assumption. It would inevitably fail to agree with experiment (because, as it turns out, g is not invariant), but the theory could still be completely consistent with itself. In other words, a theory's internal consistancy and consistency with reality are seperate issues. It's the same story with SRT and the invariance of the speed c. SRT, based on this assumption, is internally consistent. Or you could change the invariance postulate to something different (eg. a locally dependent invariant speed or no invariant speed at all) and still end up with a flawless masterpiece of a theory. There are an infinite possible number of logically coherent theories. They're just not all equally good descriptions of reality. Very easily - a flawed proof does not imply that it's conclusions are false. A flawed proof proves precisely nothing. If you want real life examples, consider the number of theorems in mathematics that started life as a mere conjecture and were only proven correct years or centuries later. Andrew Wiles' original proof of Fermat's last theorem, published in 1993, was flawed. Would you have concluded at that point that FLT must have been false because the proposed proof was inconsistent? At this point it wouldn't matter if it were discovered that Einstein proposed the Lorentz transform on the basis of a coin toss (not that the Lorentz transform is credited to Einstein). It's well known that the Lorentz transform does exactly what it's supposed to do. Where do you see a paradox in the old twin gendanken? The Lorentz transform is a straightforward bijective linear application. In addition, it is its own inverse and the speed c is invariant if the two frames are inertial - properties that make it compatible with the postulates of SRT. There simply isn't anywhere a contradiction could lurk or hide.
The assumption that the gravitational acceleration is the same for all bodies everywhere without exception would not only contradict observations, but also be inconsistent with the laws of mechanics in general. As the latter are a mathematical description of reality, you can't thus separate a theory's internal consistency from the consistency with reality (unless you are arguing yourself inconsistently). Of course if you change the laws of mechanics by means of some corresponding 'transformation', you can make any assumption consistent with 'reality', but this change would be then an inconsistency in itself. Your comparison is not really appropriate here as the Lorentz transformation is not a mere conjecture, but is supposed to follow strictly algebraically from two axioms: 1) the circumstance that the propagation of a light signal is independent of the reference frame, and 2) the assumption that a velocity dependent linear transformation (equivalent to the Galilei transformation) exists for the location of a light signal in two reference frames. Now the point is that these two axioms logically contradict each other (as shown on my page http://www.physicsmyths.org.uk/lorentz.htm ) and therefore any algebraic results based on this must be logically flawed as well. As a simplied example, I mentioned above already the equation 1=2, which is obviously inconsistent, but adopting Einstein's methods, one could make it consistent by introducing a factor k such that the equation holds true i.e. 1=k*2 . For k=1/2, the equation would thus become consistent. In this way, one could indeed make any inconsistent equation consistent. So with this kind of 'relativistic' mathematics, any equation is 'true' by default. Thomas
Not in general - only with the classical laws that deal specifically with gravity. You could easily use F=G´mM instead of F=GmM/r<sup>2</sup> (not forgetting to modify related laws accordingly, such as gravitational potential). The rest of classical mechanics is logically independent of whatever gravitational law you use, so modifying the gravitational law like this would not lead to logical contradictions. This wasn't really my point though. If I propose F=G´mM as a theory on its own, it is internally consistent (unless you want to argue that F=G´mM somehow contradicts F=G´mM). I can also make additional assumptions and see how well they'd fit with my gravitational law - either borrowing from existing theories or building my own version of mechanics. There's nothing about F = G´mM that prevents it from being an axiom in a perfectly consistent theory. No. A theory's internal consistency, consistency with other theories, and consistency with reality are all seperate issues. As it happens, relativity is inconsistent with classical mechanics (because classical mechanics is not invariant with respect to a Lorentz transformation). When it was modified to be 'Lorentz compliant' (basically with the introduction of relativistic momentum), the new relativistic mechanics turned out to fit reality better than the original classical version. Why? (also I'm not sure what you mean here). You can argue about how the Lorentz transform is different from the examples I gave, but the point I was making was a very general one. If a proof or derivation turns out to be flawed, it does not automatically follow that the conclusions are also false. Also, it's not as if the derivation you're criticising was the only derivation or proof of the Lorentz transform. One possible (crude) derivation would consist of first working out the individual length contraction, time dilation, and relativity of simultaneity formulae, and putting them together. Also, Einstein was not the first to propose the Lorentz transform - it predated relativity by several years. Initially, it was observed that the whole of Maxwell's electromagnetism (not just c) remained invariant with respect to this transform. Another similar transform, the Voigt transform, also had these properties. It differed from the Lorentz transform by predicting breadth stretching rather than lenght contraction, and a different time dilation formula. It's also not its own inverse. The postulate only refers to the speed of light, ie if for some particle (v<sub>x</sub>² + v<sub>y</sub>² + v<sub>z</sub>²Please Register or Log in to view the hidden image! = c, then (v<sub>x´</sub>² + v<sub>y´</sub>² + v<sub>z´</sub>²Please Register or Log in to view the hidden image! = c (hope those primes are visible). You seem to be implying that it is the path of a particle travelling at c that is assumed invariant. I'm not sure what this is supposed to mean, or what it's supposed to add to the first postulate. First of all, the transformation is supposed to apply to everything, not just light. Unless I'm missing something (and assuming that we know what a coordinate transformation is), the only axioms necessary for deriving the Lorentz transform are: 1) The speed of light is invariant. 2) There is no priviledged reference frame (I think this rules out the Voigt transform). I suppose there's another implicit assumption, basically forcing the reference frames to move past one another at the velocity on which the transformation is dependent. If you say you can show that the Lorentz transform is inconsistent, can you actually give an example of it failing to comply with the postulates on which it is based? Right now is not the best time for me to spend time going through your site in detail, so I'm not sure what the relevance of this example is. I'll just say that modifying the equation 1=2 (which is false) to 1 = k*2 for k=½ (which is true) doesn't make 1=2 true. It also doesn't mean there's anything wrong with the new expression 1 = ½*2.
This is a transparently specious argument. The equation 1=2k is not true for any k but only for k=1/2. If you had a theory that stated that k=3, then the theory would be wrong. Similarly with relativity, the Lorentz transform is not some arbitrary and unspecified transform that can be selected to fit the data, but rather a specific transform that makes a single unambiguous prediction about any given set-up and has no adjustable parameters. You cannot use it so that "any equation is 'true' by default". The fact that a theory with no adjustable parameters so accurately predicts reality is one of SR's most impressive accomplishments. And the fact that you imply otherwise demonstrates, at best, a profound lack of understanding. -Dale
You are missing the point I made: the motion of a body can be described by the force F=GmM/r<sup>2</sup> only in a vacuum, but not in medium where additionally the friction force arises. So applying the vacuum value for the acceleration of a body is only consistent with the laws of mechanics in a vacuum. In a medium, the laws of mechanics require additionally the friction force to be taken into account. I don't see how you can separate 'reality' from the theoretical model: you can only describe reality in terms of certain theoretical concepts, and reversely theoretical concepts in physics are meaningless without referring to reality in some way. So a theory can not really be logically consistent (with itself as well as all other related theories) but inconsistent with reality, and the reverse holds as well. As I indicated above already, the implicit assumption is that the location of a light signal is essentially affected in the same way as normal objects by the frames moving past each other , i.e. that the usual concept of 'speed' (which implies a velocity dependent linear transformation for the location of the object in the two frames) can be applied to light as well. What's wrong with it is that it is a different equation, which has been multiplied by the factor 1/2 just for the sole purpose to make the original (inconsistent) equation consistent. Thomas
Yes, the equation 1=k*2 is only true for one value of k, but this is the same situation with the Lorentz transformation: for a given value of the velocity v, it only holds for γ=1/√(1-v<SUP>2</SUP>/c<SUP>2</SUP>). In both cases, the factor only serves as an ad-hoc measure in order to make the original inconsistent equation consistent. Thomas
F=GmM/r² isn't intended as a description of the motion of any body - it's just one factor affecting motion. The law describing motion is F<sub>net</sub>=dp/dt, which applies in all conditions. The contribution due to gravitational attraction between two particles is also unaffected by the current medium. Given the distinction made between gravitational, friction, and net forces, it doesn't make any sense to claim F=GmM/r² only holds in a vacuum. "Internal consistency" just means that if a theory allows you to make a prediction in two different ways, you will not get two different answers. It has nothing to do with how 'meaningful' a theory is. Consistency is a minimum requirement of any theory proposed for consideration, and is something that can be verified by someone sitting in an armchair applying logic alone. The same is not true of a theory's consistency with reality, which can only be determined by experiment. "Meaningless" is how I'd describe a statement that contained undefined or superfluous terms. F=G´mM is not meaningless in this sense. You know what F, m, M, and G´ are, and you know what equality and multiplication are, so the statement has a very precise meaning - namely that the gravitational force between two point objects is proportional to their masses and independent of (among other things) the distance between them. In other words, 'meaningful' equates to 'testable'. Someone who knew nothing of astronomy could not discard F=G´mM as false by logical arguments alone. It's not just an implicit assumption. The propagation of light can be measured in different reference frames, using different coordinate systems - just like anything else. The whole point of a velocity coordinate transform is to describe how these measurements depend on your point of view. So you'd stick with the original, inconsistent equation? It sounds like you're against modification and improvement. 1=½*2 is true, regardless of this statement's 'history'. The only logical fallacy one could make would be to claim that this somehow made 1=2 true. There's nothing inherently wrong with modifying a theory or basing a new model on an old one. Generally, the truth of any statement has nothing to do with its history and origins in human culture - Pythagoras' theorem is true no matter how or by whom it was discovered.
Excellent, I am glad that we agree that SR makes a definite prediction and does not have any free parameters available for making any of its predictions "'true' by default". Apparently you don't know the meaning of ad-hoc. You could say that in Lorentz ether theory the time dilation factor is an ad-hoc measure, but in SR it follows logically from the postulates. -Dale
Obviously I was referring to F as the net force. So F=GmM/r² would only apply in a vacuum (as would the light signal equation x=ct as argued above). Experiments can at best give a 'consistency check' for a theory. You can't really speak of consistency in the same sense here as with the logical, internal consistency of a theory. Because the experiments might be flawed due to circumstances that were not taken into account, a theory could well be consistent with reality although experiments tell you otherwise (of course the reverse could also be the case). Anyway, it is usually not so that you simply pluck a theory out of thin air and then check if it is consistent with reality. Most theories are developed exactly on the grounds of empirical evidence, so unless logical errors in the theory are made, it should already be consistent with reality (assuming the empirical evidence is not flawed for some reason). Pythagoras' theorem may be true but the equation 1=2 is not. Modifying the latter by adding a scaling factor k such that 1=k*2 becomes true for k=1/2 is not improvement but falsification (an originally false statement has been turned into a true statement). Thomas
If you force k to have such a value that 1=k*2, then the equation is true by default. This is not any different from SR where γ is forced to have the value 1/√(1-v²/c² ) such that x'=γ(x-vt) and t'=γ(t-vx/c² ). So the free parameter here is γ. Thomas
1=2k is not true by default and γ is not a free parameter. Your silly claims to the contrary merely demonstrate your illogic and bias. -Dale
The constant k(=1/2) in 1=k*2 is applied to make the equation 1=2 consistent, and likewise is the constant γ(=1/√(1-v²/c² )) in x'=γ(x-vt) applied to make the equation x'=x-vt consistent (which as such contradicts the invariance of the speed of light). γ itself is not directly introduced as a free parameter in Einstein's derivation, but the constants λ and μ (see my webpage regarding the Derivation of the Lorentz Transformation or Einstein's original derivation (Eqs.(3) and (4) in both cases; γ is identical to the constant 'a' here (a=(λ+μ )/2))). Thomas
The equation 1=2k is consistent (only for k=1/2). It is completely irrelevant that some different equation, 1=2, is inconsistent. "γ itself is not directly introduced as a free parameter" for the simple fact that it is not a free parameter. For you to insist otherwise is to demonstrate your ignorance about what a free parameter is. The term "free parameter" in science specifically refers to a model parameter that can be adjusted to match the data. For example, suppose you have a simple linear model y = mx + b for some independent variable x and some dependent variable y. You go out and perform some experiment measuring y for a variety of x. You then do a least-squares fit to the data to determine what m and b best fit the data. Because m and b are parameters that can be freely varied to make the model match the data they are called "free parameters". Obviously, the more free parameters you have the more likely an incorrect model is to fit any given set of data, so in science a model with fewer free parameters is always preferable (Occham's razor). SR has no free parameters and GR has one free parameter, so they are very difficult to beat on this front. γ is not a free parameter since it is completely determined by the relative velocity for a given experimental set up. You can verify that the SR γ is correct or incorrect, but you cannot freely adjust it to match the data. It is therefore not a free parameter. -Dale
It was you who introduced the notion of a 'free parameter' above, and if k in 1=k*2 is one, so is γ in x'=γ(x-ct). Of course it is not a free parameter in the usual sense (which basically would require that you have an under-determined system of equations). The equation 1=2 is not under-determined but simply false, and multiplying a constant k to one side but not the other is a mathematically invalid operation (even though it results in a true equation for k=1/2). Likewise is it a mathematically invalid operation to multiply γ to one side of the equation x'=x-vt but not the other (even though it results in a true equation for γ=1/√(1-v²/c² ). So you should probably better speak of a 'cheat parameter' rather than a 'free parameter' here. Thomas
Excellent, at least we agree on that then. Alternatively we could start with the expression 1=2k and point out that dividing one side by k and not the other is a mathematically invalid operation. Similarly, we can say that the mathematically invalid operation is to divide γ from one side of x'=γ(x-vt) and not the other. As you admit yourself x'=γ(x-vt) is a true equation, so x'=x-vt is in general not true since it is not obtained from a true equation via valid mathematical operations. -Dale
The 'F' in 'F=GmM/r²' is the force on one body due to another. It is generally not equal to the net force. Claiming this law only holds in a vacuum is equivalent to claiming that the gravitational force between two masses is affected by the matter between them. Do you really want to argue that our atmosphere weakens gravity? Exactly what is your argument here? That because you can find theory X that only holds for vacuum conditions, STR must also only hold for vacuum conditions? We agree then. I was trying to make the distinction between "internal consistency" and "consistency with reality" as clear as possible. Of course the ultimate goal of any theory is to explain observations, but there's a lot more to theoretical physics than graph plotting and curve fitting. I really don't see your problem here. 1=2 is false. 1=½*2 is true. There's really no two ways about this. In the case of this derivation of the Lorentz transform you've found, it looks like Einstein started by assuming that the transform that kept c invariant took a certain form, similar to the Galilean transform, and which he used as a template. Your criticism basically amounts to "A transform that keeps c invariant exists, but I don't like how it was discovered." Again, it is well established that the Lorentz transform keeps c invariant (unless you want to argue otherwise), and at the end of the day, that's really all that matters.
If the speed of light in a vacuum is invariant, and the speed of light in a medium is not, can't you use the different speeds of light in a medium as proof of an aether?
