Markus Hanke

Fixed that for you.

The result is manifestly wrong, as shown by the maths above, irrespective of how you might have arrived at it. Again, see the maths above. The inner product of 4-velocities cannot logically exceed c^2. Again, the inner product is an invariant, so it cannot depend on \(\gamma\). It is trivially easy to formally prove that. Also, if that weren’t so, Lorentz invariance could not hold in the real world - but of course it does, as shown by experiment and observation. I don’t know what you are actually trying to show here, but whatever it is, it is physically and mathematically meaningless. I think it is fairly obvious by now that you aren’t actually prepared to listen to any of the feedback given to you here. You are just repeating the same things again and again, and ignore the fact that you have already been shown wrong. If you were actually interested in learning something, you would listen to what has been said, and use it to improve your own understanding of the subject matter. It is not enough to just blindly do algebraic manipulations - you need to also understand the geometric meaning of these objects, and how they are related to one another, and how that fits into overall physics.

You haven‘t even addressed them yet, let alone refuted anything. I have already shown you above that the absolute norm of a 4-velocity is always exactly equal to c. It can‘t be any different of course. The mistake you made is assuming that 4-velocities can be any arbitrary 4-vector, but that isn‘t true - all 4-velocities are 4-vectors, but not all 4-vectors are automatically 4-velocities. The spatial and temporal part of a 4-vector need to be related in a specific way (as I showed above), or else it isn‘t a 4-velocity. When you take this into account, you end up with an equality, not “>=“. The same is of course true for 4-momenta, since \(p^{\mu}=mu^{\mu}\). The algebraic expressions you derived on the RHS of (1) and (2) are correct, but the “>=“ inequality is not. Point (3) is trivially wrong, since the norm of a 4-vector is an invariant, and hence cannot functionally depend \(\gamma\). You are just copy-pasting the same stuff over and over again, without addressing anything that is being said to you.

I‘m sorry, but that doesn‘t make any sense at all. The tensor transformation law applies irrespective of the value of individual components in a particular coordinate basis, so saying that all tensors should be null tensors on account of their transformation law is an entirely meaningless statement. To pick just one random example, the Minkowski tensor evidently transforms as a rank-2 tensor, but it most certainly isn‘t null, regardless of what coordinate basis you choose.

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You haven’t. You are simply repeating the same statements again and again. You are right, this was my mistake, I actually thought of the photon’s 3-momentum (which is not zero!) when I wrote my post, whereas you were referring to the 4-momentum.

I will not be opening files from untrustworthy sources, and you have failed to presented it here, so I haven’t seen what you did. However, the details are irrelevant, because the very fact that you arrived at a contradiction is the error. When you start with a 4-vector - which is a rank-1 tensor -, and take the covariant derivative, the result is a rank-2 tensor. Since both of these objects are tensors, they are subject to the same general transformation law - so there cannot be any contradiction. You’re simply mapping a rank-n tensor into a rank-(n+1) tensor. If you arrived at anything else, then the error is logically yours; there is nothing to be refuted here.

What characteristic of these people would you be studying, and what would your methodology be? And how do characteristics of people relate to the ontological existence or non-existence of a deity, or its characteristics? I wasn’t referring to the text itself, I was referring to the way people interpret it. Why are you engaging in it, then? If you are fully convinced of the veracity of your own beliefs, then it should make no difference whatsoever what anyone else thinks, and there is no need for any debates. Yet you are here trying to argue your points in front of an audience that doesn’t share your worldview, and never will. The Christian God is a learned and acquired concept - you learn of it and about it from other people, or from written texts. It’s external information, not intrinsic experience. Had you grown up in an environment where that external information was absent, the concept you now believe in so strongly would never even have occurred to you. God, as the concept is understood in Christianity, is a mental, social, historic, and cultural construct.

I don’t know what you think the Bible might suggest, since there are about as many interpretations of the text as there are readers, and they are all quite different. This is true for most religions. So if the Christian God is part of the material universe, as you seem to say here, can you then suggest a scientific experiment that might show his existence and characteristics, in a way that is repeatable and independently verifiable?

There is no “discrepancy”, so the point is moot. Non-sequitur. If you arrive at some kind of “discrepancy”, then that means you did something wrong, plain and simple. Tensor calculus is not a “theory”, it’s a mathematical framework that has been extensively developed, and is fully self-consistent.

This is not what I said - you need to go back and look at what I actually said, and you will also find the maths there. Photons have no rest mass, so according to your expression, the inner product of photon momentum with itself is zero. Since the inner product can vanish only if the two vectors are either perpendicular, or one of the vectors has zero magnitude, that means that according to you the photon has no net momentum. Just repeating the same thing again does not make it any less wrong.

As joigus said, there are no conflicts or discrepancies at all. As you correctly say, the covariant derivative yields a tensor, so it automatically has the correct transformation properties. This is so pretty much by definition, because otherwise it wouldn’t be a covariant derivative at all!

\[x^2\]

This is meaningless. If you are projecting a shorter 4-vector onto a longer one, the result can never exceed the length of the longer vector, that’s just basic geometry. In other words, the longest a projection can ever be is that of a 4-vector onto itself, which, in Minkowski spacetime, is thus -1. Since the inner product is invariant, this is true for all 4-vectors in all frames. Consider a general 4-velocity of the type \[u^{\mu } =( \gamma ,\gamma v_{x} ,\gamma v_{y} ,\gamma v_{z})\] The inner product with itself is \[u^{\mu } u_{\mu } =-\gamma ^{2}\left( 1-v^{2}\right) =-\gamma ^{2} /\gamma ^{2} =-1\] as expected. So your claim is wrong. If this were true, the momentum of a photon would be zero. This is evidently false, as we know already from experiment and observation. Mathematically, you can show this in a similar manner as above. So again, you are wrong. I’ve already shown above that the inner product of a 4-velocity with itself is -1, so the magnitude of a 4-velocity in spacetime is always exactly c. Since the inner product is invariant, this is true in all frames, so it can’t be a function of the gamma factor. Also, if you look at this expression, you should notice immediately that the resultant magnitude of v does not correspond to the gamma factor you are inputting, so the expression is meaningless. You are wrong on this one, too. Yes, because all three points you have presented contradict both basic maths, as well as observation in the real world. It’s simply wrong.

Peer review is not meant to be “fair” (what does that even mean?) - on the contrary, it is designed to be as critical as possible, so as to really put the ideas within the publication to a rigorous test. It is the most effective way to tease out any problems; remember, you want to end up with something that actually works, in the sense of the scientific method. You are missing the salient point. Science has nothing at all to say as to the existence or supposed characteristics of anything that isn’t part of nature, including any and all notions of deities. This is quite simply outside the domain of applicability of science, because the very notion of “God” is not amenable to the scientific method. So science neither rejects nor endorses the Christian faith, because it deals with a different domain of enquiry. However - and this is the important point - if someone proclaims an element of their faith as being objective truth, then this claim will of course be challenged by science. Some such claims may turn out to be compatible with scientific evidence, so they are fine; others may not be, and those will be rejected. To give a simple example - if someone claims, based on certain readings of the Bible, that the Earth is ~6000 years old, then science will certainly reject this, because that claim is evidently false based on all available scientific data. So the issue isn’t faith and belief - the issue is only when people try to misrepresent their beliefs as objective, scientific facts. That’s what’s called a category mistake, and it will always be challenged.

No, a flat manifold without boundary is simply one of the topologies that naturally arise from the mathematics of the Lambda-CDM model, when certain observational parameters take on specific values. It’s just one among several possible options. It can be closed on itself - so there can be a largest possible separation between two suitably chosen points, without there being any kind of boundary. Much like (to pick a lower-dimensional analogy) a spherical surface. A manifold being finite in extent does not imply the presence of a boundary.

The mechanism is one of spontaneous symmetry breaking, so it would depend on whatever parameters appear in the Lagrangian. In this specific case it would be the speed of light, Planck’s constant, as well as the coupling constants between the various fields. The Higgs mechanism has no effect on the value of fundamental constants (other than ones related to the masses of particles of cause, which aren’t really ‘fundamental’). It only gives rise to some new coupling constants within the Lagrangian, since after the event you end up with more fields than you started off with.

This is evidently not true, since the very mechanism that creates the property of mass (the Higgs mechanism and spontaneous symmetry breaking) already presupposes the existence of at least a Minkowski background spacetime prior to said process; without this, there would be no universe as we know it today.

We are not really in a position to give any kind of medical advice or diagnosis here, especially not based on very non-specific symptoms such as the ones you describe; if you are concerned about your friend, you need to urge them to go and see a medical professional.

The appearance of a singularity in a model of physics generally means that the model has broken down because it has been extend beyond its domain of applicability. It does not mean that the model actually predicts a physical singularity to occur. In that sense, singularities - whether gravitational or at the BB - (almost) certainly are not actual, physical objects; they are more like flags saying “we don’t know yet what happens here”. Mass as a property of elementary particles only appeared at and after electroweak symmetry breaking (~10^-35s) when the Higgs mechanism kicks in; prior to that, all particles would have been massless. So yes, the very early universe contained only various forms of energy - which, however, still has a gravitational effect of course.

This seems like an awfully complicated way to do this. Why not just use Helmholtz’s Theorem? We know that the curl of the potential field gives the magnetic field (by definition!), so this is already fixed. The potential field is also invariant under certain gauge transformations (I think it’s the addition of a scalar field gradient, but I’d have to check that), hence we will always be free to make the divergence vanish, simply be choosing a suitable gauge, without affecting the curl. So in essence, under Helmholtz’s Theorem, the divergence has no physical relevance at all in this.

! Moderator Note Moved to Speculations.

I would just like to add a remark here, perhaps some readers may find it helpful. I am a regular and committed meditator - I practice several hours of formal meditation every day, and have done so for some years. Many of the perceptions described here are common and well known phenomena that naturally arise when the mind settles and becomes concentrated; in the Pāli language they are called nimittā. This can be anything from a slight tickling sensation somewhere, to pins and needles, to a sensation of something moving as a current through the body, to various pains, to full blown auditory and/or visual hallucinations, among other things. A sensation of electrical currents in parts of the body is especially common, from what I have seen. A had a period a few years back when I used to get this regularly, and the sensation of electricity sometimes got strong enough to cause me considerable discomfort, and gave me twitches and involuntary muscle spams during meditation sits. I have heard of people for whom this becomes so strong that they suffer intense pain, muscle cramps, and involuntary movements - they literally “jump” on their meditation cushions. Some people need to temporarily stop sitting because of this. As described here, with a little practice it is easy to induce these sensation at will, and control them to some extent - one can move them around the body, make them stronger or weaker, change their qualities etc etc. I cannot speculate what the underlying mechanisms are, as the human body is not my area of scientific expertise. What is clear though is that body and mind are not separate things, they are intimately connected, so it isn’t surprising that such things may occur. These phenomena are quite natural, and very common among meditators; there are specific ways and methods to address these things, in the context of an ongoing meditation practice. The general advice is to not pay too much attention to them, since directing the focus of attention towards these phenomena will strengthen them and make them occur over and over again. In many specific practice frameworks the occurrence of such phenomena is in fact taken as a sign of progress, since they naturally develop when concentration and single-pointedness become stronger. They can also become a hindrance though, because they can distract from practicing the main technique, and some people become infatuated with these sensations, as they also can feel very pleasant at times. So most of what has been described here is natural and quite well known, and not a cause for concern. This, however, is not: As someone with experience as a volunteer in the emergency services, this would have me concerned; anisocoria isn’t normal (unless you were born with it, which does happen), so I would strongly advise a precautionary trip to your doctor, to rule out other underlying issues.

There are no guarantees here. The best we can do is make the assumption that whatever finds the message has a roughly similar sensory apparatus as we do, and that their mental processes are roughly similar to our own; we can then attempt to construct a pictorial or auditory message in the most general and (to us) universal of forms, and hope for the best. Over and above that, all bets are off. The thing is that all languages are social constructs - words, sounds and pictograms mean to us what we take them to mean because everyone within our social context agrees that they do mean that, and we have been taught those particular conventions in early childhood. Even amongst us humans it can sometimes be very difficult to communicate certain ideas and concepts outside of a given social context, and our attempts at communication with other species in the animal kingdom have met with at best limited success. Communicating to an alien species that may share few or even none of our cultural and social conventions could be exponentially harder still - and potentially disastrous, should we get it wrong. In the worst case, the alien race may be sufficiently different in terms of sensory apparatus and mental processes that there isn’t even a common channel for communication, never even mind a common language. I don’t know how such an encounter would pan out.

Presumably an omnipotent being would have no need to observe the quantum system, he could have knowledge of its entire history without having to collapse it first. Since that knowledge is not accessible to us, this case would be indistinguishable from God not existing.