Note that the gradient of a scalar field is a vector, not another scalar. But if the field source begins to move, does the field gradient point toward the instantaneous or retarded position of the source? That depends on whether the field updates or regenerates instantly or with delay. There is no such thing as a gloval gradient: it is a vector field, which means that at every point of the trajectory, there is a (different) vector. These vectors do not even live in the same space: at each point of the trajectory, a tangent vector space is attached. To say what he did is simply incorrect.
Now, given that at different points we have different vectors, the question does indeed arize as to which of these enter physical laws. That is a regular question in physics, but once a particular law is consistent with empirical observations, it is accepted as a working hypothesis.
The author speaks here in terms of college calculus. Had he taken a course in geometry, he would not make such a statement, I am sure.
Physics has an issue that math does not. It is his understanding of mathematics that creates a dissonance in his view.
Incidentally, I would not want to give an impression that I not welcome someone disagreeing with Einstein' theory. TO the contrary, I would welcome such questioning and find it exciting. What seems to be happening here is a misstatement of the theory, or mathematics, is argued against. That is at best, and at worst we have some really basket cases. Nevertheless, thank you again for pointing me to that site.
Best regards, TQ.
(Please don't feel that you should supply an even more simplified explanation; while at some point I would "get it", a couple of months later I will have forgotten and start pestering you all over again.)
Popular science books are a blessing and a curse at the same time. My bookshelf includes several that attempt to explain Einstein's theory of relativity (although mostly limited to special as opposed to general relativity) in lay terms. I work my way through these books, often re-reading a sentence or paragraph several times until I feel that I've understood.
In reality it can't be said that I truly understood it at all, otherwise I would be capable of independently detecting and refuting such erroneous presentations of van Flandern's.
Is it because I'm stupid? I guess it depends on how you define intelligence. Most young boys want to be astronauts or firemen when they grow up, I wanted to become a Nobel prize-winning physicist (yeah, funny). That was before I discovered that I have no talent for math.
I wonder if there is such a thing as an indicator of intelligence that is free from bias and preconceptions.
My two lines of work are translation and simultaneous conference interpreting. I find written translation to be immensely harder than interpreting. Probably I suffer from attention deficit disorder although it has not been diagnosed. Just finishing a translation job on time requires me to exercise tremendous powers of will and concentration, and I sincerely believe that a choir of angels should appear every time and blow a trumpet chorus in my praise.
Although my customers sometimes tell me they appreciate my work, no one ever expresses admiration. But when I'm interpreting at a conference, invariably one or several participants will walk up to me and say, "Oh, I'm completely in awe at what you guys do. How do you do it?"
I'll smile and say thank you. Should I tell them that it's not work but fun and enjoyment? That instead of feeling wrung out at the end of the day, I'm exhilarated and ten years younger? I never suffer from attention lapses or boredom when I'm in the booth.
But what some people seem to think is an incredible feat actually comes easy to me.
Maybe there is not one intelligence but many, unrelated ones.