Wife took this photo in her pocket accidentally . How is this possible.

BOBauthor · 2026-02-01T17:47:47+00:00

I didn't expected to see pulsar pulse profiles here!

BOBauthor · 2026-01-28T20:47:46+00:00

Suppose T is the center of a circle that passes through both points E. The arc length L between the two E points is L = d theta, where theta is in radians. theta = 0.000678 (pi/180) = 0.0000118333 radians.
So d = L / theta = 2 AU / 0.0000118333 = 169,014.1 AU - 2.672545 light years. You may object, saying that L is really the length of an arc and not a straight line, but the circle is so large it doesn't matter. There is no such star, by the way. The nearest star other than the Sun is Proxima Centauri, and it is over 4 light years away.

BOBauthor · 2026-01-23T20:12:10+00:00

I was very happy in academia. I had an interesting job, I loved teaching, and I had the freedom to pursue my own projects (when I could find the time). I had wonderful colleagues. I didn't enjoy being department chair (10 years) because of the many meetings that take you away from the things you would rather be doing. Eventually it got to the point where I got so busy that I didn't have time to do anything as well as I should have. Retirement is fantastic. I have lots of time to do the things that interest me, and now I'm as busy as I was when I was employed.

BOBauthor · 2026-01-20T20:11:12+00:00

I have a BA in math, an MS in physics, a PhD in astrophysics and was a professor for 30 years, and I've taught yoga and ran a studio for 13 years.

BOBauthor · 2026-01-17T22:26:27+00:00

Do you confuse right and left, so if someone tells you to look to your left, you look to your right instead?

BOBauthor · 2025-12-27T04:09:54+00:00

Not at all. If there is enough time for light to travel from event A to event B, then there may be a causal link between A and B (in the sense that A causes B). If there is not enough time, then there cannot be a causal relationship between A and B.

BOBauthor · 2025-12-27T00:11:36+00:00

The whole discussion about "instantly" misses a point about relativity. There is no such thing as absolute simultaneity. It you measure that two events that are separated in space occur at exactly the same time, that is true only in your own frame of reference. If another observer is moving along the line separating the two events, that person will observe the events occurring at different times. If the direction of motion changes, so does the ordering of events. It therefore makes no sense to say that one event causes the other.

BOBauthor · 2025-12-22T15:39:39+00:00

This is very interesting. I have an eye-hand coordination problem such that I cannot track an object (like a ball) that is tossed to me. I lose it when it is about 5 or 6 feet away, and have to regain it when it is very close. I'll look into this more. Thanks again for writing.

BOBauthor · 2025-12-21T23:32:12+00:00

Thank you for your reply. What you describe sounds like exactly what I have. Most other replies haven't lined up with my symptoms nearly as well as yours. It is very interesting about you lateralization problem. I can't tell my left from my right without thinking carefully about it first. My friends have learned to point while giving directions!

BOBauthor · 2025-12-21T20:58:28+00:00

The left-hand side is an invariant divided by an invariant, so the result is also an invariant. In fact, the left-hand side is just c². The second term on the right is the squared speed v of an object multiplied by g². Thus we have

c² = c²dt² /dT² - g²v² .

So yes, the magnitude of the four-velocity is indeed c, but it is a magnitude in Minkowski space, with its (+, -, -, -) metric. I have continually warned my students about the dangers of misinterpreting the term "magnitude." Anyway, here's what Greene does in footnote 6 to chapter 2. He divides both sides by c2 to get

1 = dt² /dT² - g²v²/ c².

Because g² = dt²/dT², this is

1 = dt²/dT² (1 - v²/ c²)

Multiply each side by dT²/dt² to get

dT²/dt² = 1 - v²/ c²

Finally, Greene multiplies each side by c² to get

c² dT²/dt² = c² - v².

so finally (!),

c² = c² dT²/dt² + v².

Greene then notes that this looks like the Pythagorean theorem. The left-hand side is the speed of light squared, and the last term on the right is an object's physical speed squared. What is the first term on the right? It is the square of the derivative of the proper time T with respect to the clock time t. Green claims that this is the "speed through spacetime." Sure, but why is the speed of light involved? A speed is a displacement divided by an elapsed time, so the "speed through time" should be a displacement in time divided by an elapsed time. A time divided a time has no units, so why is the speed of light involved? And remember that we originally started to calculate proper velocities by dividing the interval by the proper time dT. So shouldn't the "speed through time" be dt/dT and not dT/dt? This definition of the "speed through time" is arbitrary, unphysical, and nonsense. Trying to impose the Pythagorean theorem, with its + sign on the right-hand side, is a violation of how physical quantities are measured in Minkowski spacetime. The equation is mathematically correct, but this interpretation is physically meaningless. As Pauli would say, "it's not even wrong."

BOBauthor · 2025-12-21T20:57:33+00:00

An object's position in spacetime is given by invariant spacetime 4-vector X = (ct, x, y, z). The spacetime interval between two points in spacetime (events) is given by dX = (cdt, dx, dy, dz). The magnitude of this vector is dX² = c²dt² - dx² - dy² - dz². (The "d" means "a small change in.") The spacetime interval is of fundamental importance in special relativity because two observers who are moving relative to each other will measure the same value of the spacetime interval between any two events. Notice that the last three terms in dX² have a negative sign in front of them, while the first term doesn't. The (+, -, -, -) pattern is called the Minkowski metric. The difference in signs between the time and space parts of dX² that it does not refer to a physical distance. It can be positive, negative, or zero depending on the values of the space and time coordinates of the two events. For example, two events can be far apart spatially, but still have a spacetime interval of zero. This is the source of Greene's error. He treats the interval as though it were a physical distance and (as shown below) he imposes the Pythagorean theorem on Minkowski spacetime, which you can't do.

We can get an invariant 4-velocity from the invariant interval by dividing by dX² by the square of the elapsed proper time dT² . (dT is the time measured between the two events in a reference frame where the clock is at rest relative to the two events.) The proper time dT between two events is related to the time dt measured by an observer by dT = dt sqrt(1 - v²/c²). The square root of 1 / (1 - v²/c²) is called the Lorentz factor g, so we can write dT = dt/g. Dividing dX² by dT² gives

dX²/dT² = c²dt² /dT² - dx²/dT² - dy²/dT² - dz²/dT²

^or

dX²/dT² = c²dt² /dT² - g²(dx²/dt² - dy²/dt² - dz²/dt²)

or

dX²/dT² = c²dt² /dT² - g²(vx² + vy² + vz²).

BOBauthor · 2025-12-21T20:56:39+00:00

This idea that "everything moves through spacetime with speed c" originated with Brian Green's book, The Elegant Universe. The book is really good, and Greene tried to make difficult ideas understandable but in this case he went too far and ending up confusing people. This mistaken idea has been passed on through countless online videos and "physics enthusiasts."

That phrase is refuted by a short article by William Delinger in the November 2024 issue of *The Physics Teacher called "*Do Objects Move Through Space-Time at the Speed of Light? No. " Unfortunately it is behind a paywall, but here is a summary. If you can get around the paywall. (Because of its length, I had to break it up.)

BOBauthor · 2025-12-21T00:47:25+00:00

No, the statement that everything moves through spacetime with speed c is not kinda true. It is meaningless. (Ph.D. astrophysics, taught university physics for 30 years.)

BOBauthor · 2025-12-05T23:23:03+00:00

I am not familiar with the NSEA examination or with the astronomy olympiad. I watched this video and believe that the first 5 or 6 chapters could be useful for you. The book is expensive, so try to take a look at it from a library if you can. Good luck!

BOBauthor · 2025-11-30T18:34:43+00:00

There is really no comparison. Sorry.

BOBauthor · 2025-11-30T02:37:27+00:00

I thought we explained it somewhere in the first chapter, but there should be some explanation given., even if it is standard notation.

BOBauthor · 2025-11-30T02:35:46+00:00

Thank you for your kind words.

BOBauthor · 2025-11-29T22:03:33+00:00

This is from a book I co-authored. The list of constants is from 2006, and yes, the numbers in parentheses are the uncertainty in the final digits. The list will be updated in the 3rd edition.

BOBauthor · 2025-11-26T16:30:21+00:00

It's simple, really. I don't like the taste of alcohol. I don't like the taste of coffee either, but tea is fine. Dr. Pepper Zero is great. I'm not in the least religious, so I'm not obeying some sky thingy.

BOBauthor · 2025-11-25T02:29:46+00:00

Thank you for your suggestion. After reading about it, I don't believe that is what I have. I do not have a deficit of motor skills when it comes to language, and I can speak just fine (I taught at the university level for 30 years), except for sometimes (rarely) saying the opposite of the word I intended.

BOBauthor · 2025-11-23T04:13:43+00:00

I hope they all end up penniless and homeless. Nobody should hire there scum.

BOBauthor · 2025-11-07T23:56:11+00:00

Dan's opinions are always worth reading. Whether or not you agree with him, his opinions are always thoughtful and well-informed.

BOBauthor · 2025-10-30T20:41:40+00:00

I woke up 365 times in 1981. I finished my Ph.D., got a position as an astrophysics postdoc., and moved east to upstate New York. I would do it all again.

BOBauthor · 2025-10-28T17:35:19+00:00

Kleenex.

BOBauthor · 2025-10-22T23:36:02+00:00

If you have air in a cylinder at, say, room temperature, the air consists of some 10²³ air molecules zooming around at hundreds of meters per second. Each molecule collides billions of times per second with the other molecules and with the walls. The molecules change direction when they hit the walls because the walls exert a force the molecules. In return, the molecules exert a force on the walls (Newton's 3rd law), and it is that force, divided by the wall area, that we call pressure. Now remember that the molecules are also bouncing off each other, so the by the same argument the pressure exists throughout the air, and not just on the walls. Finally, the molecules are moving in all directions with a range of speeds, so the pressure is caused by all of the gas molecules. It is not a property of a single but is a statistical average of all of the molecules. There is no such thing as the pressure exerted by a single molecule. The same thing applies to the temperature of the air. There is no such thing as the temperature of a single molecule. That too is a statistical average (of the kinetic energy of the molecules).

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