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1. ### B Is velocity the reason for the time dilation effect?

Thanks to all!
2. ### B Is velocity the reason for the time dilation effect?

Yes. And I want to understand how is it possible that time coordinate assigned to that event is different for the train passenger and the observer. What is reason of this thing?
3. ### B Is velocity the reason for the time dilation effect?

Yes. I used this term wrong. I don't mean about "time simultaneous" or "at the same time". I mean that when light reach to end of train with point of view of passenger then light reach to end of train with point of view of observer. Is it true?
4. ### B Is velocity the reason for the time dilation effect?

Is it true that in previous example light reach left end of train with point of view of passenger and with point of view of outside observer simultaneously?
5. ### B Is velocity the reason for the time dilation effect?

Another example is the propagation of light from the middle of a moving train to its left and right ends. From the train passenger's point of view, the light will simultaneously reach the right and left ends of the train, with point of view of an outside observer - at different times. In this...
6. ### B Is velocity the reason for the time dilation effect?

I mean another. I mean that in moving frame and in non-moving frame same event happens simultaneously. Ok. For example, light clock. Light reach to end point simultaneously in non-moving and moving frames. https://en.wikipedia.org/wiki/Time_dilation#Simple_inference
7. ### B Is velocity the reason for the time dilation effect?

I read that light reach to end point simultaneously in different frames.
8. ### B Is velocity the reason for the time dilation effect?

For example, light clock and mechanical clocks. We can simultaneously start mechanical clocks at initial point and if light reach to end point we can stop simultaneously our mechanical clocks in different frames.
9. ### B Is velocity the reason for the time dilation effect?

Time dilation is the difference in elapsed time as measured by two clocks due to a relative velocity between them
10. ### B Is velocity the reason for the time dilation effect?

Does velocity contribute to the time dilation effect?
11. ### B Is velocity the reason for the time dilation effect?

Hello! I try to understand how in different frames clocks tick and stop simultaneously but show different time? I suppose that velocity is reason of time dilation effect but I'm not sure. Thanks.
12. ### B How to measure time in reference frame with clock?

Thanks to all! My mistake was that I tried to find time of motion in moving frame but distance in this frame is 0 because clock in moving frame is at rest.
13. ### B How to measure time in reference frame with clock?

Ok. I'll try to rephrase. Let's write equation for non-moving reference frame: ##(cΔt)^2 = (cΔt0)^2 + (vΔt)^2## How to write equation for moving reference frame?
14. ### B How to measure time in reference frame with clock?

Why do you say about light beam? I mean another case. Say what time do you measure in moving frame when you travel from A to B?
15. ### B How to measure time in reference frame with clock?

I wrote that time is Δt0, is it true? But I'm confused because if we're moving from A to B then our distance is vΔt for our reference frame. And if time is Δt0 then distance is vΔt0. It's wrong result.
16. ### B How to measure time in reference frame with clock?

Sorry, it's my error. Yes. Clock in moving frame is at rest. And I want to know what time is measured by clock in this moving frame.
17. ### B How to measure time in reference frame with clock?

Ok. I ask about time in moving frame. I want to know time of motion from A to B in moving reference frame. In the rest frame time is Δt.
18. ### B How to measure time in reference frame with clock?

It represent rest frame with zero velocity with rest clock. But I mean right diagram when I say about moving frame.
19. ### B How to measure time in reference frame with clock?

No. I mean moving frame with velocity v and with moving clock.
20. ### B How to measure time in reference frame with clock?

I mean that vΔt is length of path in non-moving reference frame (time in this frame is Δt). And what is time in reference frame with clock?
21. ### B How to measure time in reference frame with clock?

Sorry. Picture in the first message is wrong.
22. ### B How to measure time in reference frame with clock?

I considered example of time dilation with light clock. I have a question about measuring time in reference frame with clock. If we know that clock move from A to B in the reference frame with clock then what time of motion is measured in this reference frame? (In non-moving reference frame...
23. ### I Several Questions About Smooth Infinitesimal Analysis

Hello. I read about smooth infinitesimal analysis and I have several questions: 1.What does "ε.1" and "ε.0" mean in this proof? (photo1) (https://publish.uwo.ca/~jbell/basic.pdf , page 5-6) 2. For what purpose do we use Kock-Lawvere axiom when we deal with law of excluded middle? (photo2)...
24. ### I Proving Continuous Functions in Smooth Infinitesimal Analysis

Hello. How to prove that in smooth infinitesimal analysis every function on R is continuous? (Every function whose domain is R, the real numbers, is continuous and infinitely differentiable.) Thanks.
25. ### B Will we get an infinitesimal x when we neglect ##x^2## in ##x+x^2##?

I read in this source: http://www.bndhep.net/Lab/Math/Calculus.htm The fact that x^2 becomes insignificant compared to x for very small values of x is a fundamental principle of infinitesimal calculus. We say x is infinitesimal when we allow its value to approach zero, but never actually reach...
26. ### B Will we get an infinitesimal x when we neglect ##x^2## in ##x+x^2##?

What is relation between ##\Delta x## and ##dx##? If we want to get term with dx then we discard ##\Delta x^2## and other high-order terms in expression. Is it true?
27. ### B Will we get an infinitesimal x when we neglect ##x^2## in ##x+x^2##?

I didn't fully understand what does it mean? What is it used for?
28. ### B Neglected terms in integral sum

I read "A treatise on infinitesimal calculus" Bartholomew Price. Integral sum, for ##2xdx##:
29. ### B Neglected terms in integral sum

Hello. As is known, we can neglect high-order term in expression ##f(x+dx)-f(x)##. For ##y=x^2##: ##dy=2xdx+dx^2##, ##dy=2xdx##. I read that infinitesimals have property: ##dx+dx^2=dx## I tried to neglect high-order terms in integral sum (##dx^2## and ##4dx^2## and so on) and I obtained wrong...

31. ### B Will we get an infinitesimal x when we neglect ##x^2## in ##x+x^2##?

Ok. We have 2 similar expressions: 1. ##2x \Delta x + \Delta x^2## where ##\Delta x## is variable and 2. ##x+x^2## where ##x## is variable. In 1. case we'll get ##2xdx## when ## \Delta x## tends to zero. In 2. case we'll get ##x## when ##x## tends to zero. But in the first case we'll get...
32. ### B Will we get an infinitesimal x when we neglect ##x^2## in ##x+x^2##?

Sorry, I forgot to say that ##x+x^2## is just expression where x is variable. Not differential.
33. ### B Will we get an infinitesimal x when we neglect ##x^2## in ##x+x^2##?

Hello. Let's assume that we have ##2x \Delta x + \Delta x^2##. When ##\Delta x## tends to zero we can neglect ##\Delta x^2## and we'll get ##2xdx##. Let's assume that we have ##x + x^2##. When ##x## tends to zero we can neglect ##x^2##. Will we get an infinitesimal ##x## as such as ##dx##? Thanks.
34. ### B What is the meaning of dx?

Hello, @fresh_42 You wrote that "In the first case ##dr \approx \Delta \, r## is an infinitesimal small change in ##r## and in the second it is an infinitesimal small piece of ##r##, so ##dr \approx h##." I have difficulty understanding this topic. How is it possible that ##dr \approx \Delta...

Yes
36. ### B Representation of infinitesimals in different ways

Hello. There are 4 types of infinitesimals: 1) dx=1/N, N is the number of elemets of the set of the natural numbers (letter N is used to indicate the cardinality of the set of natural numbers) 2) Hyperreal numbers: ε=1/ω, ω is number greater than any real number. 3) Surreal numbers: { 0, 1...