Why do pion decay and time dilation seem contradictory?

[George Jones]

But the equation in my textbook is

t = t_o/√(1-v²/c²)

Where t_o is the proper time or the time interval measured in the rest frame

Isn't $t_0$ the proper time of Alice's clock in the rest frame of Alice's clock?

 

[ghwellsjr]

So for example if Ted's clocks measure 10 seconds, as Alice moved between those two clocks, he will see Alice's clock to have measured e.g 4, 5 or 6 seconds.(I mean less than his 10s).

But the equation in my textbook is

t = t_o/√(1-v²/c²)

Where t_o is the proper time or the time interval measured in the rest frame (Ted's time) and t is that time as measured by looking at the moving clock (Alice's clock).
Now here, the equation quite visibly indicates that t>t_o, so Alice will have measured more than his 10 seconds?

Where am I misunderstanding this?

This is the same question and equation that you asked about in your first post:

According to the textbook, the clock at rest in a reference frame appears to tick slower to an observer moving with respect to that frame of reference.The time interval they measure is t' = t/√(1-v²/c²) , where t is the time interval as measured at rest in the frame of reference of the clock.

Where I gave this answer in post #3:

Are you sure the textbook says, "the clock at rest in a reference frame appears to tick slower to an observer moving with respect to that frame of reference"? Although this is true, it's also true the other way around and usually the equation you stated is for the other way around. In other words, t is for time in a reference frame and t' is for time in a second reference frame moving with respect to the first one.

I'm wondering if your textbook also indicated (maybe earlier) that to an observer at rest in a reference frame, a moving clock appears to tick slower. Maybe they are merely pointing out the reciprocal nature of time dilation for inertial observers.

 

[PrincePhoenix]

This is the same question and equation that you asked about in your first post:

Where I gave this answer in post #3:...

1- How does the equation change for both cases?

I'm wondering if your textbook also indicated (maybe earlier) that to an observer at rest in a reference frame, a moving clock appears to tick slower. Maybe they are merely pointing out the reciprocal nature of time dilation for inertial observers.

Here is the exact paragraph from of the textbook,

Time Dilation:
Suppose a clock is fixed at the origin of the rest frame of reference and indicates proper intervals of time t_o = t₀₂ - t₀₁ . To an observer moving in the frame with velocity v, past the same clock, the apparent time interval is t = t₂ -t₁. As a consequence of Einstein's postulates:

t = t_o/√(1-v2/c2)
Thus t>t_o. Apparent time interval > proper time interval. To an observer at rest a moving clock runs slower; it clicks more slowly than a stationary clock. This effect is called time dilation. For v<<c, the time dilation is negligible, as observed in ordinary situations.

Can you please clarify this with that Ted and Alice example?

Isn't $t_0$ the proper time of Alice's clock in the rest frame of Alice's clock?

And what is t?

 

[ghwellsjr]

In the Ted and Alice example, Ted has a whole bunch of previously synchronized clocks spread out over the area that Alice will travel. These clocks display what is normally called "coordinate time" but in your textbook they call them "Apparent time" and use the symbol "t". Alice's clock is moving and her clock displays what is normally called "Proper time" as does your textbook which uses the symbol t₀. The more common term is the Greek letter tau, "τ". Einstein developed the relationship between t and τ in his famous 1905 paper introducing Special Relativity near the end of section 4. It appears as:

τ = t√(1-v²/c²)

Your textbook rearranged this and changed τ into t₀ as:

t = t₀/√(1-v²/c²)

What this is saying is that the proper time of a moving clock is related to the coordinate time of the stationary clocks defining the reference frame by Einstein's equation.

The first part of your quote from the textbook is very confusing and I would say it is misleading at best and probably wrong. But since they rearranged the equation, maybe they have some other way to interpret what they are saying that they can justify.

 

[PrincePhoenix]

The first part of your quote from the textbook is very confusing and I would say it is misleading at best and probably wrong. But since they rearranged the equation, maybe they have some other way to interpret what they are saying that they can justify.

Our physics textbooks have had errors in the past. The whole time dilation section appears wrong to me now except for the rearranged equation you explained. Thanks for clearing it up. You helped me a lot.

 

[PrincePhoenix]

Found this. Maybe it'll be helpful to someone.
http://www.walter-fendt.de/ph14e/timedilation.htm