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Consider two observers, travelling away from each other, after meeting (at which time they sync their clocks). $O$ sends a photon towards $O'$ at times $t$, received by $O'$ at time $t'=kt$, where we have defined Bondi's k factor.

In the following reasoning I know I have made a mistake: So by Milne's radar definition of simultaneity, if the photon is instantly sent back from $O'$ and received by $O$ at $t_2$, then: $$t'=\frac12(t+t_2)=kt\Rightarrow t_2=2t'-t=(2-\frac{1}{k})t'$$ But $t_2$ is meant to equal $kt'$

Could someone explain to me where I have I gone wrong here?

EDIT: Basically I am trying to show that $t_2=kt'$ working from only the two assumptions that:

(i) they both observe the speed of light as c

(ii) Only relative motion is observable

![enter image description here][1] [1]: https://i.sstatic.net/4Ct1t.jpgenter image description here

Consider two observers, travelling away from each other, after meeting (at which time they sync their clocks). $O$ sends a photon towards $O'$ at times $t$, received by $O'$ at time $t'=kt$, where we have defined Bondi's k factor.

In the following reasoning I know I have made a mistake: So by Milne's radar definition of simultaneity, if the photon is instantly sent back from $O'$ and received by $O$ at $t_2$, then: $$t'=\frac12(t+t_2)=kt\Rightarrow t_2=2t'-t=(2-\frac{1}{k})t'$$ But $t_2$ is meant to equal $kt'$

Could someone explain to me where I have I gone wrong here?

EDIT: Basically I am trying to show that $t_2=kt'$ working from only the two assumptions that:

(i) they both observe the speed of light as c

(ii) Only relative motion is observable

![enter image description here][1] [1]: https://i.sstatic.net/4Ct1t.jpg

Consider two observers, travelling away from each other, after meeting (at which time they sync their clocks). $O$ sends a photon towards $O'$ at times $t$, received by $O'$ at time $t'=kt$, where we have defined Bondi's k factor.

In the following reasoning I know I have made a mistake: So by Milne's radar definition of simultaneity, if the photon is instantly sent back from $O'$ and received by $O$ at $t_2$, then: $$t'=\frac12(t+t_2)=kt\Rightarrow t_2=2t'-t=(2-\frac{1}{k})t'$$ But $t_2$ is meant to equal $kt'$

Could someone explain to me where I have I gone wrong here?

EDIT: Basically I am trying to show that $t_2=kt'$ working from only the two assumptions that:

(i) they both observe the speed of light as c

(ii) Only relative motion is observable

enter image description here

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Freeman
Freeman
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Consider two observers, travelling away from each other, after meeting (at which time they sync their clocks). $O$ sends a photon towards $O'$ at times $t$, received by $O'$ at time $t'=kt$, where we have defined Bondi's k factor.

In the following reasoning I know I have made a mistake: So by Milne's radar definition of simultaneity, if the photon is instantly sent back from $O'$ and received by $O$ at $t_2$, then: $$t'=\frac12(t+t_2)=kt\Rightarrow t_2=2t'-t=(2-\frac{1}{k})t'$$ But $t_2$ is meant to equal $kt'$

Could someone explain to me where I have I gone wrong here?

EDIT: Basically I am trying to show that $t_2=kt'$ working from only the two assumptions that:

(i) they both observe the speed of light as c

(ii) Only relative motion is observable

![enter image description here][1] [1]: https://i.sstatic.net/4Ct1t.jpg

Consider two observers, travelling away from each other, after meeting (at which time they sync their clocks). $O$ sends a photon towards $O'$ at times $t$, received by $O'$ at time $t'=kt$, where we have defined Bondi's k factor.

In the following reasoning I know I have made a mistake: So by Milne's radar definition of simultaneity, if the photon is instantly sent back from $O'$ and received by $O$ at $t_2$, then: $$t'=\frac12(t+t_2)=kt\Rightarrow t_2=2t'-t=(2-\frac{1}{k})t'$$ But $t_2$ is meant to equal $kt'$

Could someone explain to me where I have I gone wrong here?

![enter image description here][1] [1]: https://i.sstatic.net/4Ct1t.jpg

Consider two observers, travelling away from each other, after meeting (at which time they sync their clocks). $O$ sends a photon towards $O'$ at times $t$, received by $O'$ at time $t'=kt$, where we have defined Bondi's k factor.

In the following reasoning I know I have made a mistake: So by Milne's radar definition of simultaneity, if the photon is instantly sent back from $O'$ and received by $O$ at $t_2$, then: $$t'=\frac12(t+t_2)=kt\Rightarrow t_2=2t'-t=(2-\frac{1}{k})t'$$ But $t_2$ is meant to equal $kt'$

Could someone explain to me where I have I gone wrong here?

EDIT: Basically I am trying to show that $t_2=kt'$ working from only the two assumptions that:

(i) they both observe the speed of light as c

(ii) Only relative motion is observable

![enter image description here][1] [1]: https://i.sstatic.net/4Ct1t.jpg

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Freeman
Freeman
  • 775
  • 2
  • 11
  • 23

Bondi's k factor

Consider two observers, travelling away from each other, after meeting (at which time they sync their clocks). $O$ sends a photon towards $O'$ at times $t$, received by $O'$ at time $t'=kt$, where we have defined Bondi's k factor.

In the following reasoning I know I have made a mistake: So by Milne's radar definition of simultaneity, if the photon is instantly sent back from $O'$ and received by $O$ at $t_2$, then: $$t'=\frac12(t+t_2)=kt\Rightarrow t_2=2t'-t=(2-\frac{1}{k})t'$$ But $t_2$ is meant to equal $kt'$

Could someone explain to me where I have I gone wrong here?

![enter image description here][1] [1]: https://i.sstatic.net/4Ct1t.jpg