# Curved space/ spacetime

In the particular situation we were discussing, A and B are non inertial observers onboard an accelerating rocket and are spatially separated along the axis parallel to the accelerating motion of the rocket. Both A and B feel and measure proper acceleration.

O.K.

C is an inertial observer not on board the accelerating rocket.

Did I say that?

If you wish to introduce a fourth inertial observer D that is spatially separated from C, then C and D should be on a line parallel to the line joining A and B.

I'm fairly certain that C and D need only accelerate with, and in the same direction as A and B. However, since I did not accurately represent what you were calling C, would it be O.K. to let C and D be co-located with events that A and B agree are simultaneous?

The situation we were discussing is the Rindler spacetime drawn on a Minkowski diagram with one space dimension and one time dimension.

O.K.

Under those conditions, my original assertion:

remains true and can be extended to :

1) A and B agree at all times on matters of simultaneity.

Consider this: with only one space dimension to work with, any pair of simultaneous events would necessarily have to occur at the same point. Overlooking this, A and B will certainly agree upon the simultaneity of any two such events, but they will not observe the events simultaneously.

1) If A and B agree that two events are simultaneous then C and D do not agree those events are simultaneous.

How so? If A and B agree that any two co-located events are simultaneous, why wouldn't/couldn't any number of additional observers agree on the same thing?

Your counter argument is based on all the observers (A,B, C and D) being inertial observers which is not the situation that was under discussion.

Hmmm.. If I modify what I stated and let C and D simply have a relative displacement that is perpendicular to, and symmetric about the instantaneous velocity of A and B, I think you will find that all four can agree on simultaneity (although they may disagree on time).

Regards,

Bill

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"C is an inertial observer not on board the accelerating rocket. "

Did I say that?

No I said that. I thought I made it clear when I said:

We can also add a third inertial observer C and note the following:

1) A and B agree at all times on matters of simultaneity.
1) If A and B agree that two events are simultaneous then C does not agree those events are simultaneous.

I'm fairly certain that C and D need only accelerate with, and in the same direction as A and B. However, since I did not accurately represent what you were calling C, would it be O.K. to let C and D be co-located with events that A and B agree are simultaneous?
I specified that C is inertial and therefore not accelerating.

Consider this: with only one space dimension to work with, any pair of simultaneous events would necessarily have to occur at the same point.
We can have a one dimensional line and we can have two events that are not at the same point on that line, that can be considered as simultaneous by observers at rest with line. An observer not at rest with the line will not think the two spatially spearated events are simultaneous.

How so? If A and B agree that any two co-located events are simultaneous, why wouldn't/couldn't any number of additional observers agree on the same thing?

We were not talking about co-located events.

Hmmm.. If I modify what I stated and let C and D simply have a relative displacement that is perpendicular to, and symmetric about the instantaneous velocity of A and B, I think you will find that all four can agree on simultaneity (although they may disagree on time).

Regards,

Bill

Yes, if C and D are accelerating along with A and B, but I specified C and D are inertial observers and they are not accelerating.

Jorrie
Gold Member
I have a feeling this question might have to become a thread of its own :P

I agree; this thread is becoming too long. Will you start one and state the question around a ST diagram?

Jorrie
Gold Member
Well, I am using Einstein's definition of simultaneity where clocks are syncronized by sending timing light signals, as I posted earlier.

Hi kev, just a two points: AFAIK, Einstein's method for synchronizing clocks only works in inertial frames, where the speed of light is isotropic, which is not the case in an accelerated frame.

Secondly, I think "agreeing on simultaneity" in the accelerating frame does not necessarily mean that their clocks agree on what time it was when an event happened.

-J

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I agree; this thread is becoming too long. Will you start one and state the question around a ST diagram?

No need. I have figured out my mistake and it turns out you (and the prof) are right. My mistake was my incorrect assumption that the line of simultaneity joined points of equal proper time in the rocket frame. having done some detailed calculations, the clock higher up the rocket not only run faster according to the inertial observer outside the rocket but also from the point of view of the accelerated observers inside the rocket. The "line of simultaneity" only joins points of equal velocity in the time space diagram of the accelerating rocket. It is not the same as the line of simultaneity in the inertial case with constant relative motion. As you also mentioned the speed of light is not isotropic to the accelerated observers. Dang!