Is time measured, a property of the object or of the frame in which the object is observed? [closed]

Question

I'm a bit confused behind the idea of the twin paradox. Suppose we fix a given frame, then in this given frame, wouldn't all objects read the same time making twin paradox meaningless as everyone would read the same time as time dilation only happens between frames?

A similar type of doubt is brought up here

Answer 1

Is time measured property of object or of frame?

It is both. Or rather, there are two different concepts of time in relativity.

One is “coordinate time”, $t$. This is the time that you are thinking of. It is a property of the reference frame. It is defined throughout the entire coordinate chart which covers some 4D region of spacetime. Different frames disagree on coordinate time.

The other is “proper time”, $\tau$. This is the time measured by a clock. It is a property (hence “proper”) of the clock. It is defined only along the worldline of the clock, which is a 1D path through 4D spacetime. Proper time is invariant, so all frames agree on the proper time. However, although all frames agree on $\tau$, proper time is a property of the individual clock, so each different clock will have its own different proper time defined only along the worldline of the clock.

Time dilation, $\gamma$, is the ratio of the coordinate time to the proper time $$\gamma=\frac{dt}{d\tau}$$. All frames agree on $\tau$ but disagree on $t$, so they also disagree about $\gamma$. The relationship is defined as: $$\frac{1}{\gamma}=\frac{d\tau}{dt}=\sqrt{\frac{d\tau^2}{dt^2}}=\sqrt{1-\frac{v^2}{c^2}}$$

Suppose we fix a given frame, then in this given frame, wouldn't all objects read the same time making twin paradox meaningless as everyone would read the same time?

If you fix a given frame, then you still have different proper times for different objects. The twin paradox is simply the fact that if you take two events then different spacetime paths between those two events will have different proper times. This is equivalent to the fact that different lines connecting two points will have different lengths.

Answer 2

There are two related notions of time in special relativity.

Co-ordinate time is the time co-ordinate used in an inertial reference frame. All clocks that are stationary relative to the inertial reference frame measure the same time and if they are synchronised at some time $t_0$ then they will stay synchronised. The twin who stays on Earth will age according to co-ordinate time in the Earth's reference frame (for the purposes of the thought experiment, we ignore the Earth's rotation about its own axis and its motion in its orbit around the Sun).

Proper time is the time measured by a clock that travels along a given space/time path from one location in space/time to another. Proper time depends not just on the start and end points of the path, but also on the shape of the path in between. The travelling twin will age according to the proper time along the path that they follow.

If we say that the travelling twin leaves Earth at space/time location A and returns to Earth at space/time location B, then both twins move along paths between A and B, but the twin who stays on Earth travels between A and B at a constant velocity (which is zero in the Earth's reference frame), whereas the travelling twin takes a different non-inertial path. Special relativity tells us that the proper time interval measured along the travelling twin's path must be shorter than the co-ordinate time interval along the stay-on-Earth twin's path. Hence the travelling twin ages less than the stay-on-Earth twin.