In the OP post we found that if there are two events E1 and E2 separated in space and time, then there is an objective distance that can be defined between these events which does not change no matter the observer. This is the space time distance ΔS given by
ΔS^2 = (cΔt)^2 - Δx^2 - Δy^2 - Δz^2
Defining the spatial distance separating the events as ΔR, so that
ΔR^2 = Δx^2 + Δy^2 +Δz^2
The equation becomes,
ΔS^2 = (cΔt)^2 - ΔR^2
Now, let me clarify what I mean when I say that ΔS is invariant under measurement. Suppose there are two observers interested in measuring the interval separating the two events E1 and E2. But one observer O' is moving with an uniform velocity V with respect to the other observer O. Then, when O' tries to measure the time interval and the space interval separating the two events, her results will be DIFFERENT from what O will find when he measures it. Thus if O' measures the time and space separation as (Δt', ΔR') and O measures the time and space separation as (Δt, ΔR), then
Δt' ≠ Δt and ΔR ≠ ΔR'.
However, both will measure the same space-time distance between the two events. That is,
ΔS^2 = (cΔt)^2 - ΔR^2 = (cΔt')^2 - ΔR'^2
The conclusion is that the ticking of time as well as measurement of distance is different for different observers and is dependent on the relative velocity between them. But, despite this, there exists an objective entity, the space-time distance that remains a constant and uniquely identifies the location of each and every event that happens in space and time. This is what is meant when one says that space and time are relative but joint space-time is objective.
In the next post, I will look at whether it means that ideas of past-present-future are also relative or whether they are objectively real.
Note: I will present the idea of space and time according to Einstein first and then respond to the more philosophical posts. Thanks
In the last post I talked about the objective space-time distance between events which remains the same regardless of the state of the observer doing the measurement. It's given by
ΔS^2 = (cΔt)^2 - ΔR^2 where
ΔR^2 = Δx^2 + Δy^2 +Δz^2
The
physical relationship between two events is determined by the sign of
ΔS^2.
When
ΔS^2 > 0 then for any measuring observer, the two events are such that the speed of light
c^2> (ΔR/Δt) ^2.
This means that the two events are separated by a time which is less than the time it takes for light to travel between the two events. Thus there is enough time for light to travel from the first event to the second. If one thinks of light signal traveling outwards from the first event in the form of an expanding cone in space-time, then the second event falls within this cone.
It also means that there can be observers moving with a certain velocity, so that for those observers, the two events appear to occur at the same location in space.
How is this possible? Imagine one is driving a car and say the driver blows the horn once, and once more 15 minutes later with respect to the clock on the driver's wrist. For the driver, the two events occur at the same location, in the steering wheel a foot in front of his body. But to a pedestrian who sees the car moving at 80 mph, the second blowing of the horn occur at 20 miles distance from the first. Crucially, however, Einstein showed that the driver and the pedestrian will not agree regarding how much time passed between the blowing of the two horns as well. But they will agree of the value of the spacetime interval ΔS^2 separating the two events.
Now let's look at what happens when
ΔS^2>0.
Since we must have the square of the spatial distance separating the events ΔR^2 ≥ 0, the equation
ΔS^2 = (cΔt)^2 - ΔR^2
implies that for all observers the events are arranged such that
Δt^2 > 0.
Thus the time separation between the two events can never be zero. No observer sees the two events as being simultaneous. One event is always seen as occurring after the other.
The minimum time separating the two events is measured by the observers for whom the events happen at the same spatial location so that ΔR=0.
The time measured between the two events under such conditions is called the proper time Δτ and is given by
Δτ^2 = (ΔS/c) ^2.
Proper time will be the shortest time difference between the two events measured by any observer.
So what's the bottom line? Here they are:-
When square of the space-time distance is positive
1) The two events are separated such that the distance between them is less than the distance light can travel in the time interval separating the two events. Thus light and signals slower than light can travel from the earlier event to the later event.
2) There are observers with respect to which both events happen at the same location.
3) All observers will agree on the temporal sequence of the events. They will all agree that the first event has happened earlier than the second one. Thus event 1 is objectively in the past of event 2, and event 2 is objectively in the future of event 1. It is also the case that since light and other signals have time to reach event 2, event 1 can be a part of the causal chain that leads to event 2.
4) While all agree that event 1 happens before event 2, observers will differ regarding how much time separates the two events. But there is a minimum value of time separating the events, and this value is measured by those observers for whom both events happen at the same location. This minimum time interval is called the proper time between events.
Thus, despite the relativity of time, the events for whom the space time separation squared is positive and which are casually connected by light and other signals, we can uniquely determine the temporal sequence in which these events happened. For these events, past, present and future are objectively determinable by all observers even though the time differences measured will vary.
