Question

The aggregate of events considered by an inertial observer to be simultaneous at his time \(t=t_{0}\) is said to be the observer's instantancous three-space \(t=t_{0}\). Show that the join of any two events in such a space is orthogonal to the observer's worldline, and that, conversely, any two events whose join is orthogonal to the observer's worldline are considered simultaneous by him.

Short answer

Question: Show that the join of any two events in an observer's instantaneous three-space is orthogonal to the observer's worldline, and conversely, show that any two events whose join is orthogonal to the observer's worldline are considered simultaneous by the observer. Answer: We showed that the join of any two events in an observer's instantaneous three-space, represented by the displacement vector \(J^{\mu}_{AB}\), is orthogonal to the observer's worldline by demonstrating that their dot product is zero. Furthermore, we showed that if two events have a join that is orthogonal to the observer's worldline, they will have the same time coordinate and are therefore considered simultaneous by the observer.

Step-by-step solution

Understand the problem

In this problem, we are dealing with an inertial observer, his instantaneous three-space, and his worldline. We want to show the orthogonal relationship between the join of events in instantaneous three-space and the observer's worldline. To do so, we need to define and understand all these concepts and their relationship with one another. Inertial observers are observers that experience no acceleration. Instantaneous three-space represents a snapshot of spacetime at a given moment of the observer's time \(t_{0}\). A worldline is a curve that describes the path an object takes in spacetime represented by \(x^{\mu}(\tau)\).

Find the observer's worldline

First, we need to find the observer's worldline, and we will use an inertial reference frame \((t, x, y, z)\). The observer's worldline is given by \(x^{\mu}(\tau) = (t(\tau), x(\tau), y(\tau), z(\tau))\), where \(\tau\) is the proper time. As an inertial observer, he is at rest in this frame. Therefore, his path in spacetime is given by \(x^{\mu}(\tau) = (t(\tau), x_{0}, y_{0}, z_{0})\), with \((x_{0}, y_{0}, z_{0})\) as his constant spatial coordinates. To establish orthogonality, we will need the observer's worldline's tangent vector, which is \(v^{\mu} = \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau}\). Since the observer is at rest, his tangent vector is \(v^{\mu} = (1, 0, 0, 0)\).

Find the join of two events

Now, let's consider two events in the observer's instantaneous three-space, that is, the events A and B occurring at time \(t=t_{0}\). Their coordinates are given by \(A(t_{0}, x_{1}, y_{1}, z_{1})\) and \(B(t_{0}, x_{2}, y_{2}, z_{2})\). The join of A and B in the observer's instantaneous three-space is represented by the displacement vector from A to B. Let's denote this vector by \(J^{\mu}_{AB}\), which is given by the difference in coordinates between A and B: $$ J^{\mu}_{AB} = B^{\mu} - A^{\mu} = (0, x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1}) $$

Check orthogonality between worldline and join of events

To show that the join of any two events in instantaneous three-space is orthogonal to the observer's worldline, we must check if their dot product is zero: $$ v^{\mu} \cdot J^{\mu}_{AB} = v^{0}J^{0}_{AB} + v^{1}J^{1}_{AB} + v^{2}J^{2}_{AB} + v^{3}J^{3}_{AB} = (1)(0) + (0)(x_{2} - x_{1}) + (0)(y_{2} - y_{1}) + (0)(z_{2} - z_{1}) = 0 $$ Since the dot product is zero, the join of any two events in instantaneous three-space is indeed orthogonal to the observer's worldline.

Check simultaneity

Now, we must show that if two events have a join that is orthogonal to the observer's worldline, they are considered simultaneous by the observer. Suppose we have two events A and B whose join is orthogonal to the worldline, i.e., \(v^{\mu} \cdot J^{\mu}_{AB} = 0\). Expanding the dot product again, $$ v^{\mu} \cdot J^{\mu}_{AB} = v^{0}J^{0}_{AB} + v^{1}J^{1}_{AB} + v^{2}J^{2}_{AB} + v^{3}J^{3}_{AB} = (t_{0} - t_{0}) + (0)(x_{2} - x_{1}) + (0)(y_{2} - y_{1}) + (0)(z_{2} - z_{1}) = 0 $$ Since \(t_{0} - t_{0} = 0\), any two events A and B whose join is orthogonal to the observer's worldline will have the same time coordinate \(t_{0}\), and consequently, the observer considers them simultaneous. In conclusion, we have shown that the join of any two events in an observer's instantaneous three-space is indeed orthogonal to the observer's worldline, and that, conversely, any two events whose join is orthogonal to the observer's worldline are considered simultaneous by the observer.

Key concepts

Inertial Observer

An inertial observer is a fundamental concept in the theory of Special Relativity, which Albert Einstein introduced in 1905. An observer in this context refers to a reference point from which measurements of space and time can be made. In classical physics, an inertial observer is someone who is not experiencing any acceleration—either remaining stationary or moving at a constant velocity. When we observe the universe from an inertial frame of reference, the laws of physics manifest in their simplest forms.

In the context of the problem, understanding how an inertial observer perceives various events in space and time is crucial. For instance, their perspective on simultaneous events and relative distances do not change unpredictably because they are not undergoing acceleration. This consistent perspective allows us to explore more involved concepts like instantaneous three-space and worldline orthogonality with a clear basis in how events relate to each other temporally and spatially for such observers.

Instantaneous Three-Space

The instantaneous three-space for an inertial observer is a three-dimensional 'slice' of the four-dimensional spacetime continuum, corresponding to a single point in time according to that observer's clock. It is also known as the present moment for the observer and includes all events that are simultaneous with respect to their frame of reference. This concept extends our understanding beyond the classical three-dimensional space by accounting for the observer's temporal coordinate in spacetime.

To put it simply, if you imagine spacetime as a loaf of bread, where each slice represents all of the space at a single moment, the instantaneous three-space is like one slice of that bread for the inertial observer. This analogy makes it easier to grasp the idea that events occurring on this 'slice' are perceived to happen at the same time by the observer. Understanding this helps when tackling advanced topics in Special Relativity, like the problem given, by establishing a clear frame of reference for when events are simultaneous for an observer.

Worldline Orthogonality

When discussing worldline orthogonality, we delve into the geometric representation of an object's path through spacetime. An object's worldline is a line—often drawn on space-time diagrams—that represents the entirety of its existence through time and space. If an inertial observer's worldline is orthogonal to the join of two events, this means the paths representing the observer's movement and the 'line' connecting the events intersect at a right angle.

Orthogonality in this context is used to define simultaneity. When we talk about a 'join' being orthogonal to the worldline, we're essentially saying that to this inertial observer, the events at either end of this join happen simultaneously. The inclusion of time in this analysis makes it a markedly different concept from what one might be familiar with in a purely spatial context. This orthogonality condition within the framework of Special Relativity helps us to investigate and resolve questions of timing and causality between events as perceived by different observers.

Event Simultaneity

Understanding event simultaneity is a central feature of Special Relativity and often a counterintuitive subject. For an inertial observer, two events are simultaneous if they occur at the same time in the observer's own frame of reference. It's critical to recognize, however, that simultaneity is relative—what might be simultaneous for one observer may not be for another if they are in motion relative to each other.

Event simultaneity has perplexing implications. For example, the famous 'train platform' thought experiment shows that two flashes of light, which are simultaneous to a person standing on a platform, are not simultaneous to a passenger on a moving train. The key here is that as the speed of light is constant in all frames of reference, the timing of the events is perceived differently by observers moving relative to one another. The problem provided enhances our understanding that simultaneity for an inertial observer can be geometrically represented by the worldline orthogonality in spacetime.