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Causality and light cones

An event is one place at one time. Once an event is chosen as the origin, its light cone divides spacetime into regions with different causal meanings. The future light cone contains events that a signal emitted now could reach without exceeding the speed of light. The past light cone contains events that could have sent a signal to the origin. Events outside both cones are too far away in space to be causally connected within the available time.

This structure is not just a feature of a drawing. It follows from the invariant spacetime interval and is preserved by every Lorentz transformation.

Constructing a light cone

Imagine a flash emitted from the origin at \(t=0\). In one spatial dimension, the two edges of its future light cone satisfy

\[x=ct\qquad\text{and}\qquad x=-ct,\qquad t\geq0\]

In three spatial dimensions, the light front is an expanding sphere:

\[r=ct,\qquad r=\sqrt{x^2+y^2+z^2}\]

On a spacetime diagram, time is commonly drawn vertically and one space coordinate horizontally. If the vertical coordinate is \(ct\), light rays appear at forty-five degrees when both axes use the same scale. That angle is a plotting choice; the invariant causal boundary is the equation \(r=c|t|\).

The cone has four useful regions:

In Figure 1, compare the routes from O to A, B, and C. The three destinations share one time coordinate, shown by the horizontal dashed line, but lie progressively farther away. A is inside the light boundary, B is on it, and C is beyond it. More distance in the same available time requires greater signal speed; reaching C from O would require exceeding light speed.

The invariant interval

The classifications in Figure 1 compare each destination with O. The following equation instead uses A and B as generic names for any pair of events; it is not specifically the separation between the two same-time points labeled A and B in the drawing. To classify a plotted destination, subtract O's coordinates from its coordinates before evaluating the interval.

For events \(A\) and \(B\), let

\[\Delta t=t_B-t_A,\qquad \Delta\mathbf r=\mathbf r_B-\mathbf r_A\]

Using the sign convention in this course, their squared interval is

\[\Delta s^2=c^2\Delta t^2-|\Delta\mathbf r|^2\]

Some references use the negative of this expression. Their labels remain physically equivalent after all signs are reversed consistently. With the convention above:

The number \(\Delta s^2\) and therefore its sign are Lorentz invariant. Observers can disagree about the values of \(\Delta t\) and \(\Delta\mathbf r\), but they agree on the causal class.

Timelike separation

For a timelike pair,

\[c^2\Delta t^2>|\Delta\mathbf r|^2\]

so the speed required to travel between the positions is below \(c\). There is an inertial frame in which both events occur at the same place. In that frame, one clock can be present at both events and records the proper time

\[\Delta\tau=\sqrt{\Delta t^2-\frac{|\Delta\mathbf r|^2}{c^2}}\]

Every proper Lorentz transformation preserves the temporal order of two timelike-separated events. If event \(A\) can cause event \(B\), no inertial observer finds \(B\) occurring first.

Lightlike separation

For a lightlike pair,

\[c^2\Delta t^2=|\Delta\mathbf r|^2\]

An ideal signal moving at exactly \(c\) can connect the events. There is no inertial frame in which the two events occur at the same place, and the proper time along the null path is zero. Saying that “a photon experiences no time” is potentially misleading because a photon has no valid rest frame; the precise statement is that the interval along its path is null.

The surface of the light cone is unchanged as a set under Lorentz transformations. This is another way of saying that all inertial observers measure the same vacuum light speed.

Spacelike separation

For a spacelike pair,

\[c^2\Delta t^2<|\Delta\mathbf r|^2\]

Connecting the events would require a speed greater than \(c\). There is an inertial frame in which the events are simultaneous, and other frames may disagree about which event occurs first. This reversal does not threaten causality because neither event can influence the other through any signal at or below light speed.

The invariant proper distance in the simultaneous frame can be written

\[\Delta\ell=\sqrt{|\Delta\mathbf r|^2-c^2\Delta t^2}\]

It is real only for spacelike separation under this definition.

Three one-second examples

Let \(\Delta t=1.0\,\mathrm{s}\) and consider one-dimensional destinations:

  1. If \(\Delta x=0\), then \(\Delta s^2=c^2\,\mathrm{s}^2>0\). The pair is timelike.
  2. If \(\Delta x=1.0c\,\mathrm{s}\), then \(\Delta s^2=0\). The pair is lightlike.
  3. If \(\Delta x=2.0c\,\mathrm{s}\), then \(\Delta s^2=-3c^2\,\mathrm{s}^2<0\). The pair is spacelike.

The units matter. Writing only

\(\Delta x=c\)

would incorrectly equate a distance with a speed. A light-second is the distance \(c(1\,\mathrm{s})\).

import numpy as np

for distance in [0.0, 1.0, 2.0]:
    interval = 1.0**2 - distance**2  # Seconds and light-seconds; c=1.
    kind = "timelike" if interval > 0 else "spacelike" if interval < 0 else "lightlike"
    print(distance, kind)

Causal order needs two decisions

The interval class alone does not say whether the destination is in the future or the past. A timelike destination with

\(\Delta t<0\)

lies in the causal past, so a signal emitted at the source cannot reach it. For a source-to-destination query:

def causal_relation(source, destination):
    dt, dx = np.asarray(destination) - source
    interval = dt * dt - dx * dx
    tolerance = 1e-12 * max(dt * dt, dx * dx, 1.0)
    kind = (
        "lightlike"
        if abs(interval) <= tolerance
        else "timelike" if interval > 0 else "spacelike"
    )
    direction = "future" if dt > 1e-12 else "past" if dt < -1e-12 else "simultaneous"
    return kind, direction, interval


print(causal_relation([0.0, 0.0], [1.0, 0.5]))

A signal is permitted by special relativity only if the destination is future-directed and the separation is timelike or lightlike. Coincident source and destination events are a special zero-separation case, not a message sent over a finite interval.

def can_send_signal(source, destination):
    kind, direction, interval = causal_relation(source, destination)
    return kind in {"timelike", "lightlike"} and direction == "future"


assert can_send_signal([0, 0], [1, 0.5])
assert not can_send_signal([0, 0], [1, 2])

Why an absolute tolerance is insufficient

In exact arithmetic, testing whether

\(\Delta s^2=0\)

would be simple. Floating-point values near a cone boundary may differ slightly because two large, nearly equal terms are subtracted. A fixed tolerance in square metres behaves differently for laboratory distances and astronomical distances. Scale the tolerance to the magnitudes being compared and document the chosen numerical policy.

Measurement uncertainty is a separate issue. If the position and time uncertainties are large enough that the interval's range crosses zero, the honest result is “classification uncertain,” not an arbitrary lightlike label. Numerical roundoff and experimental uncertainty should not be hidden inside one undocumented threshold.

Lorentz invariance as a regression check

Apply a valid Lorentz boost to both events, recompute their separation, and classify again. Their coordinates and possibly the order of a spacelike pair will change, but the interval and causal class must remain the same.

beta = 0.6
gamma = 1 / np.sqrt(1 - beta**2)
boost = gamma * np.array([[1.0, -beta], [-beta, 1.0]])
a, b = np.array([0.0, 0.0]), np.array([1.0, 0.5])
original, transformed = causal_relation(a, b), causal_relation(boost @ a, boost @ b)
assert original[:2] == transformed[:2]
assert np.isclose(original[2], transformed[2])
print("Causal classification and interval preserved.")

For a future-directed timelike or lightlike pair, an ordinary proper Lorentz boost also preserves the future direction. A test that allows a spacelike connection after any boost indicates either mixed frames, a sign error, or an invalid superluminal rule.

Faster-than-light signals and causal loops

Suppose controllable information could travel faster than light. Its endpoints would be spacelike separated. Because some inertial frames reverse the order of spacelike events, combining such signals with ordinary Lorentz symmetry can construct a reply that arrives before the original message was sent. The specific coordinates depend on the chosen speeds, but the underlying problem is the loss of invariant temporal order.

This is why the light cone is a causal boundary, not merely the current speed record of a technology. Ordinary phase velocities or moving illumination spots can exceed \(c\) without carrying information faster than light; signal and causal-front velocity are the relevant quantities.

Scope

The classification above assumes point events and flat spacetime. Gravity curves spacetime, and global causal structure can be more complicated than a pair of straight cones. Locally, freely falling observers still find light cones and the invariant speed \(c\).

A causal classification is necessary but not sufficient for practical communication. A timelike destination may be reachable in principle even when no available transmitter, fuel supply, trajectory, or medium can reach it. Conversely, signal-processing delay and finite equipment size can make the actual arrival later than the ideal light-cone boundary. The classification answers one precise question: does special relativity prohibit the connection?

A worked Python notebook

At a fixed positive time separation of one second, locate the light-cone boundary. The question for this calculation is precise: how does interval squared change when we change spatial separation, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

A causal class and a future-or-past direction are both needed to discuss signalling. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0 to 2 light-seconds. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for spatial separation, expressed in light-seconds. The output is interval squared, expressed in s^2. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 1 - q^{2}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 1 light-seconds, the expression gives 0 s^2. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("1-q**2", locals={"q": q})
reference_input = 1.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "light-seconds")
print("Predicted interval squared:", reference_output, "s^2")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 1 s^2. The midpoint, at 1 light-seconds, gives 0 s^2. The upper endpoint gives -3 s^2. Read these together with the assumptions: they describe different choices of spatial separation within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.0, 1.0, 2.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} light-seconds -> {output_value:.6g} s^2")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows spatial separation; the vertical axis shows interval squared. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.0, 2.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Spatial separation (light-seconds)")
ax.set_ylabel("Interval squared (s^2)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does interval squared change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 1 light-seconds is -2 in those ratio units. Locally, the output decreases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = - 2 q\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 1.01
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "s^2")
print("Full expression:", full_prediction, "s^2")
print("Approximation difference:", full_prediction - linear_prediction, "s^2")

Here the changed input is 1.01 light-seconds. The tangent estimate is -0.02 s^2, while the full expression gives -0.0201 s^2. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 2.0
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of spatial separation inside the stated interval, predict whether interval squared will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: A causal class and a future-or-past direction are both needed to discuss signalling. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of s^2: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(2.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 2.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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