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Special relativity supplies a tenseless mathematical representation: events have spacetime relations, proper-time intervals and causal order whether or not a coordinate labels one event “now.” For spacelike separated events, different inertial frames disagree about simultaneity and temporal order. The invariant structure therefore does not select a global present consisting of all and only events simultaneous with an observer's current event.

That supports an argument for eternalism if one adds that what exists should not depend on an arbitrary inertial frame and that no extra preferred foliation is available. On those assumptions, restricting existence to a frame-dependent simultaneity plane looks unjustified. A spacetime account can instead represent an observer's temporal location as it represents spatial location, without introducing an objective moving present.

The conclusion is not a formal derivation of metaphysics from the Lorentz transformations alone. Relativity establishes the metric and its symmetries, not a complete semantics for existence, becoming or experience. A presentist could add a preferred but empirically hidden slicing, though this adds structure unsupported by ordinary relativistic predictions. Local becoming along a worldline or causal partial orders are other possibilities, and do not reinstate a unique invariant global simultaneity relation.

A tenseless description also does not erase every temporal distinction. Timelike-separated events retain invariant order, light cones distinguish causal possibilities, and clocks measure nonzero proper durations. The absence of a global privileged “now” does not identify past and future dynamically or explain the thermodynamic arrow. Relativity strongly constrains a global present and makes a block representation natural; whether that representation exhausts temporal reality requires additional philosophical premises.

Outside both light cones, the separation from the origin is spacelike. Thus the events in neither the causal past nor the causal future areNo causal curve can connect either endpoint to the other. These are the two green side regions in part c; the strict inequality excludes the causally reachable null boundaries.

The upper-left panel shows the three causal structure regions, with null boundaries included in the appropriate causal future and causal past. For the charged trajectory, use part b with $a=q{E}_x/m$, choosing the $x$ direction so that $a>0$. The electric field gives the constant three-force $q{E}_x$ and hence the shifted hyperbolaIt starts with a vertical tangent at the origin, stays strictly inside the future light cone, and asymptotes to $x=ct-c^2/a$. The other panels implement the coordinate and topology changes used below; they are original spacetime diagrams, not copies of source artwork.

The invariant spacelike condition is $x^{\prime 2}-c^2{t}^{\prime 2}>0$. Hence the primed-coordinate events in neither the causal future nor the causal past obeyThe green regions in part c retain their shape in primed coordinates. A particular spacelike event's coordinate time may change sign, but it never enters either light cone.

The same Lorentz transformation maps the past light cone to itself. In the primed axes the causal past isthe orange region in the upper-right spacetime diagram of part c. Future and past are not interchanged by the standard boost.

The past light cone of a spacetime event $O$ is the past-directed part of its light cone. In Minkowski spacetime, its spacetime events satisfy $|\Delta \mathbf{x}|=-c\Delta t$ with $\Delta t<0$. They are the possible emission events of light rays arriving at $O$. The interior also contains events that can send slower signals to $O$. The retarded potential of a moving point electric charge, given by the Liénard–Wiechert potential, uses the emission event where the source's world line intersects this boundary.

The joint energy-momentum spectrum lies in the closed forward light cone. In axiomatic quantum field theory, this condition permits analytic continuation of Wightman functions in suitable complex spacetime domains. Together with Lorentz invariance, microcausality and the positivity of the physical state space, it is one of the hypotheses behind the CPT theorem.

A Lorentz transformation with determinant $+1$ that preserves future-directed timelike four-vectors. Equivalently, it belongs to the identity component of the Lorentz group. For the Minkowski metric $(-,+,+,+)$, its matrix satisfies ${\Lambda }^T\eta \Lambda =\eta$ and ${\Lambda }^0{}_0\ge 1$. Such transformations preserve the future light cone, so the retarded Green function of the d'Alembert operator transforms as a Lorentz scalar.

For the homogeneous speed-one wave equation, integrate its wave energy density over $B(x_{\ast },T-t)$. The moving boundary contributes $-e$, while the usual flux contributes $u_t{\partial }_{n}u$. Their sum is $-\frac{1}{2}(u_t-{\partial }_{n}u{)}^2-\frac{1}{2}|{\nabla }_{\tan }u{|}^2\le 0$. Zero Cauchy data in the initial ball therefore force zero solution in its backward light cone.