How to build a TSMD, “time-space manipulation device'', using only currently accepted physics (GR + QFT constraints).

1) High-level design choice (pick a spacetime metric)

Any device that “manipulates time and space” in GR is defined by a metric $g_{\mu\nu}$. Your first modelling decision is the metric family you want the device to produce.

Two standard choices for TSMD thought-experiments:

• Warp bubble (Alcubierre-type metric) — local expansion behind / contraction ahead of a bubble so a ship can be carried superluminally relative to distant observers. arXiv
• Traversable wormhole (Morris–Thorne) — a throat connecting distant regions; can be used for shortcuts and time-machines in some constructions. link.aps.org

(Recent work tries to make warp constructions more “physical” — e.g., Bobrick & Martire 2021 study constraints and propose modified geometries that avoid some pathologies). arXiv

2) The mathematical method — from metric → stress–energy → physics constraints

This is the engineering pipeline.

1. Choose a metric ansatz $g_{\mu\nu}(x^\alpha;\; \text{params})$.
   Example (simplified Alcubierre form in 3+1 coordinates):

   $$ds^2 = -c^2 dt^2 + [dx - v_s(t)\,f(r_s)\,dt]^2 + dy^2 + dz^2,$$

   where $r_s=\sqrt{(x-x_s(t))^2+y^2+z^2}$, $x_s(t)$ is bubble center, $v_s$ the bubble speed, and $f(r)$ a shape function (smooth, ~1 inside, →0 outside). arXiv

2. Compute geometric quantities from $g_{\mu\nu}$: Christoffel symbols $\Gamma^\rho_{\mu\nu}$, Riemann tensor $R^\rho{}_{\sigma\mu\nu}$, Ricci tensor $R_{\mu\nu}$, and scalar curvature $R$. (Symbolic algebra systems like Mathematica/xAct or Sage are used in practice).

3. Form the Einstein tensor:

   $$G_{\mu\nu} = R_{\mu\nu} - \tfrac12 R\,g_{\mu\nu}.$$

4. Use Einstein Field Equations (EFE) to get the stress–energy needed:

   $$G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}\,T_{\mu\nu}.$$

   So

   $$T_{\mu\nu} = \frac{c^4}{8\pi G}\,(G_{\mu\nu} + \Lambda g_{\mu\nu}).$$

   This $T_{\mu\nu}$ is the matter/energy distribution (real or exotic) that would produce your metric.

5. Check energy conditions. Compute energy densities seen by observers (e.g. $T_{\mu\nu}u^\mu u^\nu$ for timelike $u^\mu$). If this is negative somewhere you need “exotic” matter (negative energy density), which classically violates the weak/null energy condition. Many warp/wormhole metrics require negative energy. link.aps.org+1

6. Apply quantum constraints. Quantum field theory on curved spacetime imposes quantum inequalities that strongly restrict the magnitude, duration, and spatial extent of negative energy (you can “borrow” negative energy briefly but must “repay” it quickly). These bounds often make the required distributions physically unattainable. See Ford & Roman and reviews/lectures on QEIs. arXiv+1

7. Estimate required total energy and stresses. Integrate the relevant components of $T_{\mu\nu}$ to get total energy and mass equivalents. Compare to plausible available energy scales.

3) Concrete worked example (toy Alcubierre-style calculation)

I’ll outline the specific steps you’d do for an Alcubierre bubble and summarize (no heavy algebra here — but I’ll show the target expressions).

Metric (repeat):

$$ds^2 = -c^2 dt^2 + [dx - v_s(t) f(r_s) dt]^2 + dy^2 + dz^2.$$

From this metric one computes $G_{\mu\nu}$. The energy density measured by Eulerian observers (the $T_{00}$ component in the orthonormal frame) turns out to include negative-sign terms localized in the bubble wall. Symbolically:

$$\rho(\mathbf{x}) \;=\; T_{\hat{0}\hat{0}} \;=\; \frac{c^4}{8\pi G}\,G_{\hat{0}\hat{0}}(f,\,\partial f,\,v_s,\ldots).$$

Pfenning & Ford calculate these components and apply quantum inequalities; they find the bubble wall must be extremely thin (order hundreds of Planck lengths) and the integrated negative energy required is enormous (original estimates vastly larger than the mass–energy of the observable universe for naive choices). arXiv+1

Van den Broeck proposed a geometric modification that reduces the total negative energy requirement enormously by making the interior big but the bubble wall microscopic; that trades one problem for another (Planck-scale walls). WIRED+1

Bobrick & Martire (2021) reframe “physical warp drives” to avoid some unphysical features of Alcubierre’s original metric and work toward energy budgets that don’t demand impossible distributions — but they still require matter distributions not known to exist. arXiv

4) Quantitative energy estimates (order-of-magnitude)

Rather than redo all algebra, here are the conclusions you must confront if you carry out the integration of $T_{00}$:

• Alcubierre (original): energy required (negative) scales to astronomical values — effectively impossible with classical energy sources. arXiv+1

• Van den Broeck modification: can reduce total energy to far smaller values (even gram mass equivalents in idealized math) but forces the bubble wall to Planckian thickness — likely outside validity of semiclassical gravity. WIRED+1

• Modern reanalyses (Bobrick & Martire): show ways to make metric choices more physical and avoid some pathologies, but do not remove the need for exotic stress–energy completely; they give a concrete program for minimizing the unphysical parts and computing realistic energy budgets. arXiv

5) The show-stoppers (why a practical TSMD is currently impossible)

1. Negative energy requirement. Classical matter has nonnegative energy density; the metrics that produce the desired spacetime almost always need regions where $T_{\mu\nu}u^\mu u^\nu<0$. There is only tiny negative energy allowed in QFT and it is strongly constrained. link.aps.org+1

2. Quantum inequalities. Even if QFT allows negative energy density, QEIs limit how much and for how long: the integrated negative energy you can create is tiny compared to what a macroscopic warp would need. arXiv+1

3. Planck-scale structures. Many fixes shrink the expensive region to Planckian thickness, which puts you in a regime where semiclassical GR breaks down (you would need a quantum gravity theory we do not yet have). if.ufrj.br+1

4. Stability and control. Even with the right stress-energy, the bubble/wormhole may be unstable, emit destructive radiation, or be impossible to form/control with any known apparatus. Recent papers study these dynamical issues. arXiv+1

6) Realistic, achievable “time-space manipulation” with current physics

Although the sci-fi TSMD is out of reach, there are real, demonstrable spacetime manipulations:

• Gravitational time dilation — clocks in stronger gravity or moving fast run slower (measured in GPS satellites; special+general relativity corrections are required daily). That’s a practical tool to “manipulate time rates.” (These are exact predictions of SR/GR and verified experimentally.)

• Frame-dragging and Lense–Thirring — rotating masses drag inertial frames (measured by Gravity Probe B).

• Casimir effect — small, measurable negative energy densities appear between conducting plates (quantum vacuum effect). These are tiny and local but are the only laboratory source of negative energy we can produce reliably. They do not scale to the amounts warp/wormhole metrics demand. arXiv

7) If you wanted to build a research model today — step-by-step modelling plan

(what a PhD project trying to make a physically plausible TSMD would do)

1. Pick a well-behaved metric family (start with Bobrick & Martire’s “physical warp” ansatz or a Morris–Thorne wormhole with tunable shape function). arXiv+1

2. Symbolic derivation: use xAct (Mathematica) or Sage GR to compute $G_{\mu\nu}$ in terms of shape functions and parameters. (Automate differentiation and tensor algebra.)

3. Solve for $T_{\mu\nu}$ from EFE and classify which components are negative.

4. Apply QEIs: use established quantum inequality bounds to check whether the needed negative energy distributions respect known QFT limits (Ford–Roman style bounds). arXiv+1

5. Numerical integration: compute total integrated energy and compare to realistic energy reservoirs.

6. Stability analysis: linear perturbation analysis (and if possible, time-dependent GR numerical simulation) to check whether small perturbations blow up.

7. Bridge to experiments: identify whether Casimir or other quantum vacuum setups could produce a portion of the required stress-energy and quantify the gap.

8) Where to read next (key papers)

• Alcubierre, The warp drive: hyper-fast travel within general relativity (1994/2000). arXiv

• Morris & Thorne, Wormholes in spacetime and their use for interstellar travel (1988). link.aps.org

• Pfenning & Ford, The unphysical nature of “warp drive” (1997) — quantum inequality constraints. arXiv

• Van den Broeck (1999) — geometrical trick to reduce total negative energy (introduces Planck-scale issues). if.ufrj.br

• Bobrick & Martire, Introducing Physical Warp Drives (2021) — modern, “more physical” reanalysis. arXiv

• Fewster, Lectures on quantum energy inequalities (review of QEI constraints). arXiv

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Bottom line

General relativity gives you the blueprint — pick a metric that does the “bending” you want, then use the Einstein equation to see what energy distribution would make it happen. When you do the math, you almost always find you need negative energy in precisely the places and amounts nature strongly forbids (or only allows in vanishingly tiny amounts). Modern papers try to shrink or reconfigure the problem, but none removes the fundamental negative-energy / quantum-inequality obstacles. So you can model a TSMD down to precise equations and numbers — though current physics says you can’t build it with any known materials or technologies.

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